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Dynamics of classical solitons (in non-integrable systems)
PHYSICS REPORTS (Section C of Physics Letters) 35, No. 1(1978)1-128. NORTH-HOLLAND PUBLISHING COMPANY
DYNAMICS OF CLASSICAL SOLITONS (IN NON-INTEGRABLE SYSTEMS) V.G. MAKHANKOV Joint Institute for Nuclear Research. Moscow, U.S.S.R
Received February 1977
Contents: Introduction I. General properties of solitons 1.1. Variational principle 1.2. Equations having soliton solutions 1.3. Quasi-particle interpretation of HF solitons 1.4. Solitons and the Fermi—Pasta—Ulam problem 2. Solitons in plasma physics 2.1. Equations describing non-linear phenomena in isotropic and magnetized plasmas and their soliton type solutions 2.2. Boussinesq equation and the stochastization problem 2.3. KdV equation and its ‘improved” version IKdV 2.4. Schrodinger equation with a potential satisfying a non-linear wave equation 2.5. Coupled Langmuir and ion-sound waves in the nearsonic region. 2.6. Schrodinger equation with a saturating non-linearity. 2.7. Soliton-like solutions of equations describing excitons in one-dimensional molecular crystals 3. Solitons in particle physics 3.1. General remarks 3.2. Solitons in a g~field theory 3.3. Properties of plane solitons due to their charge 3.4. Bound states of solitons 3.5. Approximate analytical description of bound solitonstates in non-integrable systems 3.6. Quantization of solitons
3 5 5 5 8 10 11
II 13 17 19 29 33 35 38 38 39 44 47 52 54
4. Stability of one-dimensional solitons 4.1. Some general remarks 4.2. The longitudinal stability of plane solitons 4.3. Stability of solitons in the direction perpendicular to their motion (the transverse stability) 4.4. A qualitative discussion of soliton stability 5. More-dimensional solitons 5.1. One exactly soluble two-space-dimensional problem 5.2. More-dimensional solitons in the framework of the S3 equation 5.3. Collapse of Langmuir waves (CLW) 5.4. The CLW in the dynamical model 5.5. Dipole CLW 5.6. Spheriton collapse in the system of coupled Schrodinger and Boussinesq equations 5.7. One more example of an SS-packet collapse 6. Stationary solitons 6.1. The stability of more-dimensional solitons 6.2. Some properties of spherically symmetrical solitons 6.3. Examples of stable SS-solitons 7. Long-lived pulsating solitons 7.1. Meson bubble life-time 7.2. Pulsons of the KG3 equation 7.3. Pulsons of the sine-Gordon and Landau—Ginzburg— Higgs (LGH) equations Conclusions and acknowledgements Appendices References
56 57 59 68 70 70 74 75 _19 83 86 88 88 88 99 102 112 112 14
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117 120 22 125
DYNAMICS OF CLASSICAL SOLITONS (IN NON-INTEGRABLE SYSTEMS)
V.G. MAKHANKOV Joint Institute for Nuclear Research, Moscow, U.S.S.R.
NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM
V.G. Makhankov, Dynamics of classical solitons
3
Abstract: A survey of the properties of soliton type solutions to non-linear wave equations appearing in various fields of physics is given. For the most part the description deals with non-integrable systems and remains at the classical level. Some general features associated with ~solitons— stationary states, bound states of solitons, stability, the Fermi—Pasta—Ulam problem, and so on are discussed for both one- and more-dimensional worlds. The results of computer experiments on the dynamics of the formation and interaction (in one-space-dimensional geometry) of soliton-type objects are presented at length. As an application of the general theory we speculate on solitons in plasmas and particle physics. 4 and sine-Gordon theories. Finally, new, spherically symmetric, oscillating solutions (“pulsons”) are examined in the frame-work of (p
Introduction A whole class of non-linear differential equations admits solutions in the form of so-called solitary waves. Although such solutions were obtained fairly long ago (in hydrodynamics more than a hundred years ago [1,2]) the interest in them began to arise only in the middle of our century. This fact is apparently connected with the circumstance that the solitary waves being some very peculiar class of particular solutions to non-linear differential equations were supposed to be a rare phenomenon. Moreover, the non-linearity which is responsible for the formation of the solitary waves, on the other hand, looked like automatically leading to their destruction after an interaction. In fact, there was a hypothesis that even the presence of weak non-linearities in a system always leads to an irreversible redistribution of energy among the degrees of freedom (the ergodicity hypothesis). This hypothesis was checked in a numerical experiment in the 50’s with an example of a system approximating a piece of non-linear string [31.The above-mentioned ergodization was not observed. An analogous result was obtained [4] for the Korteweg—de Vries (KdV) equation. In the problems considered non-linear oscillations arose consisting of the redistribution of energy among several harmonics near the excited one. Then the system periodically returns to the practically initial state (fig. 1.1). This result, being on the face of it paradoxical, or more precisely the attempt to understand it, is called the Fermi—Pasta—Ulam (FPU) problem. The existence by itself of the FPU problem suggests that non-linear interactions are not sure to lead to the destruction (randomization) of oscillations and therefore the solitary waves can be conserved after the interaction. Just such solitary waves are called “solitons” in the American literature [4,5]. However we do not adhere to such a terminology and understand by a soliton a solitary w~iv~having some “safety factor”, i.e., weakly changing in the interaction (see details below). In the review paper [5] sufficiently clear definitions u~ sunning and solitary waves and a class of wave equations having such solutions are given. Also various analytical rnethoc1~for the study of the behaviour of these solutions after their interaction are discussed and an imposing bibliography is given. Moreover, the main attention is paid to equations describing completely integrabie i~miltonian systems. The whole variety of physical phenomena which lead to equations permitting soliton solutions cali be divided into two different classes. They are distinguished both by the problems and by the interpretation of the results, although they come together at the classical level. The problems of the investigation of non-linear wave phenomena in real continuous media (hydrodynamics, solids, and so on) belong to the first class and field theoretic problems and particle physics belong to the second class.
4
V.G. Makhankor, Dynamics of classical so!itons
Fig. 1.1. Non-linear oscillations of the energy distribution over the spectrum in the FPU problem.
In the first case the study usually goes through the following stages: it starts at the classical or quantum discrete level then with some degree of rigour the investigators proceed to the classical or quasi-classical continuum level which determines the form of the resulting equations. The final result of such a transformation is precisely the classical soliton solution (to which through a reverse transformation one can in principle give a quantum meaning). In the second case the models under investigation are constructed on the basis of a Lagrangian formalism and some requirements following from general physical laws (relativistic invariance, symmetries, and so on). In this case classical wave equations appear in the initial stage and are the basis for the construction of real quantum objects. Moreover, unlike conventional atomic physics, the quantum soliton properties are conditioned by classical solutions as h-+0. All this determines the form of the second class of equations. These directions come together in the intermediate stage of the investigation of the soliton solutions of the classical wave equations. Therefore it is interesting to study the general properties of the classical solitons (CS), to some extent forgetting the physical interpretation. As we shall see below the existence of stable solitons in a four-dimensional one-field model with a so-called saturable non-linearity is possible. Although undoubtedly of interest within the first class, they might be rejected for the second class theories because of renormalizability conditions. Below we consider non-linear phenomena in systems allowing stochastization, i.e., in non-integrable systems. The behavior of such systems can be determined by the degree of their proximity to some completely integrable analogies; moreover, different properties are determined by the proximity to different integrable models. In this sense the integrable models can be considered as a zeroth (non-linear!) approximation to the description of real physical systems and further investigations can be performed as a perturbation series in this small deviation. Note that at present only completely integrable models can be strictly analytically investigated by various methods (we have in mind the Cauchy problem solution). As a rule, analytical methods are practically helpless (at least at the present) in the study of the evolution of non-integrable systems. Therefore, with rare exception, all the results on the evolution of ergodic (even one-dimensional) and more-dimensional systems were obtained by computer experiments.
V.G. Makhankov, Dynamics of classical solitons
5
All this makes purposeless to discuss here the ideas and results described in the review paper by Scott et al. [5].If we mention them, it is only when this is necessary for a clearer understanding of the phenomena under consideration. This review is addressed not to the experts but may be thought of as a basis to proceed further. Therefore we try to give a qualitative picture rather than to develop rigorous approaches.
1. General properties of solitons LI. Variational principle The overwhelming majority of equations having siliton solutions can be obtained as a solution of some extremum problem in field theory. Let ~(x, t) be the field looked for. Introducing the Lagrangian density ~ p~, ~), determining, as usual, the action S as the functional* ~
(1.1)
and putting the first variation of S equal to zero, (1.2) we obtain the equation for
~.
From (1.2) we have
ôSa~f a a..~’ a a..~’0 aço at aço, ax ap~ —
13
( )
—
.
The analogous procedure can be made for a system of interacting fields (for a more detailed discussion see e.g. ref. [6]). For some kinds of Lagrangians one can obtain from Noether’s theorem [7] conserved (in time) dynamical quantities such as the energy-momentum, the charge, and so on. The appearance of such integrals of motion is the consequence of the invariance of the Lagrangian with respect to some transformations of the x, t coordinates and of the field function q~(which is in turn the consequence of the appropriate symmetry of the system). However, the aforesaid conserved quantities as well as the conservation laws can be directly obtained from the equations of motion (1.3).
1.2. Equations having soliton solutions The equations possessing soliton solutions can be divided into two types: relativistically invariant ones and relativistically non-invariant ones. 1. The first type of equations is more often found in the papers on elementary particle theory. That part of such an equation which contains time and space derivatives can be (c easily the 2 = p2 + m2 = 1).obtained Passing from from the relativistic dispersion relation between energy and momentum E c-numbers E and p to the operators E ia/at, ~, ia/ax we have -~
(ii~—
v2)~+ F(p)
*
=
0,
—
—~
(1.4)
It should be noted that such a Lagrangian formulation permits a relativistically invariant representation in contrast to the Hamiltonian
formalism.
6
V.G. Makhankov, Dynamics of r’lassica! so)itons
where F, which determines the interaction potential, is some non-linear function of q’. Type (1.4) includes, for example, the non-linear Klein—Gordon equation,
(~+
—
(~
2g2jQI2~= 0
va);
~_
(1.5)
the Higgs field equation,
(~ —
m2)~p+ 2g2~~~2~ = 0
(1.6)
(eqs. (1.5) and (1.6) are attributed to the so-called classical q” field theory); the sine-Gordon equation LI~+sin~=0,
(1.7)
and many others (see e.g., refs. [8, 181). Eqs. (l.5)—(l.7) bring about the extremum of the action (1.1) with the following Lagrangians, respectively =
—
leH~~
~‘SG=
a~*a~rn2I~I2+ g2j~4, 3~(,O+mI~l—g IQI ,
(1.8) (1.9) (1.10)
cos ~,
~
where x~ {x, it} and the summation over dummy indices is carried out as usual. By using the normal procedure [6] for these equations one can obtain the Hamiltonians, which are the time components of the 4-vector of the field energy-momentum, and the momentum vector ale
(1.11)
ç
~‘40t
1 /32’
ale
‘\
(1,lla)
Sometimes systems of interacting relativistic fields are considered. E.g. in ref. [9] the stationary solutions of the system 2~=0, L1~+ ~ g flx+m2xg2xI~I20 (1.12) —
in two-dimensional space-time have been found. The Lagrangian for this system is —
2
2
2
2
2
2
I
2
2
22
c°~ —m ~ +g~~ +5(x~—x~—~L.x). The system of interacting scalar fields in three-dimensional space, L1~+
a2~2ço 0 2 21
2 Ex+axI~I+~x(x —1)=0,
(1.13)
114
with the Lagrangian le=_3*a_~(a~x)2_a2x2I~I2_~(xS_l)2,
(l.1j
has been proposed and investigated in ref. [10]. We discuss the soliton solutions of this system in section 6. Here we only note, that all the above-mentioned Lagrangians possess symmetry, i.e., they
V.G. Makhankov, Dynamics of classical solitons
7
are invariant with respect to gauge transformation -~
e’°~,
which involves the current conservation law (1.16)
apj~=O,
where Ik
=
—
j(~p*3
—
(1.17a)
coak,o*),
p~jo=i(q’*p1_~q,~~),
(1.17b)
from which it follows, that Q=fpd3x
(1.18)
is an integral of motion. Eq. (1.16) is easily obtained from the variational principle (1.2) in which the quantity ~S/e5Ois equated with zero.* 2. Non-relativistic equations having soliton solutions have been known for a long time. Among these equations the Korteweg—de Vries equation (KdV for short), [11] 2)~ + ep~= 0, (1.19) c~+ ~a(~ and its various modifications with a changed non-linear term are apparently the most studied ones. Earlier the Boussinesq (Bq) equation (sometimes it is called the non-linear string equation), [1] (ç2) = 0, (1.20) —
—
had been obtained. Finally, recently theorists have become interested in the Schrodinger equation with a cubic non-linearity (S3), [12] i~,+ p~±g2I~I2~ = 0.
(1.21)
In the theory of Langmuir plasma turbulence the Schrodinger equation with a self-consistent potential (like the Hartree—Fock equation in the theory of atoms)
~
=0
(1.22)
has been studied intensely. The self-consistent potential 1 obeys: (i) a non-uniform wave equation [13] E111 = (I~I2) —
or
(1.23)
(ii) a non-uniform Bq equation [14] —
or
(~2)
—
‘T’xxxx =
(I~I2)x~,
(1.24)
(iii) a non-uniform KdV equation [15] ~I, + I~+ *
I(~2)
+
~
=
—
(JpI2)~.
(1.25)
This conservation law is the consequence of the invariance of the Lagrangian with respect to the rotation in isotopic space (see ref. [6]).
8
V.0. Makhankov, Dynamics of classical solitons
Eqs. (1.19), (1.21) and the set (1.22)—(1.23) can be obtained from the variational principle (1.2), in which2’KdV the corresponding Lagrangians are given by = ~1~x1~t +~ai~+ 1~~~l/J~ ~ where
i~ = ~
=
=
1. * leS351(~t~ (see e.g. ref.
*
2
w
2
4
5)Hw~I ±g ~
(1.27)
,
[5]);
‘. * le~=~i(~ ~
*
2
—u5I~2 ~ I
where instead of 1 the potential u =
(1.26)
~
u,
= t~+
is
4
1 2 2 +s(u—u~),
(1.28a)
introduced by the formulae [16]
~I2.
(1.28b)
The Bq equation and systems (1.22), (1.24) and (1.22), (1.25) (we denote them SBq and SKdV, respectively) may also be obtained from a variational principle, but the form of their Lagrangian is unknown to the author. Concluding this subsection we give one more equation describing the exciton excitations in one-dimensional crystal [17] which has quasi-soliton solutions: (1.29) Henceforth the soliton solutions of relativistically invariant equations and various modifications of the Schrödinger equation are called high-frequency (HF) solutions, and the solutions of KdV and Bq type equations are called low-frequency (LF) ones. (Physical arguments for this will become clear below.) 1.3. Quasi-particle interpretation of HF solitons There are two types of soliton solutions in two-dimensional space-time: a solitary wave with the same asymptotic value at x = oo and x = + ~ and a wave with different asymptotic values. The first one is called a “bell” soliton, the second one a “kink” soliton (for details see ref. [5]). We consider some soliton solution properties which follow from the conservation laws for the case of the Klein—Gordon equation. In two-dimensional space-time we have instead of (1.5) —
(~_~+m2)~_2g2~l2=o.
(1.30)
This equation is that for a scalar field with self-interaction. The quanta of such a field are usually called constituents. Indeed, consider the field ~ obeying the equation 2)~=g2F (1.30a)
(El—m
Here as usually g2 is the coupling constant and F is the source function which can depend on an another field i/i or on q’. In conventional field theory the solutions of (1.30a) are looked for by the method of an expansion in g2, when =
c~o+g2fll2F(~,~o),
V.G. Makhankov, Dynamics of classical solitons
9
where c is the integration constant and q~is the solution of the equation for the free field (Li m2)p0 = 0. The quanta of this free field are just the constituents. Sometimes we call them the 2 is “components”. In our case the function describes the self-interaction of the constituents if g not small the aforesaid procedure does not allow one to obtain the solutions sought for. and Moreover, even when g2 ~ 1 equation (1.30) has, besides the solutions which are power series of g2 and tend to the free field as g2 0, also solutions which are singular in g2. Below we shall see that these solutions are just the solitons. To find a particular stationary solution of eq. (1.30) let us pass to the moving frame ~ = x vt and transform the function —
—~
—
=
i(w
~/i(~) exp{ —
0t Then
~fi obeys
~
—
—
k0x)}.
(1.31)
the equation 2g2i4r3 0,
At/i +
where
(1.32)
y
A = m2y2—w~, k 2. (1.33) 0= w0v, ~2 1—v We can consider to 0 and k0 as the energy and the momentum p respectively, “component” 2y2of ~2the> 0. In fact, the quantum. Equation (1.32)gives has a solution in the form of a bell soliton for A = m first integration of (1.32) —
—
A*2 + ~y2g21/14 =
const.
Supposing cit and ~ vanish as ± we have const = 0. From the condition ~ = 0 we find the dispersion relation for the component ~
A = m2y2—e2=~y2g2l/J,~,, whence it follows that the component energy 2
2
I
2
2
2
1/2
=(m y —~yg cfrm)
—
(1.34)
~2)I/2
exp{ i(t —
—
(1.35) Ef =
my is the energy of the free component quantum.
vx)} sech{(m2y2
—
2)U2(x
—
vt)}
(1.36)
of eq. (1.30) describes a bound state of, in principle, an infinite number of component quanta. In the laboratory frame this state is infinitely degenerate: the energy of each component is the same. The momentum distribution of the quanta can be obtained by the Fourier integral expansion of (1.36). In the rest frame it repeats (1.36) ~‘s(P) ~ sech(~irp/p 0), Po
A.
(1.37)
From eqs. (1.36) and (1.37) we conclude that the soliton width is entirely determined by the interaction energy of the components and decreases with increasing energy. From (1.37) it follows that the increase of the interaction energy (the potential energy) leads to the result that the components with greater and greater momenta are captured by the soliton (sometimes such a phenomenon is called condensation [19] by analogy with a superfluid). At the same time we note that the solution (1.36) is
10
V.G. Makhankov, Dynamics of classical solitons
singular with respect to the coupling constant g2 and does not reduce to the free field ~ as g2—*0. Now we consider the dispersion relation of the soliton itself as one for a quasi-particle consisting of the infinite set of components. Its energy and momentum are round (see (1.8) and (1.11)) from the formulae
E=
J
~dx
f{~~I2+ I~r±m2I~2—g2~4}dx,
=
(1.38) (1.39)
Putting q~’ =
~s(X,
t)
in these formulae and evaluating the inteerals we have
E=~(l —A2)~2(1±2A2)y, P
M
=
E
~~(l
—
A2)~2(l+2A2)yv,
(in the rest frame)
=
(1
(1.40)
—
A2)~2(1+ 2A2),
A~(/my)~1. Here we see that there is the usual relativistic relation between the energy and the momeptum of the soliton quasi-particle: E2=P2±M2.
~1.41)
A similar relation for the case of the real field = this paper can be obtained from (1.40) if one puts ~
E’ =
EIA...o,
P’
=
PIA...O, M’
~
= i/i
has been found in ref. [20].The formulas of
=
The example considered is somewhat more interesting because the soliton as classical object behaves simultaneously both as a wave ~ exp{i(vx t)} and as an extended particle with Mass M, momentum P and energy E, and during the motion this particle is effected by the Lorentz contrac~tin~ far as its width E~xcc l/y. Similar quasi-particle properties are also inherent in many other solitons, in particular the solitons of the S3 and Higgs field equations. —
1.4. Solitons and the Fermi—Pasta—Ulam problem The FPU problem was briefly considered in the Introduction. It consists in that, despite of what people expected for a long time, sometimes even non-linearities which are not “small” do not lead to the stochasticity in a system (i.e., to an equipartition of energy between the degrees of freedom), that is the system does not go to an ergodic state. First this was tested by the example of a segment of a non-linear string. Its oscillations can be approximately described by the Boussinesq equation [3]. Peculiar non-linear oscillations of the energy distribution over the spectrum arise in the system: sometimes the distribution over the spectrum expands (i.e., the energy is redistributed between harmonics which are close to the basic one), then it narrows again almost to the initial state [3] (see
V.G. Makhankov, Dynamics of classical solitons
II
fig. 1.1), i.e., the trend to thermalization is absent. A similar result was obtained by Zahusky and Kruskal [4] for the KdV equation with periodic boundary conditions. Only after numerous studies of non-linear equations did it become clear that some of them are complete integrable, i.e., they have a numerable set of explicit integrals of motion and the corresponding Cauchy problem can be solved by the inverse scattering method (ISM). Reference [5]reviews mainly such equations and the study of its solutions. From all the results of that review we need only the fact that the solitons of complete integrable equations emerge after interaction keeping their initial form and speed, and only a soliton phase shift appears. It means that such equations describe the elastic interaction (scattering) of solitons. This result was confirmed later by a number of numerical experiments [21,22]. Looking ahead somewhat we note that the success of the ISM method in the integration of some non-linear equations* gives rise to the hope (or even an optimism) that all or practically all the systems having soliton solutions are completely integrable. For example, in the review by Scott Ct al. [5] there is even the assumption that “for a non-destructive interaction of solitary waves the existence of a Lagrangian is necessary”, and “the only well-founded method for determining whether a solitary wave is really a soliton (in Zabusky’s definition) remains a direct numerical computation (the second assumption by Scott et al. [5])”. Concerning the first assumption we note that at present there at least three non-integrable examples—the Klein—Gordon equation (1.5), the Higgs field equation (1.6) and systems (l.22)+(1.23) and (1.22) + (1.24). Below we shall show that the solitons of these equations interact inelastically up to the formation of bound states, and the existence of a Lagrangian is of practically no use. The second assumption seems to us to be highly reasonable and moreover such a test has been used in many investigations of the FPU problem [22,24—27]. The inelastic interaction of solitons would indicate the ergodicity of the system which is described by the equation studied. In conclusion we note that exotic properties of equations which are connected with their integrability in two-dimensional plane space-time can disappear for a more realistic physical picture, e.g., for spherically symmetric objects [28]. We shall come back to this problem below.
2. Solitons in plasma physics There are now many original works, reviews and books [29—32]dealing with the investigation of non-linear waves in dispersive media. We discuss therefore only the most recent results. 2.1. Equations describing non-linear phenomena in isotropic and magnetized plasmas and their soliton type solutions A rich family of various oscillation modes (collective degrees of freedom) is inherent in plasmas which are specific gaseous media consisting of two or more kinds of charged particles. This large variety appears especially in magnetized and slightly non-uniform plasmas. There are only three oscillation modes in a uniform isotropic plasma, viz., two high-frequency and one low-frequency modes: *
For recent papers see refs. [231.
12
V.G. Makhankov, Dynamics of classical solitons
(i) Transverse oscillations (t) with the dispersion formula t
2
to ‘(k C
2
2 1/2 +tope)
2/me)U2, k and c being, respectively, the Langmuir frequency, the wavevector and the = (4lTne velocity of light); (ii) Longitudinal Langmuir (~)oscillations with the spectrum (tope
322
to =tope(l+~k d~) W~1Ve being the Debye length, Ve = VTe/me and the electron temperature Te is measured in energy units); (iii) A longitudinal low-frequency ion-sound oscillation mode
(de =
= kv~(l+ k2d~Y”2, where v~= VTe/mi is the ion-sound velocity. Such oscillations exist only in a non-isothermal plasma with Te5~T 1 (T, being the ion temperature). The dispersion curves are plotted in fig. 2.1. The number of modes in a magnetized plasma is more than ten. We can not discuss here all these oscillation branches the number of which increases in slightly non-uniform plasmas because there is an extensive special literature devoted to these problems [33]. An imposing series of papers appeared at the end of sixties in which the propagation of solitary waves (solitons) in plasmas was investigated. Nearly all the oscillation modes in plasma known by now were found in these works to be described in the non-linear approximation by equations having soliton type solutions. It was usually either KdV equation and equations of a very similar type or the non-linear Schrodinger equation with a self-consistent potential. This fact which is paradoxical at first sight can be simply explained: the specific feature of linear plasma oscillation is manifest in their dispersion, i.e., in the dependence of frequency to and wave group-velocity Vg on the wavevector. On one hand it is well known that wave packets disperse when propagating, on the other hand, hydrodynamics tells us that non-linearity can give rise to a steepening of the wave front, i.e., to making wave packets more narrow. The balance of these two contrasting phenomena can give a non-linear stationary wave (see, e.g., [29, 32]). We present here, as an example, the papers not discussed in [5]:Non-linear drift waves shown in [34]to be described by KdV and Bq equations, Bernstein modes by
Fig. 2.1. Dispersion curves of plasma oscillations.
V.G. Makhankov, Dynamics of classical solitons
13
the sine-Gordon equation [35].The soliton solutions of sech and tanh types have been obtained[36]for the Alfvén mode coupled equations (using the familiar technique developed in non-linear optics [37]).It was suggested [38,39] to study the propagation of cylotron waves along the magnetic field H0 usiiii a Schrödinger equation with a cubic non-linearity (S3). The latter describes whistier Naves propagating along H0 as well. There are several ways to obtain (with a certain degree of rigour) non-linear equations describing wave phenomena in a plasma, viz., a Lagrangian formalism, Karpman—Krushkal’s method [40], the reductive perturbation theory [41], the hydrodynamical approach adopted in [13] and developed in [14, 15], and many others. Some of them allow one to take into account the resonance wave-particle interaction which leads to non-local terms appearing in the corresponding equations. In what follows we shall leave apart these effects and discuss only the qualitative features of the soliton phenomenon. 2.2. Boussinesq equation and the stochastization problem The equation 2)
was
+
‘p~~
cott = (px.v + (q~ derived by Boussinesq
(2.1)
in 1872 to describe shallow-water waves. The two-parameter family of
soliton type solutions 2 =
/T
A sech {K v ~A(x
—
vt
—
x 0)},
(2.2)
where V=±(1+7A)U2,
KI
obeys eq. (2.1). Notice that the velocity v of this soliton is completely determined by its amplitude, so that the
solution (2.2) depends only on two parameters A and x0: the amplitude and the position of soliton at t=0. Equation (2.1) describes a large range of physical phenomena including the propagation of ion-sound (s) waves in a uniform isotropic plasma making allowance for the first non-vanishing terms associated with the non-linearities and the deviations from the linear dispersion law. N-soliton solutions of this equation have been found in [42]. It means that the interaction of solitons does not asymptotically change the shapes, amplitudes and velocities of solitons and consequently, that it is elastic. It is this fact, which motivates us to think that eq. (2.1) may be complete integrable [43] and describes a Hamiltonian system not permitting stochastization. The operators needed to solve eq. (2.1) by the inverse scattering method have been found in [43]. Note, that the Bq-equation can not be solved by a computer by using a sufficiently finegrid alongthe x-axis since the appropriate Cauchy problem turns out to be incorrect [24]. The improved Bq-equation (which we call IBq) can be obtained starting with the exact hydrodynamical set of equations 3V/at+(VV)V= —V~, 3n1/at + div(n, V) = 0, (2.3) 2q~= ee~ v Here, we shall, however, obtain it in a less rigorous but simpler and more descriptive way using the —
14
V.G. Makhankov, Dynamics of classical solitons
known dispersion formula for ion-sound waves: 1 ~k2
2
(2.4)
(k is measured in frequency units and the velocity is taken to be equal to unity), or
Then we have in coordinate representation for the linear part of the operator L3=~j~~(l_V2)_V2,
or in one-space dimension world (“plane” world) ~2
~2
~4
L~_~—~-~—a2at2.
(2.)
We obtain now instead of eq. (2.1) the IBq-equation Lço
—
(~2)
=
0,
(~t~j
in which ~ stands for the plasma density in the case of ion-sound wave propagation. Equation 2 = k~(l k2), (2.6) and its dispersion formula (2.4) approach eq. (2.1) and its dispersion formula, to when k e~~ In the region k ~ 1 (a wave length of the order the Debye length or less) the Bq-equation —
gives rise to an unphysical instability. This is the physical reason why the Cauchy problem of eq. (2.1) becomes incorrect since to2 <0 for k> 1. The equation with the expression (2.5) as a linear term was obtained in [21] to describe waves propagating at right angles to the magnetic field. It is convenient to write eq. (2.6) as an equivalent system using a function V which has the meaning of a “velocity”
3/ t9V
~
3
2
This system has the following constants of motion for the soliton type solution
dS
1/dt
= dS2/dt
=0,
S1
=
J
~ dx,
S2 =
J
V dx.
Notice, that the energy conservation law of the Bq-eq. [43,44] 53=~
J
2+V2+~co3—(co~)2)dx (co
is no longer valid in the above form for eq. (2.6). The Fourier analysis gives [14] (~2)~~as a non-linear term and yields in addition
V.0. Makhankov, Dynamics of classical solitons
0.36
X=O
15
~
x=560
Fig. 2.2. Formation of a soliton train from a symmetrical perturbation at rest (half a picture is shown).
Fig. 2.3. Formation of a soliton and wave train in the framework of eq. (2.5).
(~+~2)dx=0. Equation (2.6) as well as eq. (2.1) has a two-parameter family of solutions of the type (2.2) but with K2 = (1 + ~A~’t2. For the equation Lp (~,2)~ = 0 we have solitons (2.2) with K = 1 and v = (1 —~Ay”2.As should be expected the solitions of the Bq and IBq equations do not practically differ from each other when k~1,i.e., A<<1, (v—1)~l. The dynamics of various initial wave packets was investigated in the framework of eq. (2.5) via a computer by Bogolubsky [24]. The formation of solitons for arbitrary initial perturbations repeated qualitatively the picture described in papers* by Berezin and Karpman [21],except that eq. (2.6) gives a symmetric picture, for example, the formation from a packet at rest of several solitons moving in opposite directions. The results of [24] are shown in figs. 2.2 and 2.3; the evolution of a packet which was at rest initially in fig. 2.2, while the packet is moving with unit velocity along the x-axis in fig. 2.3. The most interesting result however, was obtained when investigating the collision of two solitons moving in opposite directions. The interaction of two small amplitude solitons, A ~ 1, on the basis of eq. (2.6) does not practically differ from that on the basis of eq. (2.1). But the picture alters considerably for large amplitudes of the interacting solitons. In fact, the fraction of short harmonics grows exponentially with decreasing soliton width and as a result eq. (2.6) unlike (2.1) leads to an inelastic interaction of solitons (see figs. 2.4 and 2.5). The coefficient of inelasticity grows with soliton amplitude. A modification of the IBq equation analogous to the MKdV equation yields —
Lp
=
~
(2.7)
which we call the IMBq. This equation together with the MKdV can be used to study the properties of an anharmonic lattice and of non-linear Alfvén waves. The interaction of solitons = *
±A
sechf~4—(x±vt
—
xo)},
v = (1 + ~A2)U2
In this paper the authors have studied by means of a computer the formation of solitons for the KdV equation.
(2.8)
16
V.0. Makhankou, Dynamics of classical solitons /9g
~
~o5
___
x=-30
~=3O
x=-30
Fig. 2.4. Dynamics of the nonelastic interaction of IBq solitons (A 1 = A,= 2).
s=30
Fig. 2.5. Head-on collision of IBq solitons with different amplitudes (A1 = 2. A2= I).
of eq. (2.7) describing the rarefaction and compression waves turns out to be inelastic as well [24]. We would stress here, that the hydrodynamic equations can be reduced in the case of ion-sound waves to the non-linear equation for the hydrodynamic velocity V which in a plane case assumes the form 2) LV= —(V 1~. (2.9) The interaction of its solitons
2)112, (2.10) ~+(1 +~A is also inelastic [24] (see fig. 2.6). This result at first sight is in a contradiction to that which was obtained in [45] via a computer simulation of the exact system (2.3). There, the authors came to the conclusion, that the interaction of s-solitons is elastic when their amplitudes are less than a critical one. As we saw the inelasticity of soliton interaction is fairly displayed only at sufficiently large amplitudes. A rather complicated analysis is needed to show this effect at not so large amplitudes (see, for example, [251). Finally, it should be underlined that the improvement of the Bq-equation has first arisen from a pure mathematical point of view (the correctness of the Cauchy problem) and then this “improvement” V= ±Asech2{~~ xjf(x~ vt—xo)}~
V
V
—
—
—————-——
Fig. 2.6. Interaction of solitons for eq. (2.9).
V.0. Makhankov, Dynamics of classical solitons
17
turned out to be not only natural from physical considerations but also to give a new qualitative result; the inelastic interaction of solitons and the stochastization of the system. We shall below in an analogous way “improve” (or “make worse” as one prefers) the KdV equation. 2.3. KdV equation and its “improved” version IKdV The “improvement” of the KdV equation is no longer so obvious and descriptive as it was for the Bq equation. But from a computational point of view (in a computer sense) such an “improvement” is sensible and has a right to be. First, a few words about KdV equation having in the laboratory frame the form (2.11)
4,t+~,X+(q,2)X+~XXX0,
which has been used to investigate firstly shallow water waves [11] then long ion-sound waves in plasmas [5,46] travelling in the positive direction along the x-axis, and many other linear wave phenomena (see ref. [5]and above). The most frequently used form of the KdV equation is + p,~= 0, (2.12) p, + a(~2)~ which is (2.11) written in a coordinate system moving along the x-axis with the velocity of linear waves for k—*0. Equation (2.12) can be easily obtained by the transformation ço—*p(x—t) and describes a completely integrable Hamiltonian system [47]. Its N-soliton solutions had been derived by the inverse method to give non-destructive (elastic) soliton collisions which result asymptotically only in a phase shift of the interacting solitons. Equation (2.11) has been derived in the long-wave limit with the linear spectrum being to = k(l k2). The basic assumption employed in deriving this equation that the whole perturbation propagates only in a positive direction along x-axis, breaks down when k> 1. Such an incorrect description of short wave behaviour on the basis of eq. (2.11) does not lead to the incorrectness of Cauchy problem as in the case of the Bq equation.* We shall “improve” the KdV equation to obviate this short-coming. At the same time our equation will describe the evolution of long waves at any rate no worse than KdV but will be more convenient from the computer study point of view. We take the dispersion relation in the formt to = k(1 + k2~’to obtain instead of eq. (2.11) —
Pt +
~
+
= 0.
(~2)~ —
(2.13)
This equation has been suggested in [49] to investigate tidal waves. In the coordinate system moving with unit velocity along the x-axis, we have ~ + (q~)x+ 4Dm~
= 0.
(2.14)
The evolution of arbitrary initial perturbations and a process of soliton interaction have been studied on the basis of eq. (2.14) in [25] by using a computer.** *
The dispersion, w = k( I
—
k2) leads to an ‘unpleasant” stability condition for computer studies of the KdV equation, r ~tch3 (where c is a
numerical factor of order of unity, while r and h are the temporal and spatial grid steps). This formula clearly coincides with that for the KdV equation when k 4 1. ** The stability condition for a computer study in the case of eq. (2.14) was shown [25]to be not r < ch3 as above but r ~ h.
V.0. Makhankov, Dynamics of classical solitons
18
We have for eq. (2.14) the constants of motion
~fr =
0,
S~=
J
dx,
S~=
J
(~2
dx,
+ ~
S2 =
J (~ —
and the two-parameter family of soliton solutions
A
1/2
4D=Asech2{(2(2A+3)) (x_vt_xo)},
v=~A
(2.15)
(note, that v = I +~Ain the laboratory frame). It has been pointed out in ref. [25],that the evolution of initial packets studied on the basis of eq. (2.14) is, in general, similar to that for KdV equation which has been simulated in [24,21]. A noticeable difference arises only when one considers sufficiently short (k ~ 1) and powerful (A ~ 1) solitons. The picture when a large amplitude soliton (A1 = 6) overtakes a small one (A2 = 1.5) has been examined resulting in the emission of a rarefaction wave of a very small amplitude (A~ 1.7 x 10~) which lagged behind the solitons. Many various manoeuvers were employed providing the display of the inelasticity of the interaction and the computational errors to be less than 0.014%. The effect of inelasticity was found to be about 0.3% of maximum soliton amplitude with the level of computational errors being less than 10% of the effect. Such an effect for head-on soliton collisions within the framework of the IBq equation reached about 10% at A = 2. The difference may be explained by the fact that the “distance” between the IBq and Bq equations is larger than that between the IKdV and KdV equations*. An analogous situation took place in the framework of an IKdV with a cubic non-linearity instead of quadratic one (IMKdV): 4Dt +
(4D~)~ + 4Dm
=
0,
(2.16)
whereas the MKdV equation ~
0
is completely integrable according to Wadati [48]. Equation (2.16) has soliton solutions 2)”2 (x vt x = A sech{(2 + A 0)}, —
2,
—
v = ~A
(2.17)
and the inelasticity of their interaction grows by more than ten times in comparison with that of eq. (2.15). When defined as AineilAmax the coefficient of the inelasticity Kinei depends on Ama,, as shown in fig. 2.7. Thus, solitons of the IBq, IMBq, IKdV and IMKdV equations can show the inelastic interaction which means that these equations describe dynamical systems in which stochastization and hence irreversible energy exchange between degrees of freedom are possible. It is meaningful now to turn to the definition of a soliton which so far has not been formulated very clearly. We broaden it so as to include localized solutions of the corresponding non-linear equations *
Such an explanation was suggested in ref. [43~.
V.0. Makhankov, Dynamics of classical solitons Xuiet
19
% 2.6g
2.75
2. 1
/ ~ ~ Fig. 2.7. Inelasticity coefficient of soliton interaction versus amplitude for eq. (2.16).
which, however, can inelastically interact and even fuse. It seems, therefore, to be more natural to define a soliton as a solution of a non-linear wave equation (or a set of equations) confined in a space at all times and having a finite energy (or a rest mass) (see subsection 1.3). The second requirement is of use when one considers solitons to be classical models of particles [10, 19,49]. All the equations mentioned in this and preceding sections describe, from the plasma theory point of view, the so-called low-frequency branches of oscillations (for example, ion-sound, Alfvén, and many others). From the point of view of a given quasi-particle interpretation their solitons are made of massless constituents the velocity of which tends to unity. Including the interaction makes the components tachyons (t’> 1). Once the frequency of these oscillations tends to zero together with wavevector (to cc k) it is natural to call them low-frequency ones. In subsequent subsections of this section we shall discuss high-frequency solitons which have components with a rest mass of, for example, that of Langmuir waves. 2.4.
Schrodinger equation with a potential satisfying a non-uniform wave equation
Here we examine solutions of the system (l.22)+(l.23) in the plane geometry: i4D
5 +
4D,,,,
—
~I~4D = 0,
L11 = ~
(14D12),
(2.18)
and compare them with the solutions of the S3 equation i4D1 +
4D~,,+
=
4DJ24D
0.
(2.19)
Both the system (2.18) and eq. (2.19) can be obtained starting from hydrodynamic equations that describe Langmuir plasma turbulence. In this case ~ and 1 are respectively the amplitude of the high-frequency electric field and the density variation (perturbation). However, as above, we obtain eq. (2.18) less rigorously but in a simpler way by using the dispersion relation for Langmuir plasma waves 3
,2
fiSk = Wpe+~topeK
with ~
2
(2.20)
tie,
being the Langmuir frequency, and de =
Ve/tope.
V.G. Makhankov, Dynamics of classical solitons
20
A high-frequency wave propagating in plasma creates a Miller force [63] which expels some of the plasma particles from the region where its amplitude is larger, Fcc_~~~(IEI2). As a result, the local plasma density is decreased in the larger amplitude region and the wave propagates in the perturbed plasma with a density n = n0 + Sn. Upon substituting this relation into eq. (2.20) one obtains
2(d~°~)2 + ~w~]Sn/n
tok =
+
~w~k
0.
(2.21)
When obtaining eq. (2.21) we took the density perturbation into account only to first order in Sn and neglect higher-order terms. Let us write eq. (2.21) in the form (0)
3
SWtokWpe
(0)
~tope(de
(0) 2
1 (0) )k 2 +2Wpe Sn/n0,
then changing to the coordinate representation with the help of the formulae Sw iô/8t, k i3/3x, we have 2/3x2 1)4D = 0: (2.22) (i3/3t + 0 the equation for the high-frequency field envelope of -oscillations. In eq. (2.22) we use the following dimensionless variables -+
—
—*
—
2 —x x=~/,~-,
2
t~I~Wpet,
2
~
~=4I~t 1Sn
‘~
2
Mp~n~T’ 3 E
and = me/mi,
Vs =
(T/m~)”2.
We have now to write the equation which governs t, the plasma density perturbation. In a quiescent plasma it is clear that the wave equation is
(~
=
—
0.
(2.23)
In the presence of the high-frequency field the Miller force appears which in turn leads to a source function on the right-hand side of eq. (2.23). This function can be simply found if we recall that the Miller force expels plasma from the region of the high-frequency field, Sn cc El2 therefore we finally have the second equation of system (2.18). The so-called quasi-static limit of (2.18) is sometimes considered which one can obtain by neglecting the term 32c1/3t2 (the limit of infinite ion mass), then —
2
2
= _~(I4Dl2),
or for soliton-like solutions ~t~_l4DI2, and we come to the S3 equation, formula (2.19). It should be underlined, that the above method for getting eqs. (2.18) is valid only in the plane
21
V.0. Makhankov, Dynamics of classical solitons
Fig. 2.8. Quasi-decay instability growth rate versus wavenumber of perturbation.
geometry. In a more general case a more rigorous approach gives the system (see [13])
(~
V2)cI = V2(IVci’12), (2.24) div( 2iV~/i5 VV2cir + 4Vci’) = 0, where cit is now the amplitude of the high-frequency potential, and not the field (we shall return to these equations in section 5). Both eqs. (2.19) and (2.18) describe the instability first examined in ref. [50] and called there the modulational instability. Its initial stage is satisfactorily depicted by the following dispersion equation [25] (Nk—Nk 1.~)dkl =0, (2.25) —
—
—
~
~
K Ve
~
K meve
totokl+tok,_K+1S
where w5,~is the ion Langmuir frequency, the solution of the linear dispersion equation (2.20), k the wavevector of the Langmuir wave, K the wave number characterizing the modulational amplitude, viz., the wavevector of the low-frequency perturbation and Nk is the 1-quantum number. Equation (2.25) can be obtained from weak plasma turbulence theory [25] and also directly from (2.18), if one takes a slightly modulated 1-wave [13] tok
4D4D0+S(pexp(—iwt+ikx),
4oc
exp(—itot+ikx).
The growth rate determined by eq. (2.25) as a function of wave number is depicted in fig. 2.8. There are many names for this instability in the plasma physics literature the modulational, the modified decay, the quasi-decay, or the two-stream instability, and so on. In fact, eqs. (2.25), (2.19) and (2.18) describe the instability of a gas of Langmuir plasmons in relation to their adhesion similar to what takes place in a gravitating gas. The analytic investigation of the dynamics of this instability is quite complicated, except for the solution of the Cauchy problem for the S3 equation. It is because the S3 as well as the KdV equation describes a complete integrable Hamiltonian system so that the inverse method can be applied [12]to look for the asymptotic behaviour of an arbitrary initial packet.* A whole series of papers have recently appeared, in which the attempts were undertaken to examine the dynamics of solitons in the framework of (2.18) by analytical methods. The most interesting of them, from our point of view, are those by Karpman [52]and by Thornhill and ter Haar [53].In the former one the formation of a soliton from a near-sonic initial packet of small amplitude 1-waves is studied on the basis of (2.18). In the latter, an analysis of the conservation laws for eqs. (2.18) found in [54] —
dS1/dt=0, *
1=0,1,2,
In fact, computer simulations have shown a much more complicated picture [25].
(2.26)
V.G. Makhankov. Dynamics of classical solitons
22
2dx;
s
S
1
=
S2
=
f J
(2.27)
l4DI
0~s=J
{i(4D4D~_4D*4D~)+2~u}dx; 2+~l {l4D~l
(2.28)
2+~2+~u2}dx;
(2.29)
4Dl
with ~I~ +1 u~=
0,
was used to examine the features of the interaction of Langmuir solitons with each other and with ion-sound waves. Computer investigations of the dynamics of the formation and interaction of Langmuir solitons in the framework of both eqs. (2.18) and (2.19) and a comparison of their results was first done in ref. [25] and later in [51, 54]. In ref. [25] a quasi-particle interpretation of these phenomena similar to the one, which was discussed above, in section 1, for the Klein—Gordon equation was also suggested. However, we deal now with non-relativistic constituents (plasmons) and non-relativistic solitons. The rest mass of these solitons is determined mainly by the integral S since the Langmuir frequency is much greater than the dispersion and non-linear terms. We can therefore take the first integral as the total number of Langmuir quanta (plasmons) conserved by virtue of eq. (2.26) in all interactions.* The second integral represents the momentum and the third one the energy of the soliton. Lastly, eqs. (2.18) picture the system of two interacting fields with the second one being a massless one transmitting the interaction between plasmons. To proceed further, let us consider some relations following from eqs. (2.27) and (2.28) for the soliton-type solutions of eqs. (2.18) and (2.19). The first one we supply with index d (dynamic), the second with s (static) =
A sech{~(x
I~~l4Ddl7’
—
vt— xo)} expfi(~x—cL~t
—
y
~d4v2~4Y,
~,)},
(2.30)
=l—v,
where the soliton velocity v is measured in units of the ion-sound velocity v. and coincides with the group velocity of Langmuir waves, = 3koV~/wpeVs, Vg
=
A sech[~(x
—
Vt
—
xo)} exp{i(~x
—
fi,,t
—
~
(2.31)
Both families of solutions (2.30) and (2.31) are four-parameter ones (A, v, x0, in contrast to the low-frequency solitons of the KdV and Bq equations, which are two-parameter families (A, x0). The solutions (2.30), (2.31) are sometimes called the two- and one-parameter families, respectively, i~)
*
This
fact is also a consequence of U(l)
symmetry.
V.0. Makhankov, Dynamics of classical solitons
23
forgetting the parameters x0 and ~ which determine the position of soliton at t = 0 and its phase. It should be noted that for a completely integrable equation such as the KdV, S3, sine-Gordonequations and others, it is important to take these parameters into account since only they alter after the soliton interaction. We shall, hereafter, deal mainly with equations which deviate from completely integrable ones since despite of the elegance and the large number of the latter, they seem to be albinos in a normal population. The integral S2 for eq. (2.19) was obtained in [12] and simply coincides with the Hamiltonian H
S2~=
f
(14DXl2
—
‘I4DI~)dx.
(2.32)
The first integral has the same form in both cases. Upon substituting (2.30) in (2.27) and (2.29), (2.31) in (2.27) and (2.32) we have (see [51,53])
Sd
2V’~m,
=
m
=
Aly,
(2.33a) (2.33b) (2.33c) (2.33d)
2)y6},
S2d = \/~{mv2/2_~m3(l 5v —
S,,Sd~y, S2s =
\r2y{mv2/2—~m3y2},
or ~S,,(v2—j~S~), S 2 j~y6S~(l 5v2)). 2d = ~Sd(v One can see from formulae (2.34) and (2.35) that the integrals S and S
(2.34)
S2s =
—
(2.35)
—
2 for a formed soliton are firmly associated. In the static approximation there is only one collective degree of freedom, hence both S and the high-frequency part of S2, i.e., are conserved and the formation of a soliton from an initial packet takes place only with S and S2 being connected by relation (2.34). Thus, given 5, the value of S2 is uniquely determined; if not, viz. S2(t = 0) S2(s) the initial packet can, generally speaking, oscillate infinitely around its equilibrium position [25]. Figure 2.9 taken from [25] depicts the time-dependence of the amplitude for a packet which is initially at rest. One can see that first it narrows with its amplitude increasing, then it passes through the equilibrium position due to inertia (remember the rest mass of the plasmons), that is, the soliton state, and turns back, and so on, while the amplitude of the packet did practically not alter in the whole computer run. In the framework of the KdV equation an arbitrary initial packet at once starts to break up into a number of solitons, which is determined by the number of bound eigenstates of the Schrödinger equation with a potential which has the form of the packet under consideration (see the inverse method and details in [5]),and a weak background which carries the energy 52,,
:~
excess. The S3 equation shows a long oscillating processaround the equilibrium (soliton) state, except for
Fig. 2.9. Amplitude of a Langmuir wave packet as a
function of time (at the point x
= 0)
in the quasi-static approach.
24
V.0. Makhankov, Dynamics of classical solitons
o.5~ .1
0
Fig. 2.10. Dependence of packet amplitude on time (in the origin) in the frame-work of system (2.18).
x Fig. 2.11. Formation of a soliton from an initial packet.
N-soliton initial packets. This is the main difference in the dynamics of arbitrary initial packets for the completely integrable S3 and KdV equations which is associated apparently with the fact, that the linear spectrum for the KdV equation is of a quasi-decay type, i.e., the decay w~(k1)+ w~(k2)= w~(k1+ k2)
(2.36)
is only just forbidden by virtue of the fact that tos(k) cc k and can become allowed in the presence of a non-linear interaction. As a result, we have small amplitude trains of ion-sound waves when solitons are formed from initial packets. The Langmuir wave spectrum, as can be easily seen from the S3 equation, is of a non-decay type with +
to~’,
2w~
(iS~1+~2
tope
being forbidden strongly. That is why apparently there appears no background of Langmuir waves~(in the case of the Klein—Gordon equation with a cubic non-linearity as we shall see below, the situation may alter). The computer simulation results confirm the theory of the S3 equation: 1-solitons emerge from the collision without changing their shapes and velocities. But we have a completely different picture if we study the evolution of initial packets and the interaction of solitons with each other on the basis of the system (2.18). The soliton now forms from an arbitrary initial packet during the time of a few oscillations with the energy excess carried off by s-waves (1 field). Figure 2.10 shows the plot analogous to that of fig. 2.9. but for the system (2.18). It is also seen from fig. 2.11 that, when oscillating, 1-plasmons emit s-waves until the soliton state is reached. It is very interesting to note, that although the oscillations of 1-plasmons in the self-consistent well of the soliton continue, the emission of s-waves ceases. It means, that in the Classical system considered there arises a prohibition for emission as occurs in a familiar quantum theory of atom, so that a classical object displays not only quasi-particle properties but also certain quantum prohibitions (see also [10]).
*
In principle, even in the S3 model the energy excess may be carried off by very small amplitude solitons but the smaller the amplitude of the
soliton the slower this process goes since r ~
V.0. Makhankov, Dynamics of classical solitons
25
!P12 ~
/fr --
____ 0/
AJL
-<2>
x
/1
Fig. 2.12. Coalescence of two identical Langmuir solitons followed by s-wave radiation.
Fig. 2.13. Interaction of t-solitons with different masses (a light soliton hits a heavy one).
~I2
~
l’~
Fig. 2.14. Interaction of solitons with approximately equal masses.
)~-~
Fig. 2.15. Interaction of solitons of different masses (a heavy soliton carries with it a light one).
26
V.0. Makhankov, Dynamics of classical solitons
With the help of a computer simulation the dynamics of the interaction of -solitons with each other and with ion-sound waves has been studied in [51, 54]. In ref. [54] most attention was paid to a study of “elementary events” of collisions of solitons with various masses and it was found that only solitons of masses very close to one another interact effectively. The greater the difference between the masses of the colliding solitons, the smaller the effect of their interaction which gives rise only to changing the soliton energies contained in the integral S2. The number of solitons in such interactions is conserved, as a rule, but only the shapes and relative velocities of solitons change, except the case when a heavy soliton colliding with a light one can absorb it. Figures 2.12—2.15 [54] give pictures of these processes. In figs. 2.16—2.17 the interactions of ~-solitons with compression and rarefaç~tion sound pulses are shown. In the same paper it was found that when two identical solitons collided head-on they either pass through each other or fuse depending on their amplitudes. The analysis of soliton adhesion conditions was in detail carried out in ref. [511to obtain the region of parameters of solitons (namely, velocities and amplitudes) where two identical solitons adhere. This region is depicted in fig. 2.18 in the m, v-plane above the solid curve; it was determined on the basis of a series of numerical experiments. In the same papers the effect of boundary conditions on the process studied by a computer was shown to be significant, leading, e.g., to the adhesion phenomenon becoming reversible if the system is either placed in a potential well reflecting s-waves [54](boundary conditions with fixed ends t(— L) = c1(L) = 0) or with a feedback (the periodic boundary conditions) [51] (see fig. 2.19). It was also shown in [51] that the interaction (as well as the formation) of e-solitons passed through two stages. The first, the rapid stage, is associated with the attraction of solitons which start to interact and with the fusion of them into one soliton, which is then in an excited state; in plasma theory terms, this stage is usually called the aperiodic instability. The second stage, which is a slower one consists in 2 P1
/
/
/
1,:
A
Fig. 2.16. Interaction of (-soliton and s-pulse of compression.
‘I
/
/
‘
\,i
Fig. 2.17. Interaction between -soliton and s-pulse of rarefaction.
V.G. Makhankov. Dynamics of classical solitons
27
Fig. 2.18. Ares of fusion of two identical solitons in the mu. c-plane.
the emission of the excess energy (mass “defect”) in the form of acoustic waves; if the compound particle does not manage to break up into two, it goes over to the ground state (for details see [51]). Notice, that the analytical results of ref. [53] obtained by studying the constants of motion confirms qualitatively what was found in the computer simulation [51, 54]. At the end of this section we discuss briefly the whistler solitons studied in refs. [31,55, 56]. When propagating along the external magnetic field helicon waves (w) have a spectrum (iS
= (1)HekOC/Wpe,
I’f~
J\~~ 31~ ~4.33I0
(1
ml
II
i~.l0
a
t-O.8
~‘~_
~
I
~
I
I
2 3,4j~
~
31210a
2
!~I~A~ /
I
Fig. 2.19. Reversibility of a process of two soliton fusion under 2 ~‘ periodical boundary conditions. ~/ ~ ‘I ~jIj - S6.,0
28
V.0. Makhankov. Dynamics of classical so/ilons
which implies that
depends strongly on the wavenumber k0 and in that sense, the helicons differ qualitatively from Langmuir plasmons; indeed, they have no rest mass, which constitutes the main part of Langmuir soliton energy. Therefore, as was shown in [551only in the case of packets which are sufficiently narrow in k-space, one can obtain a system resembling (2.18): ~4Dt +
to
4Dxx + ~t~4D = 0,
=
(2.37a)
fr (l~l~)~
(2.37b)
where ~ is the complex envelope of a helicon field amplitude which is defined by the relation h,,
ih~=
—
exp{iw0(k0)t},
~
where 2/W~,e(flo),
wo(ko) =
c
toHe =
eH/mec,
toHek~C
is the velocity of light, and the following notation is introduced: t =
X = (top2eVs/WHeC2)X, =
(Wp2eV~/WHeC2)t,
(w~tek~c4/w~ev~)Sn/no, 14D12 = (k~to~ec4/41rnomiv~we)I4Dl2.
It should be underlined, that the opposite sign before the term of interaction in eq. (2.37a) leads to the fact that the helicon solitons move faster than sound waves, v’> v,,, and have a density hump instead of an 1-soliton density pit. The soliton solution of system (2.37) has a form similar to (2.30), except for the expressions for 1 and v which now are =
14D12/(t)
—
1),
V
v’;Ivs =
2(toHekoC2/to~eVs)>
1.
Moreover, the system (2.37) has just the same constants of motion, except for the sign of the term proportional to 1 in S 2, namely,
J{I4DxII4Dl+}dx,
S2
=
c11
+ u,
and =0.
Then we have 2—~y6S2(v2—5)), ~y2 = v2— 1. S = 2V2A/7, S2 = ~S(v Note, that in this case the energy part determined by S depends on the soliton velocity and vanishes formally as v —*0:
E= WHeCV 5
w~
4
The condition for the validity of the system (2.37), iXk ~ k0, implies that A~v/V2y and
S2>O.
V.0. Makhankov, Dynamics of classical solitons
•28.
29
~_~__~i~
l~I~
~
~ 2.0
II
f-2.06
~
___________________
~/\ ~ ~
t=6.75
t=3.513
~
~
Fig. 2.20. Formation of a supersonic helicon soliton.
Fig. 2.21. Head-on interaction of two identical w-solitons.
The formation and interaction of whistler solitons were studied numerically in ref. [55] (see figs. 2.20, 2.21). There the role of ion-sound waves in such processes was found to be decisive in spite of the w-solitons being hypersonic. Moreover, although decays of the type w —* w’ + s are forbidden within the framework of the weak turbulence theory [56], head-on collisions of solitons and their interaction with ion-sound waves give rise to a new (soliton type) channel for the dissipation of energy contained in the w-mode. We have been purposely dwelled on the results of ref. [55] since the influence of ion-sound oscillations on high-frequency dynamical processes is often ignored. For example, an erroneous statement has been made in [54] that there is a wide region of parameters, I
2.5. Coupled Langmuir and ion-sound waves in the near-sonic region A first look at the solution (2.30) is quite sufficient to see its short-coming (inconsistency): the density of the plasma perturbation 2
*
2
‘~Y 4Di We emphasize that the S3 equation has formally e-soliton solutions in the parameter region mentioned and it is in this sense that the
quasi-static approach is reasonable.
30
V.G. Makhankov. Dynamics of classical solitons
together with the constituent energy, and the inverse soliton width, yA/”/~,tend to infinity as v —* 1 with the soliton mass m approaching zero at the same time. This fact was first noticed in refs. [44,51,58,591. In [14] the inhomogeneous Bq and in [58] the KdV equations were suggested to replace the second equation of the system (2.18). In that case the perturbation of the density obeys, more consistently. the ~d,
equation ~
—
—
/3(t2)~+~
or I~+ ‘1?~+
+
~3(~2)
2)+~~ a4~~5++ = (l4Dl a’I~,,~ = — (14D12)x,
(2.38)
which equations must now be satisfied by the potential in the Schrödinger equation (a and /3 are small parameters proportional to me/mi). Approximate solutions of system /
‘4Dm +
4D~,,—
=
0,
4\
I
2
011 —
~ — ~—‘~
2
(1
2
)=
U ~
2
(l~l),
(2.39)
accurate to the linear terms in = ~me/mi have been found in ref. [14].the exact soliton solutions ~t the above systems were obtained in [58,60,61] by various techniques. It follows from (2.33b) that the faster the soliton the larger the relative contribution of the kinetic energy to total energy. Besides, ~n [51]it was found that fast solitons with S 2>0 interacted like sound pulses loaded with high-frequency field but not vice versa. At the same time in [51] the description of the plasma density perturbation (i.e., the ion-sound waves) was linear, so that as v 1 the soliton interaction dissappeared completelv,* On the other hand, a consistent description of the ion-sound waves using a Bq (or rather an IBq) equation gives a two-parameter (see section 2.4) family of solitons, i.e., there appears as compared to (2.30) an additional connection between the amplitude and the soliton speed — which is larger than the sound speed. It was shown in [58,601 that bound Langmuir-sound(1—s) solitons, which are stationary solutions of the set (2.39) (or of (2.38) plus the Schrodinger equation) form, like the s-solitons, a three-parameter family of solutions of the form —~
—
—
A tanh{B(x — Vt — x0)} sech{B(x 2{B(x — vt — 6A sech V = V—A, A2 == 48A2, A = =
—
Vt
—
~v2,
—
x0)} exp{i(~vx — 1~t—
~)},
(2.40)
(2.40a)
with A the eigenvalue of the Schrodinger equation being determined by the formula (2.41)
which means that the coupling between the amplitude and the velocity of solitons appears again. Note, that the solutions (2.40) as well as (2.30) are subsonic, V2 < 1, giving rise to A <0 and to attraction among the constituents in the soliton.t An elegant method of solving the system analogous to (2.39) (for a near-sonic w-soliton) was developed by Bogolubsky in [611.We give it in the appendix. * The
author has been recently informed by Zakharov, that the system of the Schrodinger equation with (out + O/Ox)4 = —(3/ox)IcoI2 governing the
selfconsistent potential is completely integrable [23]. ~ Actually, from (2.30) it follows that A = [1—~v2 (eigenvalue) is given by the potential energy of the constituents in the soliton.
V.G. Makhankov. Dynamics of classical solitons
31
The system (2.39) has the constants of motion [44,60, 62] S=fl4Dl2dx,
f =f
—
S~=
~{i(4D4D~ 4D*4D~)+ ~u} dx,
S2
{4DXI2+~c2+l
(2.42)
~
which, as before, can be taken as a high-frequency quantum number, momentum and fs-soliton energy. Note, that in contrast to slow 1-solitons, the solution of (2.40) is a density cavern loaded with a high-frequency field the energy density of which has a two-hump shape (“camel” shaped). We have envisaged in previous section the dynamics of the formation and interaction of 1-solitons and in fig. 2.18 we depicted the region of adhesion of two identical 1-solitons moving towards each other in the m, v-plane. Consider how the picture of formation and interaction will change for 1-s solitons (2.40). First, a few general comments. Turn again to the m, v-plane (or, which is the same, the V—A, v-plane). Relation (2.41) corresponds to the line I in fig. 2.22. Points lying below this line give the four-parameter family of 1-solitons. Under but in vicinity of the line I, this 1-soliton family loses its stability (for instance, the point A) for it is found to be in a region of influence of the solution (2.40). Due to the conservation laws the soliton can only decrease its velocity v; hence it will, emitting s-waves, go off the line I along a straight line parallel to Ov towards small v where the effect of the solution (2.40) weakens. When it is given by a point above the line I, say B, the Langmuir wavepacket will move (in parameter space) along this line either emitting s-waves or breaking down into several packets (possibly both simultaneously) depending on the initial data. A similar movement in parameter space will occur due to the interaction of different types of solitons. Following the paper by Thornhill and ter Haar [53], Watanabe [62], starting out from the constants of motion (2.42), has found the selection rules for the processes of interaction of 1, ls-solitons and s-waves: 1+14-*1+ls,
t+s’E-*l+Is.
Somewhat earlier in ref. [611such processes were studied by means of a computer. There the Bq-equation
Fig. 2.22. Dynamics of ts-solitons (packets) in the A, v parameter plane. The curve I corresponds to the relation (2.41). A and B are possible initial points.
32
V.G. Makhankov, Dynamics of classical solitons
was replaced by the IBq equation (LI
3
ax2at2)~
8 (~2)= 8 (14D12)
—
in the system (2.39). Soliton type solutions to such a system coincide with those given by (2.40). The stability of two-humped solitons was verified by numerical experiment for v = 0.95 on the basis of the coupled Schrödinger and IBq equations. The solution (2.40), being conserved with high accuracy, propagates up to distances many times longer than the size of soliton. The head-on collision of two identical ls-solitons (with v = 0.95) turns out to be inelastic. Just after the solitons come in contact with each other there occurs an exchange of energy and momentum among the plasmons; the inside humps lose some of the plasmons and the outside ones gain them; finally there is a strong throwing-out of supersonic 1-plasmons (see fig. 2.23a). The fraction of these plasmons is rather considerable,. about 20 per cent of integral S is carried away by them. The remaining part of 1-plasmon comes together again in two diverging two-humped packets, which seem
~~ x—-z2
~1v~c
(~
\\
l~ \.~ X=—2.2
A
(~
-L2240
~
\\
1
//
ItI
~ X=2.2
“
2
t=e.8
—1 z5.#O~ ~
f—a
~
x~2.2
‘I
_122.102
~
/
IPI~3.6
\‘1/...~...
2.1 -O.82.i05
~
A
~
—2.5IO~
I~
~
~I/’ I
_,.12.,02
I
1
( a)
‘‘
~
—1. 18~i02 ( b)
Fig. 2.23. (a) Head-on collision of Es-solitons: (b) Interaction between an Is-soliton and a compressional s-soliton.
V.0. Makhankot, Dynamics of classical solitons
33
to turn into smaller amplitude Is-solitons. A similar effect of 1-plasmon acceleration has been seen in Is-solitons interacting with s-pulses. Already for an amplitude A,, = 0.4cI~,,an s-soliton knocks about 40% of 1-plasmons out off an Is-soliton, and these free plasmons are again supersonic (fig. 2.23b). A rarefaction pulse interacts with an Is-soliton significantly less strongly. Recall the opposite picture that took place in the interaction between 1-solitons and s-pulses. At the end of this subsection note, that even a small correction proportional to in the low-frequency field equation turns out to be able to rearrange the solution for the high-frequency field. New two-humped Is and Iw-solitons, members of a three parameter family, were shown to be less “conservative” in interactions than their four-parameter 1-soliton and two-parameter s-soliton brethren. We see in this example that a coupling of fields can enlarge the efficiency of soliton interaction. In addition to the dynamical effect mentioned above, the solutions (2.40), (2.41) have one more attribute distinguishing them qualitatively from an 1-soliton (2.30). The latter exist in the whole area of the parameters A, v on the left from the line v = I but Is-solitons exist only under the line I. This means the larger the soliton mass, m cc 5, the less its velocity and as v—*0 we have —
A
(2.43)
i.e., the mass of a soliton at rest (and hence its amplitude) is limited from above. Having used (2.40) we get [62] S = 4A3~/(27)”~, which together with (2.43) gives 4(3~ \5\/~J
m ~ m ~s
=
(3V3
\ 10
\/~\~3/2 me!
There exist no stationary soliton-type solutions to the system (2.39) with a soliton mass exceeding ~ Thus, non-linear and dispersion terms in the low-frequency field equation have given a restriction on the mass (amplitude) of high-frequency solitons.* As we shall see below, an analogous restriction on the soliton mass arises in the framework of the Schrodinger equation with a so-called saturating non-linearity. 2.6. Schrodinger equation with a saturating non-linearity Let us consider here as an example the properties of the stationary solutions of the equation i 4D,
+ 4D~+
‘4D(1
—
exp( — E14D12)) = 0,
(2.44)
which gives the S3 equation when expanded in powers of , if 4D124 1/e to order ~(e’) in . Here, as before, = ~me/mi. Equation (2.44) is more correct than S3 if one examines the behaviour of sufficiently large amplitude packets of I-waves, 14D12 l/e near the stationary state. Such stationary solutions satisfy the equation
— *
(~ —
2}= 0.
— ~‘exp{—~fr
It should be noted here, that the system (2.39) strictly governs coupled Langmuir and Ion-sound solitons only in the region v—si.
(2.45)
V.G. Makhankov. Dynamics of classical solitons
It is convenient to change from A to ep(x, t) = ~/i(x)exp(i~2t).
_~2,
i.e.,
Upon integrating once one gets
c~=(2I)~2l(~2)+ The soliton boundary condition, ~/i—*0 as x—*±~, gives c point, where = 0, we have
1/2 and utilizing that at an extremum
t/,~
/i
I
2
2
2
T(~1m+eXp(~,&m)
Wm
~
Note, that the derivative I =
1
1~2
((2)~2(~2)+)
can vanish only if (2.46) and we come again to a condition like (2.43) which is needed for soliton solutions of eq. (2.45) to exist.* We shall not here discuss the properties of the solutions of the above equations in more details.** We only want to emphasize, that higher order in non-linearities (i) can essentially change the stationary solution for the high-frequency field as well as (ii) lead to an upper bound for the soliton amplitude (mass). This, despite of all the distinctions between system (2.39) and eq. (2.44), displays their common features. As we shall see below conditions of type (2.46) are intrinsic not only to two-dimensional problems but also to many-dimensional ones. We refer readers wanting to become more familiar with the results of studies of equations such as (2.44) to recently published original papers (see, e.g., [64—68,129]). Of most interest from the plasma theory point of view is to find the magnitude of the constant a in the inequality 2
JAJ=~
It turns out to be very susceptible to a change in the model. The transition from the system (2.39) to eq. (2.44) changes a from 3120 to unity; replacing in (2.39) the Bq equation by the IBq equation also changes the magnitude of a (see appendix). On the other hand, its magnitude determines the width i~x of the soliton and hence the characteristic wavenumbers (momenta) of the constituents of the soliton and, finally, the efficiency of linear Landau damping [60]. Various numerical experiments [691 including non-one-dimensional [70] ones show the production of an anomalous high-energy tail in the electron thermal distribution. This may mean that the width of I-wave packets is at least larger than several (5 to 6) times the Debye radius de and a <1. * Equations such as (2.44) or (2.45) were applied by many authors to examine plasma turbulence as well as self-focusing of laser beams ins medium with a saturating non-linearity [64—67]. ~ Not so long ago there has appeared a work [129]in which the authors investigate also solitons in the framework of a model with a low-frequency saturating instability but in a form which is more general than (2.39) and (2.44).
V.0. Makhankov, Dynamics of classical solitons
35
2.7. Soliton-like solutions of equations describing excitons in one-dimensional tnoleCular Crystals A series of papers has appeared recently in which it is shown that various excitations in solids can be treated on the basis of equations like the Schrodinger equation. This refers, in particular, to excitons occurring in one-dimensional molecular crystals [71,73, 74] and in a-spiral biomolecules [72]. In refs. [71,72] the description of exciton chains was reduced to an S3 equation. However, as was pointed out in [73,741, the Heitler—London approximation is usually used to obtain the S3 equation, which implies dropping from the initial Hamiltonian a small, but as we shall see below, very essential term.* The exciton spectrum was found both for a model with rigidly fixed molecules [73] and a model which took into account an exciton—phonon interaction [74]: w(k) = ((~+ 2~cos ka)2 + (2js’ sin ka)2)°2. Equations for the creation and annihilation operators were also obtained; it was shown in [741that the classical analogs of these equations have the form ‘P
~4Dt=
—
(X4D~— ~
4D14D12,
—
(2.47a)
and i4D~=
—
4D* + aço~+ 4D~+
(2.47b)
4Dj4DJ2
We see, that (2.47a) differs from the familiar S3 equation by the term ~ but usually a I (for a definition of a and scale on which ~, x and t change, see [75]). Let us consider some properties of soliton type solutions of the systems (2.47). Multiplying (2.47a) by ~ and subtracting from it (2.47b) times ~ we get the “current” conservation law m9.Ia *2 2 * * ~I4DI~—I~(4D —4D )+(q ~ (2.48) ‘~
2
This yields the integral of motion (exciton number) (2.49) where, as before, S =
f çJ~dx. However now the current j
consists of two parts which we denote by
= ~(a)+ ~
(2.50)
where j
•(a)
=
*2 —I. 2ia(p ), i(4D~4D 4D*4D~). 2
—~
~(r) =
.
(2.51)
(2.52)
—
Relation (2.52) is a well-known current which usually appeared in the Lagrange field theory (see subsection 1.2). The sense of such a splitting up will be clear, if, using the conventional procedure, one tries to derive the energy integral. After some algebra it is easy to see that
2 i~ (ko+1
— ~I~I~) =
fi
(~~I~2 — ~) +
i( 4D~4D~~q~~)+ ~I2j(~} —
—
‘P12 ~
* Moreover, in obtaining the S3 equation one changes from Pauli to Bose operators which is an incorrect procedure and which leads to an uncontrollable error.
36
V.0. Makhankov, Dynamics of classical solitons
which was a constant of motion representing the energy of excitons in the a—*0 limit, now is no longer conserved. Therefore S~ associated only with the energy of the excitons cannot be thought of as the total energy of the system. Remember, that the terms ap~ and a4D~describe a reversible resonance interaction between excitons and molecules which may lead to a broadening of the spectrum line. Proceeding in the same fashion one can obtain i~(4D*4D+ ‘P~’P)= 2a(~+ 4D*2) ~a{
a(4D*4D~+ 4D4D~)+ I’PxI2_~I4DI4
—
Finally, integrating (2.53) over x yields 2~i~ dx, dS~°~ = 2a dS~°> = l4DI
J
J
(tp~+
4D*2)
4D*4Dxx
—
4D~4D}.
dx,
(2.53b) (2.54)
where 5~~) = if (4D*4D~ ~ dx is the momentum of a packet in the a —*0 limit. That is why we have put a superscript (0) on S~and S 2. After changing to the conventional “polar” representation of a complex function —
—
pçlie we have dS~°~/dt = 4a
J
dS~”/dt= ~a —
{(.~/,2
J
cos 2i~ ~
~,282)
sin 28} dx,
—
8X~//’cos 28
dx.
(2.55)
Let us now consider qualitatively possible soliton solutions of the system (2.47). In the a ~ 1 limit, to zeroth order in a, we have for the soliton solution the expression (2.31) in which 2—~A2. fl. 1+~v Employing such a solution one can, in principle, obtain corrections of first, second, order in a (e.g., by the method of successive approximations). Let us, however, try to find some particular solutions of the exact system (2.47). To do this, let 8 = kx tot, ~ = x Vt; instead of (2.47) then we have —
—
+ (to
—
1
—
k2)~/i+
~3)2
—
a2k2~fr2= (a2
—
(v
—
2k)2)iJi~,
2 2 —— a2ki/iI/i~ a k ~/~‘ (v 2k, I/fE + (v 2k)tfr~(t/f~ + Ai ti + Ijf~)~ 2
\2
—
an
2
—
(2.56) 2
57)
—
1
This system has as a particular solution a packet at rest governed by the following equation —
(~O
+
‘P3
(2.58)
= ±a~,
where the upper sign relates to the right-hand side of space (x ~ 0) and the lower one to the x ~ 0 half-space. Due to the x x symmetry of the initial system and hence of (2.58) and because we have ~ in eq. (2.58), the derivative ~ should be continuous and equal to zero at the origin. Bearing this in mind we discuss the equation -+
‘P+~’P+’Pa’P+.
—
(2.59)
V.0. Makhankov, Dynamics of classical sol(tons
37
Fig. 2.24. g,,, g-phase plane for eq. (2.55). 9D
__ ~ Fig. 2.25. Potential in which the fictitious particle moves.
Fig. 2.26. Solution of eq. (2.55) with a = 0.
This equation governs the oscillation of a point particle moving in a non-linear potential 12
U~~4D
14
+~4D
(2.60)
under a frictional force a4D~while ~ is the “position” of the particle and x is the “time”. To put it in other words, one may define I
2
2
14
e(4D)~{q,~—4D +~4D}~T+u as the energy of the “particle” and rewrite (2.60) in the form = (~+~ —
‘P
+
4D~)4D~ =
—
a’P~.
(2.61)
Figure 2.24 taken from [19] shows the contour curves of constant in the ‘P04D~-plane,and fig. 2.25 shows the potential u(4D) as function of the particle position ~p.The points F1 and F2 are those at which the energy has its minima. The point S is a saddle point. The solutions of (2.60) depend on the “initial” values of ~ and hence on e at x = 0. Now let a = 0. Given ‘P(0) such that uI~0= 0, i.e., 4D(0) = V2,* ~. = 0 there is one half of an oscillation from point F. to S corresponding to half a soliton ~This value corresponds to the point F, in fig. 2.24.
V.0. Makhankov, Dynamics of classical solitons
38
Fig. 2.27. Soliton solution spectrum for the case of a non-vanishing dissipation, a 0.
(2.31) with v =0, i.e., A
(see the curve C1 in fig. 2.26). Given initial values 42(0) in the segment [1, V2] we have oscillations of the C2-curve type. Finally, when 4Do(O)> V2 we arrive at oscillations of the type shown by the curve C3. The “dissipation” term ap~changes the picture. There are now, apart from a non-oscillating (nodeless) solution, three families of oscillating solutions (which have one or more nodes). The nodeless solution (curve C1, a normal soliton) starts now from point ç,~higher than 42(0) = V2 and the greater a the higher the initial point will be. Solutions of the first family C~ approach ‘P = 1 as x ~, for the second family ç(co) = I ; lastly, the third family of solutions being of the soliton type tends to zero as x It is quite natural to join the nodeless solution and the third set into one family as they have a similar asymptotic behaviour and to count C’~the nodeless solution belonging to C~family. Then we may arrange solutions from C~in an ascending order ‘P~(x),n = 1,2,3,. , where n 1 stands for the number of nodes. This means that even with 4D(c~)= 0 we have apart from the convex solution of the kind (2.31), the whole family of oscillating solutions ç~(x).The nodeless solution of (2.59) is approximately = ‘Pmax = V~
—
—*
—* ~
..
—
exp{ — ~ax}AV2 sech(Ax). To end this section we note that although we are unable to solve eq. (2.56) in the general case, k 0, one may suppose that such a solution will possess features somewhat similar to those of the C~ family, but being in motion.
3. Solitons in particle physics
3.1. General remarks The quasi-particle properties of extended solutions mentioned earlier have for many years attracted the attention of research workers [76—78]. The intuitive idea of particles as localized field bunches are older and start from the work of Born and Infeld [781.However, only recently the interest in such particle-like solutions of non-linear field theories has grown (one might even say exploded) again with a new force. There are apparently several motives for this explosion of interest and one of them is the *
See curves C~.i = 1.2,3.4 in fig. 2.27.
V.G. Makhankov, Dynamics of classical solitons
39
accumulation a large body of mathematical information about one-space-dimensional classical solitons Step by step there have been appearing the requirements which classical solutions must meet as a basis for a construction of quantum field particle models. The theory is based on the variational principle, which assumes the existence of a Lagrangian and hence other properties of Lagrangian systems. One may consider various combinations of fields as well as a single field with self-action, especially in the one-space-dimensional case. In the three dimensional case rather strong restrictions arise due to the renormalizability requirement. As a result we can have in the Langrangian a not higher than 4th-order polynomial potential. We shall see that, therefore, stable static solitons may occur only in a system of two or more fields. One of them is usually connected with spontaneous symmetry breaking. The natural requirement such a theory should also meet is that there are no field singularities (see [19]and papers cited there). Finally, the theory must be relativistically invariant. To begin with, define a classical soliton as a solution of the equations of such a non-linear theory which (i) has a finite and non-zero rest mass, (ii) exists in a finite region of space, and (iii) is stable (i.e., lives for a sufficiently long time). A final requirement refers mainly to the one-space dimensional theory: we require that (iv) solitons may interact inelastically. The first three requirements coincide almost completely with those given by Lee in [49] (see also [10])except that our definition includes pulsating solitons as well. This definition differs radically from a more narrow one given by Zabusky to describe solitary wave solutions of the KdV equation interacting elastically with each other. Such a narrow definition is of no use in particle physics since, as was reasonably pointed out in [10], “such a highly restrictive definition would automatically exclude all the four-dimensional local field theories that are of interest to particle physicists”. The soliton solutions are sometimes treated as “soliton bags” [9] defining “a potential or a self-field preventing a certain particle or constituent field quanta to tend to its asymptotic free field in remote past or future or as the coupling constant goes to zero”. And although those “bags” have been used to construct dynamical particle models the authors, probably motivated by the success of Gardner, Greene, Kruskal, Miura and Lax’s inverse method associated their theory, in fact, only with completely integrable equations. It is in this sense in which their approachdiffers from the one developed by Finkelstein, Lee, Vinciarelli, Bardeen et al. [122—123],Dashen et al. [83], and many others [80]. .~
3.2. Solitons in a (p4 field theory We have briefly discussed in subsection 1.3 soliton solutions of the Klein—Gordon equation with a cubic nonlinearity (KG3) and seen that they had certain particle-like properties. Here, to display characteristic features of such solutions we consider some one and two-field theories in two (one space and one temporal) dimensions. We restrict ourselves to non-linearities not higher than cubic in the equation (quartic in the Lagrangian) the so-called 424 field theory. 1. Let us begin with Landau—Ginzburg—Higgs (LGH) equation for a real (i.e., neutral) field (3.1) *
Another one is the way in which these solitons may be quantized as well as work on the theory of “topological” solitons (see the review paper by
Rajaraman [80]).
40
V.0. Makhankov, Dynamics of classical solitons
Fig. 3.1. Kink-soliton of the LGH equation. ~ is the energy distribution density in the kink.
its two-parameter soliton solution for sublight (v x(x, t)
m =
—
Im’y tanh~ç,~ (x
—
ut
—
‘1 x0)1,
< C)
constituents,* (3.2)
is of the form of a step (“kink”). This implies different boundary conditions at two infinities x = ± x, = ± m/g, and distinguishes the kink from a bell-shape-soliton of the KG3 equation. It is now convenient to rescale the variables as follows: ~ = (g/m)~,I = mx, t = mt; the results are then invariant with respect to the parameters m and g. The energy density for (3.1) is I
2
2
1
2
2
—1)}. (3.3) defined by (3.2) and (3.3) are plotted in fig. 3.1, where one can see that the kink-soliton energy, as for bell-soliton, is concentrated finite 2 = M2in+aP2 alsoregion holds:of space. It can be easily verified that for the soliton (3.2) the Einstein formula E ~C”~{x1+x~+s(x The functions x and
M=q-~-,
~
EMy,
P=Mvy.
The potential of the LGH system has the form depicted in fig. 3.2a, the corresponding one for the KG3 equation in fig. 3.2b.t We can also see that the kink starts in one of the two minima, goes over the barrier and finishes in the other minimum (track of a traveller passing over a hill). A bell-type soliton starts and finishes at the same level of the relative minimum (the track of a traveller climbing from a valley to the top of a hill seeing on the other side a bottomless pit and hurrying back). These characteristic features of kink and bell solitons determine as we shall see below, their stability. 2. Systems of interacting fields are sometimes considered which have soliton-type solutions. Note here refs. [9, 10, 122, 123] where relativistic equations were studied and [15,51, 60] devoted to non-relativistic systems. As we have seen above from the example of Langmuir solitons a system of interacting fields gives a considerably richer world of phenomena associated with the formation and interaction of solitons. Moreover, this occurs also for equations which are completely integrable in the self-action approximation.** *
~~15 Interesting to note, that constituents of this field have imaginary initial masses that seemed at first paradoxical and repelled theoreticians.
However, after the appearance of papers which initiated theories with spontaneous symmetry breaking, the interest in such equations started to grow (see refs. [130]). t Note, that the potential function a being introduced in subsection 2.7 which governs the motion of a “particle” differs from a real potential V of the system by its sign, i.e., a = — V. **The most striking instance is the interaction of s-solitons because in the self-action approximation both the S3 and Bq equations are complete integrable.
V.G. Makhankov. Dynamics of classical solitons
(a)
41
(b)
Fig. 3.2. Potential energy density in (a) a kink as a function of the field
x: (b) the hell-soliton
as a function of the field versus
~.
Consider as an example soliton solutions of systems with the Lagrangians ~/.3 ~
~SK’P
I
~
I
2’\
2
2
22
x)~
(3.4)
and 2
-~KKH’P1I
—l’P~I —m 2
2
2
421
2
+g
I IcoI X+~(X 1X~/.L x). 2
2
2
22
(3.5)
In the first case we have interacting complex Schrodinger and real Klein—Gordon fields: (3.6a) 2
2
2
LI~+~t~—g~4Dj
The
=0.
Lagrangian gives rise to interacting real and complex Klein—Gordon fields 2ço g2çox = 0,
~KK
—
U4D
+
m
(3.6b) (3.7.a) .(3.7b)
Following [9] we use a parametrization to find solutions to the systems (3.6) and (3.7) ‘P(x, t) =
~I~(h)
exp{iA(~)},
~(x, t) = I/i(f),
(3.8a) (3.8b)
where h=k v
=
f=
1x—w1t, k2x w2t, k3x — to3t, —
with k, and oi1 (i = 1,2,3) being constants. Upon substituting (3.8) and (3.9) into (3.6) one arrives at 2~~I/i = 0, ~ ±2A’) A~I+ g
~
(3.9a) (3.9b) (3.9c)
(3.lOa)
42
V.0. Makhankoi.’, Dynamics of classical solitons
(3.IOb) (3.l0c) Then as in the case of the Schrödinger equation (see subsection 1.3) we search for a particular solution of (3.10) assuming the function exp{iA(v)} to be a plane wave, viz., A’(v) = const.
(3.11)
We then have from (3.lob) (3.12) or 4’ = 0 which is of no interest. Relation (3.12) yields inw~
/to
1
to1w2
A =-~--1--v=ml-i--x—-~—i-—t ~i~2
\“i
t(1K2
which means that k0~p
=
rn(w~/k~), ~
=
rn(w1w2/k1k2).
(3.13)
Using (3.12). (3.lOa), and (3.l0c) one obtains (3.14a) (3. 14b)
2~2= 0. (k~ w~)I/i” In the case where —
—
+
~
g
to
± k3,
3
we obtain the exact solutions to (3.14) putting (3.15)
which gives f = h, or k1 = k3 and to1 = to3. Then eqs. (3.l4a) and (3.l4b) coincide if a compatibility condition is fulfilled leading to the following dispersion relation for the constituents: p
2
2m
2
2
~/2m _p 2/,LK, 1—p/rn 2m
22
2
and 2
12
K
where
2/rn2. —p Note, that in subsection 2.4 we got a formula analogous to (3.16) for a massless second field. -2
~
p2
A2
2
and the non-linearity played effectively the role of mass with ~
=
A2/2.
V.0. Makhankov, Dynamics of classical solitons
43
The solutions of the system (3.14), ‘p= ±~~_?~sech2{~(x — Vt — xo)} exp{i(px
=
—
(3.17a)
t)},
sech2{~(x Vt — xo)J~~
~
(3.17b)
—
are singular with respect to the coupling constant g and can be interpreted as follows. The x field defined by (3.l7b) is a self-consistent potential governing the solutions of the Schrodinger equation.* Its solutions (3.17a) describe a finite (extended) non-relativistic particle consisting of constituents with a binding energy, =
~2/2m,
—
which is determined by the mass ~zof a ,y-”quantum” and may be found from formula (3.16) putting V = p/rn = 0. This formula displays also the inconsistency of the system (3.6), since one of the interacting fields is relativistic but the other one is not. This fact is more clearly seen from the difference between the soliton velocity v and the group velocity, de Vg=•~[~=
/
2
2
V1 1j—----flf
The validity condition for the above system is therefore ~ m, that is, p ~ m, V Vg. The parametrization (3.8), (3.9) and the relation (3.11) applied to the more consistent (in the above 2 = (p2/2)< 1, mentioned sense) system (3.7) gives for V ‘~
‘P(X 2t)
= ±~T
x(x, t)
=
sech2{~(x —
j~rsech2{~(x
—
Vt
Vt
—
—
x0)} exp{i(px
—
t)},
xo)},
(3.l8a) (3.18b)
and the dispersion formula 2
=p +m —p~ 2
2
2
hence — Vg~
2<
The
V
2
__p_
p
I V condition leads now to 2
i.e., the mass of a “quantum”-carrier of the interactions is less than the mass of a “charged” 42-field “quantum”. The binding energy of the latter in a soliton is defined by the formula 2 — 2_ m. =
\/m
An analogous consideration applies for constituents with imaginary masses; we then shall get kink-solitons instead of bell-ones. *
“Quanta” of the x field can be interpreted as carriers of interactions between the “quanta” of the complex g field.
44
V.0. Makhankov, Dynamics of classical solitons
As an example of systems consisting of a complex 4D-field (with the zero boundary conditions at both infinities) interacting with a scalar k-field with a degenerate vacuum, we give the models suggested in [10] and [79]. The first one is described by (1.14) or in the plane geometry (broken symmetry) by 2X2’P0, 319 ‘P11”’P~~+a
x
2xI’Pl2+~x(x2—1)0;
(
1~—x~~+2a the second one by
LIi4D+(m2—1+X2)4D+2423=O, Ux + I + 2142 I2)x + =o. (—
320
~
-
One can easy verify that both systems have as exact particular solutions the following solitons x =
tanh(Kx),
(3.21)
‘P
V2 A exp{
—
i(tot
—
8)} sech(Kx),
where A = (1 4K2)~2/2K, K2 = a2 to2 in the first case, and A = (1 2K2)~2,K2 = m2 to2 in the second one. The solutions (3.21) exist, if to2 = K2 = a2/2 for the (3.19) model, and if max(0, m2 ~)~ to2 ~ rn2 for the model (3.20). This means, that (3.21) is a solution of (3.19) only in a single point to2 = a212 from a total region of the existence of soliton-type solutions 0~to2~a2,and thus the amplitude of this solution is fixed. In the second case we have a range of variation of the frequency to and hence of the amplitude A. This means that such solutions, like those in the case of the complex KG3 equation, have an internal degree of freedom (U(1) symmetry of the 4D-field). The mass of a soliton of eqs. (3.20) is 42 2K )/K), (3.22a) M =charge 2(~K is+ rn(1 and the —
—
—
—
—
2
—
Q = 2to 1—2K2
(3.22b)
This soliton may be considered as a “bag” of the x-field in which charged “mesons” of 42-field are locked in (see analogous models in [79,85, 122, 123]).* 3.3. Properties of plane solitons due to their charge We have early defined the charge of a soliton Q=iJ(’P*’Pt_ 4D~’P)dx as the constant of motion associated with the U(1) symmetry (the global rotation in isospace) called sometimes the internal symmetry [79]. It means that in the Q 0 case we have an added internal degree of freedom as compared to the neutral scalar field and as we see later it effects the stability of solitons. *
Using the Lorentz-invariance of the system (3.20) one can easily get solutions of the type (3.21) for moving solitons.
V.0. Makhankov, Dynamics of classical solitons
45
Following refs. [79,84] we illustrate the main properties of solitons due to their charge. Consider a model with the Lagrangian — a ~p*a(p V(I42l2), where we assume that V~0and V-+rn214212 as =
—
I4Dl—~0.
The equation of motion then is 142
+
=
0,
(3.23)
and the energy of a soliton
E
f
~ = I~I~ + Ico~l2+ V(1
~ dx,
2).
(3.24)
421
1) Varying E with respect to
~,
but keeping ~ and Q fixed we find that
ç(x, 1) = ~I~(x, t) exp{ — i8(x, t)} is the lowest energy solution if 1
8~(x,t) 1(x, 1)
4D(x, t) =
~ç~= 1(x) exp{
—
i(wt
—
=
=
const, and 8~= 0. Therefore
0,
8~)},
(3.25)
and r=~+to2~2+
Q= 1
(3.26)
dx,
12
+
V(q2),
to2~I 21 —
(3.27)
~
V(~42)= 0.
If one regards, using (3.27),
to
(3.28)
as a function of Q and a functional of c~,then eq. (3.28) is easily verified
to be a consequence of the variational principle (ôE/ô~t1)IQ= 0,
(3.29)
from which the relation* dE(Q)IdQ
=
to
(3.30)
follows. 2) The solution of the form (3.25) has a finite energy only if the 4-field tends to zero at both infinities and if to2 < m2. The first statement follows from V ~ 0 and formula (3.26). The second one is obtained by employing (3.28), the first condition, and the fact that V = m21(p12 as ~ 0. Then for large xl =
*
(m2
—
w2)~t
Note, that (3.28) as well as (3.30) may be obtained by varying the functional F = E — wQ with w being a Lagrangian multiplier (for details see [10]
and below).
46
V.0. Makhankov, Dynamics of classical solitons
so that the function exp{— KlXl}
= C
vanishes as lxl—*~,only if to2< rn2. 3) The energy of a soliton is always less than the energy of a plane wave packet with the same charge Q. We can enclose the system in a linear box of length l—*cx. Then, from (3.27) and (3.28), one can find, neglecting terms O(l_3/2), the plane wave solution —‘
(~~) 1/2
‘Ppi
exp{i(kx
=
—
wt)},
to2 =
k2+ in2,
with an energy E
0~= toQ which is a minimum for k
E~ ~ Emin =
=
0, and
Qrn.
2< rn2 (in fact, since both the energy and the Lagrangian include only conjugate From the condition to pairs, ~ and 42*, one may simply assume to ~ 0) and relation (3.30) we get
dE/dQ
= to <
rn.
Consequently, for any Q 0, E(Q)
=
J
to
dQ < Qm.
(3.3 ~)
The following four properties of charged solitons will be formulated here without proof. We shaH discuss them later in more detail. 4) The value of to is in the three-dimensional case always greater than a certain tomjfl which follows from a virial theorem. 5) Though possible values of to lie in the range tomin < to < rn, the charge Q has only a lower bound. As an example, see formula (3.22b) which gives Q ~ as K —*0. 6) In the K—*O case (w—*rn —0) the solutions of eq. (3.28) tend to those of the KG3 equation for which —‘*
E =
Qmfl _~(Qg2)2± O(Qg2)4},
V
m 2~2(l 2g2~2+
Qg2~1,
if —*
—
as —*0. In this sense the frequency range to —~rn 0 may be considered as the one near “free” field-plane waves in a large box. It should, however, be stressed that soliton solutions differ principally from free field ones by virtue of the third property (formula (3.31)). Note also, that in the K—*O limit the size of a soliton increases without bounds. 7) There exists a conserved, so-called topological charge in the theories with a degenerated vacuum (see, e.g., the kink-soliton of the LGH equation) —
Q t0~,=
x(x
= +
~) ~(x —
=
—
cc).
V.G. Makhankov, Dynamics of classical solitons
47
3.4. Bound states of solitons 1. The bound states of solitons have been examined at length in the sine-Gordon theory [80—83]
xt
2 sin
—
x~+ m
x =0.
An exact doublet-soliton solution has been obtained:*
x
=
4 arctan{tan a sech z’ sin(z” + 8)},
(3.32a)
where m’yV’l A2(x = myA(Vx t),
=
—
—
Vt),
(3.32b)
(3.32c)
‘—
2
arcsln[m 2(1 A )], (3.32d) and A = /my (see (1.40) in subsection 1.2). The solutions such as (3.32), representing a kink-antikink bound state, have been given various names; for instance, “mesons” (Perring and Skyrme [77]). “breathers” (Ablowitz et al. [81])and “bions” (Caudrey et al. [81]),and many other ones (see 180,811). The mass of such a bound state which, following Caudrey Ct al. [81], we call a “bion”, varies from zero up to two soliton masses, viz., a =
—
0~Mb~2MS~ l6m,
(3.33)
depending on the internal state. Note that due to the complete integrability of the sine-Gordon equation “bions” can not appear as a result of a collision of solitons (a more detailed discussion of sine-Gordon bions for the interested reader can be found in ref. [80]). 2. We shall now consider bions (hereafter we use bion without quotes) for the Higgs field equation (3.1) which is not completely integrable. This last statement follows, for instance, from the fact that the collision of solitons in the framework of eq. (3.1) can give rise to the creation of bions [26]. Examine the evolution of the initial state
x
1~(x,0)
=
tanh
~3’= (x —
Vt
—
x0)
—
tanh
(x
+
Vt + x0)
—
1,
(3.34)
which is a state consisting of a kink and antikink moving towards each other (see fig. 3.3). If the distance between the solitons is large, so that V2x0~1/y,
(3.35)
then the function (3.34) is a solution of eq. (3.1) with exponential accuracy. The difference
2)x 2~, m 1~ + g can be easily obtained. In the centre-of-mass system =
F(~)~ (Li
I
x+vt+x
3Lt~~
~,,_
=
+tanh *
—
0 —tanh x—vt—x011/ x+vt+x0\/ x—Vt—x0\ j 1yanh ~ )I~,,tanh ~ ~,,_
x+Vt+x0 — —tanh x—Vt—x0 — —l V2 V2
This is probably of immediate concern for the complete integrability of the sine-Gordon equation.
(3.36)
48
V.0. Makhankov. Dynamics of classical solitons
///~/ ~
/
~- I
\~ ~
__
/
__
Fig. 3.4. Bion field function x in the frame-work of the LGH equation at various instants of time.
Fig. 3.3. Initial state for the interaction of kink and antikink solitons.
Since the exact solution of the above initial problem has not so far been obtained, some steps have being undertaken to study this problem by computer by Kudryavtzev [26] and by Bogolubsky, Getmanov and the author. We place the solitons at rest at relatively small distance apart, x0=1/V~ they attract each other and an oscillating state similar to a sine-Gordon bion arises. In fig. 3.4 the function x(x, t) is shown for various times and fig. 3.5 depicts X(x, t) as function of t at the origin. The system is seen to achieve a quasistationary oscillating state in quite a short time, so that the period of oscillations, T = 5\[2/m, does not alter practically for a time of the order of hundreds reversed masses. This means, that radiation at infinity due to these oscillations is very small (some calculations give the energy decrease per period to be 0.5%). An analogous state has been obtained as a result of the head-on collision of two (kink—antikink) solitons moving with equal velocities, v = 0.1, [26]. In order to clarify this phenomenon let us follow ref. [26] and consider how the potential energy of a two-soliton system such as (3.38), +2 ~(x~ 1)2] dx, (3.37) E~=
J
—
~ [(a~x1~)
depends on the distance between the solitons I
2 describing the kinetic 0. Note that the term (öxXjn) energy of an oscillating motion of constituents in the soliton (i.e., a motion of the “quanta” in the =
2x
______
Fig. 3.5. Plot of function function of time,
x(0~t)
in the centre of the bion as a
Fig. 3.6. Potential energy of a kink—antikink system as function of the distance between the solitons.
V.0. Makhankov, Dynamics of classical solitons
49
0
x Fig. 3.7. Field function for solitons passing through one another. It is not even approximately a solution of the Higgs equation.
well), is a fraction of the potential energy of the whole soliton since E~=
EI~
0
(Ek + EP)JVO.
An approximate plot of the function E~(l)is depicted in fig. 3.6. It can be readily understood as follows: when 1 ~ 1 the potential energy is a constant and equal to the sum of the soliton masses:
~
=
/ 4v2m
3
—~-—--—~-
=
2M5
(3.38)
when / —*0 (overlapping solitons) E~tends to zero with I. After solitons begin to diverge E~grows with jl~remaining positive. This becomes obvious if one pays attention to the fact that the function x~is no longer a solution of eq. (3.1) after the solitons have passed through each other, and the longer the distance between them the greater the value of 5~*Therefore from figs. 3.6 and 3.7 it follows that neglecting radiation solitons repel one another and will diverge. Thus in the absence of radiation two types of motion may occur: an infinite one if the energy exceeds the sum of the two soliton masses and a finite motion in a well for smaller energies. Radiation changes the picture. The creation of a bion may take place (and the motion becomes a finite instead of an infinite one), if an excess energy equal to or larger than L~E= E(x!fl)
—
2M~
(3.39)
can be emitted during the period of one oscillation. The radiation power has been evaluated in ref. [26].It tends to zero with ‘rad
tanh
ç~=(i
—
J/J as
tanh 1)
(3.40)
It is this which ensures us that such a bound state is quasi-stable and that the bion is long-lived. The above estimate of the radiation power (0.5% for a period) gives the following lower bound for the bion life-time 3/m. Tb~200T= \/2 l0 The upper bound for the velocities of solitons which can collide to produce a bion state is thus Vlhresh =
*
0.12,
The distance between solitons which have passed through one another is given a negative sign.
50
V.G. Makhankov, Dynamics of classical solitons
in accordance with the value that has been independently obtained in computer experiments carried out at Dubna. 3. Shirkov has recently suggested and examined the whole class of equations of the LGH type with polynomial potentials in a Lagrangian of higher degree than He has noticed that for the potential ~.
U(~,n)=~—fl (1-g~)~/fi lwiwn
(3.41)
(l+y~)~
1wjw~n
one can obtain a quadrature solution (see subsection 2.6):
f
d42
=x—x0. (3.42) J \/2U(ço)+c~ There are 2n + I absolute minima for such a potential (stable equilibrium position; see fig. 3.8) symmetric around the central one ~ = 0: 1, k1,2,...,n. ‘Po=0, ‘P~k=±g~ The integral (3.42) is for c 1 = 0 reduced to 2x,
2
1”k
‘P c e± fl (1 —g~(p2)~ =
The potential
(1—g~u)fJ(1+y1u) 11(1 —g~u)
(3.43) u=g2
-
U( 42,
1) yields
~~2~±2x
2~2~ I four solitons (see fig. 3.9) with equal masses:
which describes for g
M=J(~
42~+U)dx=2JU(42)dx= JV2Ud42.
In the case m
=
0, 2X)h/2~
M =
(344)
~
(I + e± We have then four pairs of solitons at n = 2, m = 0. Two of them “tie” the central vacuum ~ with the first one p~g~ = I the two other ones the first vacuum with the second one p~g~ = 1. In the particular =
—
Fig. 3.8. Potential having n + I stable equilibria.
V.0. Makhankov, Dynamics of classical solitons
SI
P -~
—--
~-=-------—----
Fig. 3.9. Four solitons for the potential U(~,I).
case g1
=
‘~
Fig. 3.10. Eight solitons for the potential U(~,2).
\/~g2 g the masses of these solitons are, respectively,
2. 5/24g and their solutions
M1
~/
=
g~p~ = I
±(1 +
M
2,
2
=
l/24g
2 e~)112.
(3.45)
The general structure of such solutions is depicted in fig. 3.10. These results may be qualitatively generalized as follows: the potential U(
42, n) defines n pairs of four solitons. A soliton of the kth foursome “ties” the (k — 1)th vacuum with the kth one and has a mass Mk
gf
\/2U(42)dço.
k-~I
This class of models is apparently intermediate between the 424 and the sine-Gordon ones. In the n cc limit the potential assumes a periodic structure and solitons become identical. Moreover, one may assume that for U(42, n) there exist a finite set of bound bion-like solutions which is transformed into a continuous spectrum of the (3.33) type in the n —*cc limit. This assumption was verified by computer by Katyshev and the author and we found: (i) that collisions of solitons in the U(42, 1) and U(42, 2) models are inelastic, and (ii) that bions of the type given in fig. 3.4 may arise as in the Higgs model. The inelasticity of the soliton interaction and hence the radiative emission is very small so that the collision of solitons with different masses are like those of billiard balls. An analogous investigation was carried out (by computer by Getmanov) to show the possibility of the creation of a bound two-soliton state in the frame-work of the KG3 equation. Such states were, indeed, found [27]. This fact is no surprise, if one takes into account that we had a picture of non-linear oscillations of an f-wave packet in the non-relativistic limit, i.e., the S3 equation. That picture resembles the bion state (see fig. 2.9). However, due to the complete integrability of the S3 equation the creation of bions as a result of soliton collisions was forbidden. Including the second field ~ led to the dissipation of the non-linear oscillation energy and to the appearance of bound soliton-states becoming in the limit a large mass soliton. In the present case the weak radiation is possibly associated with the quasi-decay type of the small-oscillation spectrum, 2 = k2 + m2, —*
to
since the mass term may be compensated by a non-linearity.
V.0. Makhankov. Dynamics of classical so!itons
52
3.5. Approximate analytic description of bound soliton-states in non-integrable systems Here we discuss a method of approximately solving the equations of the 424 field theory which can be developed since these equations are “near” the integrable sine-Gordon equation. The basic idea of this method has been prompted by computer experiments on (i) the Fermi—Pasta—Ulam problem, (ii) the scattering of solitons in the Higgs, Klein—Gordon, Boussinesq and Korteweg—de Vries models and has been surmised from time to time by various authors. Its outline is the following: the behaviour of a system may be defined by its “vicinity” to a completely integrable model. Such a “vicinity” might be verified, for instance, by means of the scattering of solitons. The smaller the coefficient of inelasticity of the interaction of solitons in a given model, the nearer this model is to the corresponding completely integrable analogue and with the higher accuracy many-solutions of the latter approximate solutions of the former one. This is especially related to the bound soliton-states. This idea was used to find approximate many-soliton solutions in the framework of the ~ field theory in ref. [27].In an earlier paper [82] an analytical approach was developed to investigate non-linear self-localized oscillations of the KG3 equation on the basis of an asymptotic method. However, that technique was valid only if the amplitudes of the oscillations were sufficiently small (in the region near “free” field solution when K 0).* Following [271we consider (3.46) L1~+m2sin~0, -+
with the one-soliton solution
x =4arctanet~,
~
—
rny(x— Vt)+6,
(3.47)
and the equations Li~+ m242(1
0,
(KG3)
(3.48a)
1142—2m242(1—422)0,
(LGH)
(3.48b)
_2422) =
(the coupling constant has been removed by a i~calingg2422—~~ Solitons of eqs. (3.48) are, respectively, = ±sech
z,
(3.49a)
±tanh
z. The transformations
(3.49b)
= ±sin ~,
(3.50a) (3.50b)
‘PH =
42k
‘PH =
±cos
~hange the one-soliton solution (3.47) of eq. (3.46) into the solutions (3.49) of eqs. (3.48). The main assumption is that the transformations (3.50) applied to exact multi-soliton solutions of (3.46) give approximate solutions of corresponding equations (3.48), apart from small inelastic effects. This hypothesis has been verified by computer simulations. We give a few examples. Collisions and bound states of a soliton—antisoliton configuration in the barycentric system for the sine-Gordon model are x=4ar *
11 sinh myVt an~— [V cosh m’yx
.
A similar work has been done by Dashen, Hasslacher and Neven 1831 but for the Higgs field equation.
(3.51)
V.G. Makhankov. Dynamics of classical solitons
53
Using (3.50b) one can get the relation — ~PH—
sinh2(m’yvt) v2 cosh2(myx) sinh2(myVl) + V2 cosh2(rnyx)’
(3 52)
—
which describes an elastic collision of the kink-antikink solitons of (3.49b) quite satisfactorily. As t—*oo, we have ‘PH
—~
±{tanh[my(x vt)+ 8]— tanh[my(x —
+ Vt) —8] ±I},
6
=
In v.
The relation (3.52) pictures well a situation taking place in a numerical experiment by Kudryavtsev for V> V(hresh = 0.2. When, on the contrary V < VthreSh, an energy greater than a sum of the kinetic energies of the colliding solitons is carried off by the radiation and as we have seen a bion is formed. The bion of the sine-Gordon equation may be described by the formula (3.51) if one puts v—*iu:
x
=
4 arctan{i~(x,t, a)},
(3.53)
where ~(x, t, a) = tan a sin(mt cos a tana=l!u,
—
6) sech(mx sin a)
0~aE~ir.
Application of (3.50b) gives the formula (3.54)
which describes the bion of the Higgs equation, if the parameter a is taken as a slowly decreasing function of time. A quite similar operation may be applied to (3.53) by means of (3.50a) to obtain a KG3 equation bion: 2~
(3.S~
42k~2+l.
Such a bion has been found in computer experiments carried out at Dubna. This state turns out to be stable for sin a ~ 0.2. The oscillations (3.55) are followed by a slight radiation, the intensity of which decreases with amplitude. As the result of such radiation large amplitude (heavy) bions subject to eqs. (3.54)~and(3.55) transform into small oscillations around the vacuum fixed by the boundary conditions (~ = 1 for (3.54) and q~= 0 for (3.55)). The spectrum of the “free” small oscillations in the Higgs model is of the form toi~,. = k2 + 4m2. The relation (3.54) shows that the bion frequency lies in a forbidden region of the small oscillation spectrum, viz., tob < and tends to the lower bound of spectrum as a 0. Thus even a very light bion can not decay into free oscillations. Light bions of the KG3 equation have been studied in works by Kosevich and Kovalyev [82] and of the Higgs equation in that by Dashen, Hasslacher and Neveu 183]. But the series obtained by the above authors are asymptotic and hence their results have rather denoted the possibility of the existence of such oscillating solutions. Numerical experiments [26—27]have confirmed the reality of such states. —
0lj~i
—~
54
V.G. Makhankov, Dynamics of classical solitons
3.6. Quantization of solitons From the particle physics point of view the greatest interest is to develop the classical level of a description of particle-like solutions into the quantum one. This problem has now its history. One of the earliest work in this direction was done by N.N. Bogolubov [88]. But it had no followers at that time probably because the classical theory of solitons was at a very initial stage. At the present time a fast flow of papers devoted to these and associated problems has sprung up. The main results of these researches up to the begin of 1975 have been summarized in a good review by Rajaraman [80]printed in this journal. Furthermost development of the theory was reflected in the proceedings of the Conference on “Extended systems in field theory”, Paris, summer 1975 [142], as well as in the rapporteur talk by Faddeev at the Eighteenth International Conference on High Energy Physics, Tbilisi, July 1976 (see also his review paper “Searching for many-dimensional solitons” in the Proceedings of the Fourth International Symposium on non-local, non-linear and non-renormalizable field theories [140]. Therefore without giving technical details we summarize here only a few results (undoubtedly well-known to specialists), as the subject of the present paper is the classical theory of solitons. To begin with, let us note that quantum solitons are descendents of classical ones and in a weak coupling approximation, g2 ~ I, quantum soliton (QS) properties are with good accuracy described on a classical level. The masses and the form-factors (dimension) of QS approach their corresponding classical values and remain finite, as g —*0. This is radically different from conventional quantum objects (bound states), i.e., atoms and molecules which occur only at the quantum level. First some general remarks [49]: 1. It is convenient to construct a theory of QS if all non-linear couplings would be defined by a single constant g2. In that case, for example, let (3~42)~—~ V(g~),
~=
where V has a minimum at ~= ~
0, so that
V(g42)=~m2422+O(g(p3)+O(g2424)+~,
(here ~ may stand for a system of interacting fields). In the g —*0 limit we have the linear equation L1(p2 + m2p = 0 with solutions which are the usual plane waves (free field). 2. The classical soliton solution, as we have seen above, is singular in g therefore we may define ‘Pci =
(pig.
Then
and ,.1f~=
—
~(a,~
— 2 V( 42) 42) will not depend on g and neither will the soliton solution
~,
as long as
h.
In fact, for
g 0. (This fact was repeatedly pointed out earlier.) 3. The quantum action Sq =
Sci
is proportional to ~ or
~
2 plays a role similar to that of
One may expect that g
V.0. Makhankov, Dynamics of classical solitons
55
small g the quantum soliton solution may he written as a series in g2 like that in h. Moreover, the leading term of this series corresponds to the WKB approximation and tends to the classical limit as
g—*0. 4. A method of an expansion of quantum field operators near the classical soliton solution [10, 18, 80] has been applied to quantize stationary solitons and “charged” Q-solitons reducing to them:
{-~
ç(x)
=
+
~ q~(t)~tn(x)} e’~’.
Two types of divergences, the ultraviolet and the infrared ones, naturally appear when calculating the energy of QS. The first one may be treated with a correct procedure of introducing the counter terms when we subtract the vacuum energy [80,85, 86], which is apparently possible for renormalizable theories. The latter being associated with the translational modes may be removed by means of either certain artificial manoeuvres [114] or by the method of collective coordinates [10,84] (see [80, 131] as well). In this way the mass of a kink-solition in the Higgs model with allowance for quantum corrections associated with zero-point oscillations is M_2\1’2m
3m
3g2
—
+
m
V2~r 2\/6~
The first term in the g —*0 limit is the energy of the classical soliton. Taking the translation mode into account gives rise to the correct value of the soliton energy [10,80]:
E~ VM2+P~,
P,, =~2rrn/L,
where P~is the soliton momentum, L is the length of the box. An analogous formula for a kink-soliton in the sine-Gordon model,
is
8m3 m8m M—---7-—-——-—--, g ir y
—
y—
g2/m2 2
I—g/8irrn
The quantization2 ofwith a Q-soliton in a weak coupling theory leads to theofsoliton masscharge to coincide its classical one. However, the eigenvalues the total Q as ina the lowest order in g generalized momentum conjugate to the angle variable i~ (Q = i~8/M) are all integers: 0, ± 1, ±2, ±3 Applying, e.g., the Bohr—Sommerfeld quantization rules gives in fact 5.
E~ E~ —
1
dE/dn
=
whence, using the relation (dE/dQ) =
f dE n=j Eo
to,
we get
Q(E,)
F,,
—~=
I j
dQ=Q(E~)—Q(E0),
Q(Eo)
which proves the suggestion. 6. The mass of a quantum bion in the sine-Gordon model may be obtained with the help of the
56
V.0. Makhankov, Dynamics of classical solitons
WKB method (for arbitrary g2) [80]: Mb(N)=~~~sin(L~—7),N1,2,...<8ir/y. 16 The continuous spectrum of a classical bion becomes a discrete quantum one for 8ir/’y> 1. The quantum features of a bion are very remarkable. In its lowest state, N = 1, the bion corresponds for small values of ‘y/i6 to a free constituent and in the case N > 1 it represents a bound state of N constituents. A more detailed description is beyond the scope of our review so the reader is referred to the paper by Rajaraman mentioned above [80] and to works by Faddeev and coauthors [81]. 7. The formfactors of QS may be obtained on the basis of the classical solution using Fouriertransforms: (p!
0~(O~ 0)Iq) =
where
4.
p
—
~—
J
exp{—i(p
—
q)x}~(x)dx,
q is the momentum transfer
Stability of one-dimensional solitons
4.1. Some general remarks One of the most important and natural requirements for solitons both in plasma theory and in various field models is the condition of stability. We have already mentioned that the soliton must have a sufficiently long lifetime from the point of view of the processes under consideration. In other words, the soliton lifetime must be well above a characteristic soliton interaction time. In some papers the more complex requirement of absolute stability of the classical soliton is put forward; moreover the requirement lies at the basis of the definition [49]of a soliton as “a classical solution of a non-linear local equation which is non-dispersive in space at all times” and, generally speaking, has no singularities [19]. In the case of initial perturbations having the soliton symmetry we call the soliton stability a “longitudinal” one. In the opposite case, when the trial perturbations have a smaller symmetry we have a “transverse” stability. For plane solitons the latter imply a study of the stability with respect to soliton perturbations which are perpendicular to its direction of motion. The first investigations of the existence and number of particle-like solutions were mainly related to the longitudinal stability. Thus, in refs. [76] soliton-like solutions are considered of the equations 42”
=
42(1
—
421X)
with the boundary conditions 42(0) = ‘P(co) = 0. For n> I the absence of Liapunov stability of all possible solutions to this problem was demonstrated. Further work in this direction was devoted to the stability of plane solitons of the sine-Gordon and KdV equations. This problem is discussed in detail in the review paper by Scott et al. [5]. The paper [891 is related to the Klein—Gordon equation. Since the publication of this paper a considerable amount of work on the longitudinal and transverse stability of different types of solitons has appeared. We note merely papers on the transverse stability of solitons of the KdV [90]and Schrodinger [91,92] equations. These papers use both analytical and computational methods. Finally, we must emphasize that one cannot consider the soliton stability problem as a completed (historical) one. Papers on this topic appear with a rising frequency. Therefore our review does not
V.0. Makhankov, Dynamics of classical solitons
57
claim a completeness, but is rather an attempt to demonstrate some methods which are efficient in these problems. It is natural that as before we do not describe the results of the review by Scott et al.
[51. Some stability problems of more-dimensional solitons are discussed in section
5.
4.2. The longitudinal stability of plane so/itons The purpose of this section is to remind the reader the method used for the relativistically invariant Klein—Gordon equation with a cubic non-linearity and the Higgs field equation as examples. 1. We have = 0. (4.1) ‘Pt, V242 + m2p — 2g2~42~242 Let us consider this equation in a one-dimensional geometry. If the field ~ is real, we have —
2
23
L142+tn 42—g42 =0.
(4.2)
This equation has a two-parameter family of solutions ‘P,,(x, t)
m sech{m
=
7(x
—
Vt
—
x0)}.
(4.3)
We say that the soliton solution (4.3) is stable when we write the solution of (4.2) in the form
(p =
42,,+~/,where
II ~II ‘~
at t0
ll~lI~
we have in some metric that I~/’Ilremains bounded for all t from zero to infinity. In ref. [93] it was found that the solution (4.3) is unstable. Let us find analytical solutions describing the evolution of this instability in the soliton rest frame and the instability growth rate 1~.Introducing the dimensionless variables I -~tim, x x/m, 42-4 m’P/g and linearizing eq. (4.2) near the solution (4.3), we get for small perturbations the equation* —*
ITh/i+(1 —3q~)~=0.
51R(x), we have an eigenvalue problem
Separating variables, ~Ji(x,I) = e’ ~ +(E+3ço~)R= 0, E——(1+fl2),
R—*0,
Jxl-*co.
It is easy to see that this problem has a solution related to the translational symmetry 42(x) (see, e.g., [5,86]): R
2(x), 4(x) ~ d42,,/dx, that is, R5(x) =
sinh(x)/cosh
E
1
=
—
1,
—*
p(x
+
/)
fi = 0.
Since this solution has a node, a nodeless solution with E0 < E1 must exist. One can find it by the method summarized in ref. [1191 2(x)’
R0(x) = cosh *
E
0 = —4,
El =
±V3,
Here and below we study the linear stability only, since one can conclude the occurrence of an instability from the linear approximation.
V.0. Makhankov, Dynamics of classical .so(iton~
and show that the problem has no bound solutions except R0(x) and R~(x)(see [119]).This means that the correction to the solution (4.3) can be expressed as 2(x) eSlI, cJ ~ 1, ~4’(x, t) = c sech where the instability growth rate is El = V3. In the laboratory frame the growth rate is smaller: ~iah
=
2. For the Higgs field equation,
x~—V2~—m2~+g2~3=0, we have rn Iym X~=±-~tanh~-~(x--Vt—x 0) Letting again x
=
x. +
~i
we get for q~ithe equation
One of the stationary solutions to this equation is the function 2(x/V2), cosh which has no zeros and k?~= 2. — Separating the variables ~t’= e~’R(y),y = x/\/2, we see from the equation =
=
—
cosh2
~R~ +
y R
(El2 + 2)R = —
E
0R, 2 = 0 is the state with minimum energy. For other states E 2 + 2 < 2, or that El 9> 2 and El Therefore the function ~t~’ is oscillatory: ~/‘~ exp( ±i~flIt). 3. We return now to eq. (4.1). It is convenient to write the solution of (4.1) in the form
112<
0.
—
=
B 0 sech(7B0(x — vt — x0)) exp[
±
—
B~(vx
—
t) +
ii9j.
(4.4)
Here we introduce the amplitude B0 which is related to A0 by
B~=1—Ag, and changes within the interval 0 ~ B0 ~ Bmax = I. As B0 I the soliton solutions (4.4) are reduced to the solutions (4.3) which are unstable with respect to longitudinal perturbations. Let us consider the limit B0—+0. In this case introducing the function 42(x, t) which is a slowly varying function of time, —‘
ç=e
/ x ‘I~~=,t
and neglecting ‘P,~,we get instead of (4.1) the Schrodinger equation with a cubic non-linearity (S3) i~,+ ~
+
I~I~ =0.
Soliton solutions of this equation are stable with respect to longitudinal perturbations. This is confirmed by a direct numerical calculation (see [51]and section 2.4) and follows also from the complete integrability of S3. Hence as B0—*0 the solitons (4.4) are stable too, since in this case
V.G. Makhankov, Dynamics of classical solitons
59
In ref. [891it is proved that the solutions of eq. (4.1) of the form ip = f(x) eAi are stable with respect to small longitudinal perturbations for A2> An analytical proof of the instability of these solutions is given in section 4.3. Numerical experiments [94] on solitons (4.4) for eq. (4.1) which are at rest, performed at Dubna, confirm the instability in the region A2 < Bearing in mind that for the solitons (4.4) A2 = y2(l B~) we get finally that these solitons are unstable for ~.
~.
B
—
2))112,
(4.5)
0> (~(l+ v
i.e., for a sufficiently large velocity close to unity.
V -4
1 one can obtain stable solitons with an amplitude arbitrarily
4.3. Stability of solitons in the direction perpendicular to their ,notion (the transverse stability) Let us consider the transverse stability of solitons for the KdV, Schrödinger and Klein—Gordon equations by varying the action with a Lagrangian integrated over the longitudinal coordinate x, and a class of trial functions restricted to soliton-like functions.* I. Let us illustrate this method by the example of the KdV equation, 42,
+
+
~2)
q~ = 0,
(4.6)
which is derived by the variation of the action S = f ~ dx dt, where .~(4/J, 1) ~ + + t,,i/i~+ and ~ = 4k,,, f/I = The terms arising in the equation at a weakly oblique propagation of solitons, i.e., propagation at a small angle to the x-axis can be easily found from the linear dispersion equation which in the laboratory frame is —
_l
‘~
~2
~2
Itj~+Ity
~i/2 ‘~
to_(1+k2)1I2_~1+k2+k2)
—k Il
ik2
—
2
j2
lit
x+2k2
y
2k2
In a system moving with unit speed in the x-direction we have
to’=w—k~=1(1 —3k~).
(4.7)
Here the value of a = I corresponds to the positive dispersion and a = I to the negative one (see, e.g., [901).Eq. (4.7) is obtained under the assumption that k~~ k~,i.e., the angle i~ = k~/k~ is small. This means that one may consider perturbations across the soliton motion with a wavelength which is well above the soliton width. Note that for a quasi-longitudinal soliton in a magnetoactive plasmas the first term of (4.7) vanishes (see [95]). Using (4.7) we get the desired equation —
42, +
+
‘PXXX
J
+
~
dx’ =
(4.8)
~
which one can sometimes rewrite in another, more convenient form: ~ *
+
~
+
~
+
+
=
0.
For Schrfldinger-type equations with a self-consistent potential such a problem is solved in ref. [92].
(4.8a)
60
V.0. Makhankov, Dynamics of classical solitons
Eq. (4.8) is derived by the variation of the action with the Lagrangian 2= ~{cI~I~,+
+ 24~~b~ + ~,2 +
~a1’~çb~ + ~a[I~
—
tt3~(x=
(4.9)
±cc)]}.
However, when one varies S = f 2’ dx dt dy and gets (4.7), one must bear in mind that only the leading terms of the expansion in the small parameter ~ = k5Ik~are taken into account. We rewrite (4.9) in a more convenient form: 2’ = ~ Here 1
=
+ ~423+
24242~~ + 42~+ 2a424235 + ~aq~I~
+ + ~a~I~42~Y+ ~[I~—
I~(x=
±cc)]}.
(4.9a)
f~pdx’. Moreover, when obtaining (4.9a) we used the relations
= l:I:i~~~ +
+
=
ills = tI~xxy,
in the third, fourth and fifth terms of the right-hand side of eq. (4.9), respectively.* As we saw above the family of soliton solutions to eq. (4.6) is 2{A ~= 12A~sech 0(x M0)}, where M0 = 4A~t. —
(4.10) (4.l0a)
The function (4.10) is also a solution of eq. (4.8). The study of the transverse stability of a plane soliton of the form (4.10) means that we put some perturbations on the function (4.10) and investigate the behaviour of this perturbation in time as t—*cc. If this perturbation remains bounded as t—*cc, the soliton solution (4.10) is called (transversely) stable with respect to the perturbation under consideration. If the perturbation increases unboundedly with time the soliton (4.10) is unstable. Here we restrict ourselves to small perturbations of the soliton amplitude A0 and its phase M0, when the functional dependence is given by (4.10). Suppose that A and M are functions of a transverse coordinate y and time t. Substituting the trial function 2 sech2(A(x — M)) = l2A into (4.9a) and integrating over x from x = —cc to x = +cc we get ~
—~M,A3+~A5— a(~—+4)AA~_ a~A~M+~ (ç_2)A
Varying the action S
= f
A~+2~ A3MY2.
~ dy dt over A and M independently we have the Euler equations
~
AA~=
(411)
-~-A~-a-~-A~M +~-~-~-A~M0. ,9t i9y 5 t9y For the study of the dispersion properties of this system we linearize it near the soliton solution (4.10) putting
A=A 0+6A, *
M=M0+6M.
Also note that eqs. (4.9). (4.9a) and (4.8) make sense only for solitons in our broad conception, i.e., ~ goes sufficiently fast to zero as x-*±x.
V.0. Makhankov, Dynamics of classical .colitons
61
In the zeroth order approximation in ÔA and SM a relationship between A0 and M0 in the form (4. IOa) arises naturally. Linearizing the system (4.11) in SA and SM and expanding the required functions in Fourier integrals we get the dispersion equation ak~—~A~)[8A~+~
=
(1 — 24A~)].
(4.12)
This equation describes transverse instabilities of ion-acoustic type solitons both in isotropic and in magnetized plasmas which were investigated also in the papers by Kadomtsev and Petviashvili [90] and Spatschek, Shukia and Yu [95],respectively. In the first case we have j2
/
to
2
8
221 ak5A0~1+a~-~-~ 0
which differs from the formula of the paper [90] by the second term in the brackets of (4.13). As we see easily, for a = 1 (negative term only a frequency If a = for —1 (just in this 2(k~)thr 72A~= 72k~.
(4.14)
For the instability of the ion-acoustic solitons in magnetized plasmas with Te ~ T 1 we have from eq.
(4.12) 4 21 ait ~ (4.15) Here, as in the first case, we get an additional term in the brackets and, moreover, a numerical factor in the formula for the growth rate in comparison with the result of ref. [95].We note at the same time that in this case the instability arises for waves with negative but not for those with positive 2
32
to
.
dispersion.
From (4.15) it follows that (k~)thr>3A~
3k~,
i.e., the instability threshold of (4.15) lies at the boundary of the applicability region, of eq. (4.8). It is interesting that this statement does not refer to the instability of (4.13), since (4.14) holds for A0 < 1, but the condition for the use of (4.8) is A0~l. Note the important fact, that the instability growth rate Im to is proportional to k5 and vanishes together with k3. An approximate dependence of the growth rate on k~is represented in fig. 4.1. This testifies to the longitudinal stability of the ion-acoustic solitons. 2. Using the described formalism we consider the stability of solitons for the Schrodinger equation with a cubic non~linearity* 2 142, + V 42 + 421242 = 0, (4.16) *
Note, that the longitudinal stability of soliton solutions of this equation was easily shown by a direct computer experiment [51].
62
V.0. Makhankov, Dynamics of classical solitons
for which the action reaches an extremum value for the Lagrangian 2+ 2’ = —~i(42427— ~ V! As in the preceding section take a trial function in the soliton-like form
~
(4.17)
—
=
V’2A sech(Ax) ~
(4.18)
and study the soliton stability in its rest frame. Integrating 2’ with the function (4.18) over the longitudinal coordinate we get =
4{A~i, A4~—f3(A~/A)+ ~A3},
(4.19)
—
where
I J
(1_~tanh~\2d
ir2+
cosh~ ) Varying over ~ and A we have 132
\.
36
~
12
11 ~
(4.20)
3y A
0.
A,-2-!-Acby
Putting again
(4.l9a)
~ = ~+
64 and A
=
A 0+ 6A, we get from the system (4.20):
(i)
in the zeroth-order approximation the conventional relation
(ii) in the first approximation in 64 and SA the dispersion equation 2 = —4A~k~(1 — f3k~/A~).
(4.21)
The instability (4.21) resembles surprisingly one for the case of the KdV equation (formula (4.13)) with the exception of the numerical coefficient and the sign of the dispersion. An approximate dependence of the growth rate of (4.21) is the same as the one in fig. 4.1. A linear growth for small k5 becomes a plateau for k~ A0ff3. The instability disappears for perturbations with wavelengths which are estimated from the formula k~>A~If3.
(4.22)
This fact is very interesting and is evidence in favour of Langmuir wave collapse (see [13] and section 5*). Taking into account the interaction of Langmuir and acoustic waves introduce practically no change in the picture of the instability, at any case for its initial stage [22]. 3. The Klein—Gordon equation with a cubic non-linearity for a scalar field with the parameters m and g, 2
2
23
42,,—V42+m ço—g(p =0,
(4.23)
* Note, that although an equation describing more-dimensional oscillations differs from (4.16) nevertheless this fact does not effect the result qualitatively (see, e.g., [92]).
V.G. Makhankov, Dynamics of classical solitons
63
Im (~)
Fig. 4.1. Approximate plot of the growth rate of a KdV soliton with a positive-dispersion instability.
is, as we saw earlier, derived by the variation of the action S÷=~J{’P~_(V’P)2_ m2’P2+~g2424}dxdydt. Becauseof the relativistic invariance of (4.23) it is sufficient to study only the stability of a soliton at rest: ‘P~=V2~sech(mx).
(4.24)
We vary the action with the Lagrangian 2’
~{~—
~
42~m2422+~g2’P},
and take the trial function in the form =
V2A sech Bx.
=
-~-
(4.25)
Then (A~- A~)—
(A,B,
-
A
2B 5B5) +
~
(B~ B~)- ~A
The equations of motion take the form A
2 5B~-A1B, ~ll
~
B~—3B~ A ~AB -
-
2m
-
m2
4~- ~g2A4 +
~.
-
+
~g2
~-
-2
(~-~-~f~)
+
-
~-
~
-
Putting A = A SA and 2. B = B0 + SB, we get in the linear approximation in SA and SB (4.25a) k~ — 2.93m Hence it follows that even for k 5 = 0 the solitons (4.24) and (4.3) are unstable, i.e., a variation of the amplitude and width of the one-dimensional soliton leads to its instability. The instability is stabilized for wavelengths of perturbations ~( = ilk5 which are the order of the soliton longitudinal dimension, 1/rn =~x.The transverse stability of the envelope soliton (4.4) for eq. (4.1) is found by the trial 0+
2
64
V.G. Makhankov, Dynamics of classical solitons
function B
=
sech(Bx) e’4.
(4.26)
In this case we have ~ (~ - ~)B
+ ~ B~- B~- B + ~B3.
Hence by taking the variation we find
+ B2 =
0.
Putting again B = B 0 + SB
and ~
=
~
+
~ we get the soliton relationship between B0 and 4~and the
dispersion equation 2),
(4.27)
k~+ (~± + 4(1 B~)k~f3)iI where z~= I 2B~.Since the condition for the existence of the soliton (4.3) is =
(~2
—
—
0~B~1,
it is advisable to consider the three possibilities (i) ~
(ii) ~
and (iii) ~
In the first case, B~>~, we have an instability when k,,2
< (k~)thresh
—
(B~
(4.28)
~).
As B 0—*
I we get an instability of the (4.25a) type.* If B~= we have
k~±~= k5B0.
=
(4.29)
For k~< B~/f3eq. (4.29) has an “unstable” root (Im to > 0) which is proportional to \/k~for small k5. Finally in the case of B~<~we find from eq. (4.27) that an “unstable” root arises also, if k~
~-1—2B~
(4.30)
we have (4.30a)
2’
Im to *
=
Ic5 (1 — 2B~)”
Note that even the value of the growth rate (4.25a) coincides satisfactorily as k,,
.-
0 with the one obtained in subsection 4.2, ft = V~.
V.G. Makhankov, Dynamics of classical solitons
Tm
65
(.3
(n)
(It)
(I)
_______________ 1’~tk,~.L
~<*
~ ~ ‘K5I~,~L
~
‘Kth~~L 4(~
Fig. 4.2. Growth rate of the KG3 soliton instability as function of the transverse wavenumber.
which because of a factor 2 arising when we change from d2/ 2 to id/dt is the same as (4.21).* The approximate behaviour of the growth rates Im to as functions 19t of k 5 in the regions (i), (ii), and (iii) are represented in fig. 4.2. In these figures one can see a characteristic fact: if the curve of Im w(k,,) starts at the origin there is a “longitudinal” stability of the soliton, at any rate with respect to perturbation of the types considered. In the case (i) when B~>~ the function Im w(k5) is not zero for k5 = 0, which means longitudinal instability of the soliton (4.4). It is interesting that this result coincides with one obtained by Zastavenko [89] in a completely different way. (In the paper [89] only the instability threshold for a soliton of the (4.4) type with respect to a variation of the amplitude B0 was found.) Because of a Lorentz transformation we have formula (4.5). 5. Let us study the “transverse” stability of solitons (of the kink type) for the Higgs field equation. For the real field we have the equation (m = g = I) (4.31)
(~_v2_1)x+x3=o,
which is the zeroth variation of the action with the Lagrangian 2+ l)2}. (4.31a) 2’ = (V~) To begin with we investigate the simplest case when only the soliton shape is perturbed and the vacuum values of the field ~± = ±1 are conserved. Here the trial function has the form (2
—
x~= ±tanh(B(y,t)x). Inserting it into (4.31a) and subtracting the vacuum value of 2=~fdx we get after the variation with respect to B i I
~
(C
2
C
2 \
i
3
—2
),
L) 0\Ui
i.e., B~=
~,
ayi
and for SB = B
(~_~)oB+~SB
—
B0
we have the equation
=0.
It is easy to check that the function = ±A tanh(x/\/2) 213t2 = 1— 2i(a/~t). *
]n fact, changing from g to g e’ we have 8
66
V.0. Makhankov, Dynamics of classical solitons
also leads to stability: (~_~)oA+~SA=0, where SA = A 1. Here we note that the variation of the soliton amplitude gives rise to a simultaneous variation of the vacuum values of the field and hence demands in principle an infinite energy. However, because of the stability of the vacuum perturbations (the dispersion relation is to2 = k2 + 2) we can subtract from (4.3la) the corresponding part of the Lagrangian which is relevant to the vacuum perturbation and vary the remaining finite value of 2,, = 2’— 2’vac where —
i
2
2
2
i
4
sx}I~==.
2vac~{XtXy+X
Now it is also interesting to note that the use of the trial function
x~= A tanh(Ax/V2)
(4.32)
as a soliton-like solution leads to an instability: w2=—~+k~,A
0=l.
This probably non-physical instability arises because of an artificial stiff coupling (resonance!) between oscillations of the soliton and the vacuum introduced into the system by the choice of the trial function x~in the form (4.32). Actually breaking this coupling, i.e. choosing a trial function
x~= A tanh(Bx) leads to the dispersion equation 2 = (1.69±0.l41)+k~,
(4.33)
to
which has no unstable roots. Now consider the general case of the complex Higgs field. The equation, the Lagrangian, and the soliton solution (for V = 0) are* (434 ‘Pu v’P 42+4242—u, a
,,.
2
2’ = 42,,
i
2
2
—
2
I
4
IV’Pl + 1421 H’PI },
(4.34b)
= ±exp{±iA
0t}V1+ A~tanh(\/~T+A~)x),
(4.34c)
respectively. The choice of the trial function is determined by the form of the solution (4.34c) and by the feature noted above (the possibility of resonance). Therefore 4’ A tanh(Bx), (4.35) = e and the variation is carried out with respect to 4,, A and B, independently. The Lagrangian integrated over x is ~=
~
—A
2B —~-+~-}. 5B~)+~~(B~— B~)—~A
*
An equation of the type (4.34a) can describe supersonic Langmuir solitons.
(4.35a)
V.0. Makhankov, Dynamics of classical solitons
67
At first, let all the derivatives with respect to y be zero (“longitudinal” instability), then, varying with respect to A, B and 4, we have, respectively, A,, B,, A 4,, 4 A 8A3 2~ 5AB -2 + ~ -~-+2 —~A -0, 2 A2 2 A4 A,, 4 2 B,, ~çb,yA +~~A~A ~-0, A —
2 -
-
-
2
2
a A24,, at
B
2
—
— -
In the zeroth order of perturbation theory we have =
A 0,
with A~= A~+ 1, B~= ~ In the first approximation in SA = A we get a homogeneous system of equation, from which we find 2 = 3.69A~ —2 ± \/~.97A~ — 11 .95A~ + 4.
—
A0, SB = B — B0,
and 84,
=
4,
—
(4.36)
to
The right-hand side of this equation is always greater than zero since A~= 1
+ A~.As
A 0-*0 the value
of A0 tends to unity and eq. (4.36) coincides with (4.33) with k5 = 0. And finally, if k5 0, the dispersion equation is 4i+ (4 + 12A~ + 8A~ — ~A~k~)to~ + 4A~A~k~ = 0, (4.37) + ~(43+ 94A~)to where w~= k~. For the simplest case of k 5 ~ I it is easy to find three roots for to~.The first two large ones coincide with (4.3), and the third (small) one is —
2i ,~2
to
~2
2 —
‘~
I A0~,A0 +3A~+2A~’
or 2
i.e., we obtain a stable solution again. In the general case it is necessary for an investigation of eq. (4.37) to calculate its discriminant which is always negative (for example, for A0= 1, D = —124+ l24k~—l9.2k~—2.35k~)and eq. (4.37) has no complex roots for to~. Besides a detailed study of eq. 2 ~ 0 (the equality is possible when k (4.37) performed in [94] shows that for its roots to 5 = 0). This means that the solitons of the complex Higgs field are also stable, both in longitudinal and transverse directions, at any rate with respect to the fairly general types of perturbations considered.*
*
Note that this result is in contradiction to Potyakov’s statement [86].
68
V.0. Makhankov, Dynamics of classical solitons
4.4. A qualitative discussion of soliton stability The soliton instabilities considered in the preceding sections have a clear physical interpretation. Let us start with the longitudinal instability of the bell solitons of the Klein—Gordon equation with a cubic non-linearity. As we saw in subsection 4.3 the soliton (4.4) is “longitudinally” unstable for amplitudes B~> Let us consider what this instability means, for example, from the point of view of a quasi-particle interpretation.* A scalar complex field ~ has a charge ~.
Q
=
i
J
~
—
~
dx,
which is conserved together with the energy integral E
=
J {l’P,I2~l’P~l2~
m2I42I2—g2I42j4}dx.
For the soliton (4.4) these quantities are Q=8~~AVl_A2,
E
=
8~-Vi
—
(4.38a)
A2(l —~(1 A2)),
(4.38b)
—
respectively.t The functions Q(A) and E(A) are represented in figs. 4.3a and 4.3b. It is shown that
(a)
(-s)
Fig. 4.3. (a) KG3 soliton charge *
Q as function of the frequency s
(constituent energy). (b) Soliton energy as function of w.
Below we follow ref. [961. Here we have again introduced the quantity (phase) A which is related to the amplitude by A = I — B2.
V.0. Makhankov, Dynamics of classical solitons
69
E esc
0.5 Fig. 4.4. Soliton energy as function of
a Q.
both Q(A) and E(A) have maxima at the point where A = i/V2. In fact, equating dQ/dA and dE/dA to zero we get A~.= ~in both cases. Further, we have ~m
~lA~-~It2~,
EmElASII2~V~, g Q—~-----~Q, E-~---~E. -
The soliton (4.4) is stable in the region where
where the condition E < Qrn is always satisfied (see section 3 and ref. [84]),i.e., the energy of a bound state is smaller than the sum of the energies of its components. Let us consider a possible evolution of the state S~with charge Q1, energy E~and, naturally, soliton phase A1 in the instability region. From eqs. (4.38) one can eliminate A and express E as a function of
Q: E =
(‘ -s-V ~
)
421/2
~i
±
—
Vi —
4~2)}~
(4.39)
This function is shown in fig. 4.4. The lower curve E,,~corresponds to right-hand parts of the curves Q(A) and E(A) starting with A ~ 11V2 and the upper curve to the left-hand ones. One can see from (4.39) that the spectrum E(Q) is not a decay one, i.e., E~(~1)
Qi =
Q2 + Q3,
therefore the decay of the soliton E~Xinto two solitons E~X and E~
2’~
Imto =~(I—2B~)=
31
to a one-soliton stable state which lies on a curve with the same charge and the energy excess is
70
V.0. Makhankou. Dynamics of classical solitons
radiated in the form of a neutral field. Once more note here that an analogous picture arises for the soliton solutions of an equation with a logarithmically diverging potential [961. Similar considerations remain, unfortunately, valid only for longitudinal instabilities when there exists a region of stable solutions. As we saw above (subsection 4.3) the soliton stability region disappears for two-dimensional perturbations of a scale which is in any case greater than the soliton dimension. In this case the interpretation given above is inadequate. Most probably this means that two-scale solitons of the KG equation (one scale determines the soliton dimension and another one its energy localization domain) of a bubble type will be unstable and begin to break into smaller formations as occurs for solitons of the non-linear Schrodinger equation [92].The latter phenomenon is well known in plasma theory as the Langmuir wave collapse (see below and [43]). The problem of the stability of one-scale solitons (including pulsating bion-type ones) remains unresolved in the meanwhile. Apparently just such an instability is responsible for the failure of the self-channelling of a laser beam propagating across plasmas (the dispersion formula is to2 = k2 + to~e). The kink-type solitons exhibit a surprising feature to remain stable even in the case of a complex field when these solitons are in fact, a rotating mixture of soliton and antisoliton states. Therefore, we think that just such formations are the most interesting one from the point of view of constructing extended particle models, provided one succeeds in finding longitudinally stable solitons in threedimensional space. In any case it is necessary to have a Higgs field among the interacting field as a stabilizing factor [10,86].
5. More-dimensional solitons From the title of this section it follows that we shall now be dealing with solitons in the real three-dimensional space. This material is still developed least of all and hence is less ready to be reviewed. Research work in this direction is only slowly become strong both in quantity and in quality. Since there is no general mathematical theory of non-linear partial differential equations various less or more exact soluble models are now being studied. The greatest number of papers are devoted to considering stationary solutions which are sufficiently symmetric to make the problem one-dimensional in a mathematical sense, i.e., spherically and cylindrically symmetric models are being examined, As regards the dynamics of “more-dimensional” solitons, viz, formation and interaction processes, computer experiments are still in the foreground. If in considering stationary states the latter are only auxiliary in nature (like a slide rule) they become a competent tool for theoreticians (together with pen and paper) for studying dynamical problems. All of this, namely, an incompleteness of the material, a variety of models and approaches, an unstable terminology, and a growing flow of publications puts naturally its imprint onto the choice of material as well as the mode of presentation of the following sections. We apologize to those authors whose works are not discussed here. Moreover, some of the problems discussed below will probably be reviewed in separate papers* in which the reader would find a more complete bibliography. 5.1.
One exactly soluble two-space-dimensional problem
We have earlier declared that our main task is discussing non-integrable equations. Here we violate ~We have heard of several such papers which either are in preparation or about to be published [23].
V.0. Makhankov, Dynamics of classical solitons
71
this rule and following Hsing—Hen—Chen [97] consider two-dimensional solitons of the Kadomtsev— Petviashvili equation* 42~± ‘Pyy + ~
+
+ 42xm
(3422)
=
0,
(5.1)
which describes perturbations in a weakly dispersive and a weakly non-linear medium. It seems to be more natural, to begin with such a simple example of an exactly soluble system that the reader can feel some specific features of more-dimensional problems. Let us formulate briefly the statement of the problem following Lax: given two commuting operators (the so-called L—A pair)
Ai =
~i,
0,
(5.2)
which are linear with respect to required function ç, there are two coupled eigenvalue problems LxPr~A~I1, A~Pza’h1r,
(53)
with A and a independent of time. The operator equation (5.2) is then satisfied on a class of functions ~ which are solutions of eq. (5.1) and the spectral problem (5.3) may be employed to obtain a Bäcklund transformation. The operators L and A for eq. (5.1) have been found in [98]. L=
a~+42—ba5,
A=
4id~+ 3i(42a~+ d~42)+ 3ib
(5.4) dx + i(a~+ a,).
(5.5)
Putting b = ±l/V3one gets the upper sign before the second term in (5.1) (i.e., negative dispersion), while b = ±i/V3gives the minus sign and positive dispersion. Examine first transversely stable solitons: ~
+ 42~,,+ 42~+
(3~~)~ + ~
=
0.
(5.6)
One might proceed from (5.6) to the equation with positive dispersion by the substitution y —*iy.t We show that eqs. (5.3) may be considered as a relation between the solutions of (5.6). Putting = In ‘I’ and q?.. ~ we rewrite the system (5.3) as follows: (5.7a) 4i(4,~~~ + 3~i~çli~ + ~)
+
3i(24~~4,~ + I~)±iV34~+ i~+ i~,= a,
(5.7b)
and eliminating s1 we get ~
Thus, for every pair
(5.8) (s/i,
a) satisfying (5.8) there exists a pair
(—
i/i,
—
a) satisfying the same equation
* Eq. (5.1) may be obtained from (4.8a) by rewriting (4.8a) in the laboratory system and dropping the last term which stabilized the transverse instability of solitons in positive dispersion media. 2,, and the transformation y-*iyItleads to k~—~—ik,. makes the result obtained in subsection 4.3 clearer, since the sign in formula (4.13) is determined by that of k
72
V.0. Makhankov, Dynamics of classical solitons
with the lower sign in the fourth term. For this new pair we have a new solution of eq. (5.6); therefore (5.9a) ~
(5.9)
Taking the difference and sum of eqs. (5.7) and (5.9), integrating the results over x and thus eliminating* the constants A and a (for details see [97])we get the Bäcklund transformations (~-
‘F)2~
2(~+ ‘F)~~
J
(~- ‘F),,
dx’
=
0,
(5.10)
~
J(~-’F)tdx=0~
24i~~ =(F—’F)~.
(5.11)
The double sign appearing before the derivatives with respect to y reflects the fact that there may exist waves propagating in both directions along the y-axis. Eq. (5.11) which is an intermediate step in getting (5.10) is needed below. Equation (5.10) allows us to construct a new solution ‘F of (5.6) once we know one solution, ‘F. Since ‘F = 0 is one of those solutions we can start just from eqs. (5.3) which are now (5.l2a) 4s/im
+ 4’~+
s/i,
=
0.
(5.l2b)
The general solution of this system is 2y+ i(4k3 = ~ s/’~exp{ikx ~ V~k Eq. (5.11) then gives for ‘F=0 ~4~=24i~~
k)t} = ~ s/it. e~.
(5.13)
=2(s/i~/s/~r)~,
and hence ~ =2 ~
—
~&kke~)-
(~
~kk:’)(~~ke~)
(5.14)
(~e4) s/’k
is the required solution of eq. (5.6). A cnecial choice of s/i 8k,-iAi +
5kjA’
1 results in special solutions. For instance,
one gets
putting s/’k = = 2A~±) sech2{A(±)[x ±2V~A
1_1y 4(A~±) + 3A~)+ ~)t]}, —
where
A(±)= ~(A1+ A2), *
A(_) = ~(A1 — A2).
The possibility of eliminating these constants implies that they a,e not essential in constructing new solutions.
(5.15)
V.0. Makhankov, Dynamics of classical solitons
73
WaVe ~rorst ~ane
Fig. 5.1. Angle between the wavefront plane of a two-dimensional KdV soliton and the abscissa axis.
The solution (5.15) approaches in the A(..)-+0 limit the conventional one-dimensional soliton of the KdV equation (see (4.10)) if one takes into account that the coefficient before the non-linear term in (5.6) differs from that in (4.8a) by a factor ~, which leads to (5.16)
‘PEq.(5.6) = ~‘PEq.(4.8a)~
The relation A() = 0 implies A(±) = A 1 = A0, and using (5.16), the above statement follows. The soliton (5.15) has a plane wave front obeying the equation x ±2V3A()y = constant. The angle between the wave front plane and that in which x = 0 is determined by (see fig. 5.1) =
arctan2V~A(.),
(5.17)
and lies in the region (5.18) To see this, let for example A1 A ~ A2 then (5.15) may be rewritten as follows 2sech2{~A[x ±\/~Ay— 4(A2 + ~)t]}, 42* ~A so that ~*zzarctan~.J~A~*V3A, since A~1.
(5.19)
(5.20)
This means that i~ is bounded with the soliton amplitude and lies in the range (5.18). Positive dispersion. This case corresponds to the transformation y-*iy. We then have from (5.15) an exact solution of eq. (5.1): (p
=
2A~÷~ sech2{A(±)[x ±i2V’~A()y— 4(A~÷) 3A~...)+ ~)t]}, —
(5.21)
which is singular (recall the linear instability of a soliton in the positive dispersion case) at the points of intersection of the following lines 2V~A(÷)A(..)y = 1T(~+n),
n =0,1,2,...,
(5.22)
and x
= 4(A~) +
3A~)+
This singularity implies apparently that there is no non-linear stabilization of the instability (4.I3),* ~This result is in agreement with that obtained in [99]when studying resonance interactions of wavepackets and contradicts one obtained in [100] for the non-linear dissipation of an unstable KdV soliton.
74
V.0. Makhankov. Dynamics of classical solitons
which resembles very much the collapse of Langmuir waves which we shall discuss in the next section. In the negative dispersion case the multi-soliton formulae may be obtained as well. However, we do not do this here and refer the interested reader to refs. [97, 1001. 5.2. More-dimensional solitons in the framework of the S3 equation Both S3 and KdV equations are completely integrable and therefore Bäcklund transformations exist to construct solutions of the S3 equation. We give now a comparative analysis of the solutions of these equations disregarding their specific features. As we have seen solitons of the KdV as well as of the S3 equation may be unstable but for opposite signs of the ratio ~ Moreover, the dispersion formulas (4.13) and (4.21) are remarkably similar except for the numerical coefficients which are determined by the particular form of the one-dimensional solution and the different degree of the non-linearities. This is because the structure of the function to(k,,) is the same for both equations. This is of no surprise if one sees how those relations are obtained. In fact, after integrating over x the solutions depend only on y and t. This dependence is governed by the approximate equations
±J
42, +(~~
~
dx
=
0,
i42,
2±p,,,, = 0.
+
‘PI421
The first equation resembles the second one if one takes t it and to —ito (which is valid for solitons at rest). This in turn implies that the sign before to2 in (4.13) must be changed to correspond to (4.21), so that the instability of solitons for the KdV and S3 equations occurs for opposite signs of the ~,,,, term. “Positive” dispersion in the S3 equation gives rise to (in the transverse direction) stable solitons* since making the substitution y iy, k,, —ik,, we have from (4.21) to2> 0. By analogy with the KdV equation we may therefore suppose that a stable soliton moving at an angle to the x-axis is described by —~
-*
~=
—*
—*
V8A,A
2 sech{(A1 + A2)x ±(A1
—
A2)y} exp(i4A,A2t),
as can easily be verified by substitution into the S3 equation with “positive” dispersion. This formula is valid if IA1 A21 ~ IAi + A2j. For the normal S3 equation (“negative” dispersion) one arrives at a solution which is singular in the points —
A()y
=
~(n
+~),
x = 0.
Thus the equation
142~+’P~~’P,,~+I42I420 has stable more-dimensional solitons as solutions [101] while the equation
2 i421+42~~+42~~+I42I=042
may have singular solutions. In the following subsections of this section we shall consider the collapse of Langmuir waves in various one- and two-field models. *
See for example ref. [loll. The additional double integration over x of the ç,,, term does not alter our discussion.
V.G. Makhankov, Dynamics of classical solitons
75
5.3. Collapse of Langmuir waves (CLW) Pinching of a wavepacket in non-linear media was first met in a study of the propagation of powerful laser beams through matter and plasmas [102]. An analogous phenomenon has later on appeared in theoretical investigations of the dynamics of Langmuir turbulence in plasmas [13]or more precisely in clarifying the mechanism of dissipation of the “Langmuir condensate”,—the turbulent energy accumulated in the long wavelength region of the spectrum. As we have seen (see [50]) Langmuir turbulence turns out to be unstable with respect to a coalescence of plasmons. This instability resembles the well-known gravitational one. The regions with a higher density of high-frequency energy become the centres of its condensation and of a further collapse of energy bunches (wavepackets). The problem of the Langmuir wave collapse has now its history. Zakharov by analogy with the self-focusing of light beams suggested this idea in 1972 and it called forth a lively discussion going on up to the present. Many papers [70, 103] especially in the Soviet literature are devoted to this problem.* There are both analytic and computational works. It should be emphasized that the problem of collapse has two somewhat different aspects: the mathematical and the physical ones. A basic question, from the point of view of turbulent plasma physics and its application is whether the CLW can provide a transfer of the energy of the Langmuir oscillations (e.g., due to Landau damping) to the bulk of the electrons or whether this phenomenon is not effective enough and will result only in the appearance of high-energy tails in the distribution function. It is expedient to separate the mathematical aspect from the physical one in order not to affect any thermonuclear projects which are at the present time realized or developed. We shall chiefly deal with the mathematical aspect and examine the possibility of CLW in various models described by the appropriate systems of non-linear equations. These models may naturally be improved from the physical point of view. The latter statement does possibly not refer so strongly to the finite size particle method [103], although this method is also not free from its own defects [106]. We start from the simplest model, viz., the Schrödinger equation with a cubic non-linearity. 1. The CLW is described in the 423 approximation by the equation (5.23) V2(is/i, + V2,/i) div(IVs/iJ2Vi/i) = 0, —
where s/i is the high-frequency potential envelope, and not the field envelope as it was in subsection 2.1 (see (2.24)). If the initial packets are spherically (SS) or cylindrically (CS) symmetric, eq. (5.23) can be reduced to 2 n—-l 2
l(pt+VrrcoyçP+Ic0I420,
(5.24)
and n = 2 corresponds to the CS and n = 3 to SS geometries. The field ~ = —Vs/i is subject to the condition 42(0) = 0 at the origin. Eqs. (5.23) and (5.24) have, respectively, the first integrals S =
S
IVs/~I2
—
2 *
f d3r, d3r, f {1V2s/i12 ~Iv~r}
(5.25a) (5.25b)
We know only a few works performed in the USA which are concerned in some way with the collapse of f-waves [64,92, 104,105].
76
V.G. Makhankov, Dynamics of classical solitons
or
S
=
=
J 112r2 J
dr,
(5.26a)
{I(r~I2+ 2142 12_
‘r2I I~}dr
(5.26b)
in the SS case, and
S
=
s 2
=
J
~
J
dr,
(5.27a)
{vV~~I2 +
~
~
dr
(5.27b)
in the CS case. To show some qualitative features of the CLW we discuss it in the SS geometry. In one of the first works towards a numerical study of the CLW an arbitrary two-scale wave packet (the energy containing layer width z~rmuch less than the size of the packet R0, that is, a “bubble”) turns into the quasi-plane soliton which moves to the origin accelerating (see fig. 5.2) and oscillating about the “soliton state”. Such a dynamics is natural for packets, when the instability condition S2~0holds. Packets with S2> 0 disperse. Such a behaviour may be realized with the help of the constants of motion 5, S2 and following the discussion of refs. [13, 1021. Let us consider 2)S D
(r
=
J 14212r4
dr.
Making the Ansatz (5.24), we arrive at =
6S 2 —2
JKr)r12
dr
—~J114r2 dr
<
6S 2,
Fig. 5.2. Initial stage of the SS-collapse of an f-wave packet in the S3 model.
V.G. Makhankov, Dynamics of classical solitons
77
or, upon integrating over t, D<3S2+Ct+C2.
(5.28)
As D> 0 (and hence C2 >0) this inequality puts an upper bound t0 on possible values of t, where t0 depends on the initial conditions, i.e., on the constants C1 and C2: (529)
6JS2j
0
It follows that the solution of the initial value problem exists for only a finite time and gives rise to a singularity as t t0. The value of t0 depends on the sign of C1 which may be thought of as the initial radial packet velocity. If C~> 0, the packet disperses for small t, while in contrast C1 <0 leads to its contraction; t0 enlarges or decreases correspondingly. We have for a packet, initially at rest, —~
Ro/ 2 (C2/3~S2~)~ 42m,
t0
i.e., the contraction time of a packet is defined by its major radius and stored energy (42m being the field amplitude so that 42m 42max). We have already noticed that the evolution of two-scale initial packets starts, as in the planar case, with the formation of a self-consistent quasi-planar spheriton* and that self-consistent packets move to the centre with an appreciable velocity [107, 108]. In this connection it is convenient to change in (5.24) to the function =
r42,
which gives 2~2~0.
(5.30)
j~t,r+
The solution of this equation up to terms of order o(~r/R 0)and phase multipliers is the spheriton =
Ae~’sech{V~R(r_Ro)},
A =~(2_~A2).
(5.31a) (5.31b)
Therefore ~=
where A
A e~’sech{~(r_ R0)}. = Af R0.
or
(5.31c)
The condition i~r~ R0 can now be rewritten as
A>>V2/R0.
One can ascertain from (5.25a) and (5.31a) that the following relations hold for contracting packets: 2 = const AIR02I, AR0 *
Though one-scale packets with z~r R
0 evolve in a more complicated way, the picture resembles qualitatively that discussed above [108,109].
78
V.0. Makhankov, Dynamics of classical solitons
a V
Fig. 5.3. Approximate plots of the potential energy. the effective kinetic energy, and the total energy of a soliton as functions of its radius R~.
and
Ar0 = A1R01
(5.32)
(A, and R0, being, respectively, the initial amplitude and average radius of the packet). The integra)~.S’ and S2 may be evaluated with the same accuracy and we have for the spheriton S~= S/R~, 2 — S2)
S
R~S0I= R022V2A,
S2
R02S~PI+ 2S~~ = ~ (2 + R02
(5.3~. (5.33b
v
In (5.33b) the two first terms are related to the effective kinetic energy, the last term to the poterIiaI energy. The collapse condition, S 2 < 0, for packets initially at rest is 2>96R02, (5.34) S
or
A2> 12fR02. This condition is, indeed, sufficient but not necessary for the collapse to begin. To show this, let us consider how the kinetic and potential energies of a packet at rest with a given S depend on its position R 0. In fig. 5.3 one can see the function S2(R0) which has a single maximum at the point R~ defined by the condition dS2/dR0 = 0, whence 2= S2/48, (5.35) R~
which differs twice from the value given by (5.34). One can see also in fig. 5.3 that if the packet for a given S is on the right of the point R~it will disperse infinitely, and a packet with R 0 < R~will contract. Thus, the collapse condition can be written as* S2< S~~(48fS)>0.
(5.36)
We can examine the initial stage of the packet contraction (dispersion) using the geometric optics *
A similar inequality should exist for arbitrary packets in three dimensional space since in getting (5.28) we dropped negative terms.
V.0. Makhankov, Dynamics of classical solitons
79
technique [109].We take the field ~ in the form = tF(~)exp{i(kr
—
~ = r — v(t)t,
A(R, t))t},
and assume A to be a slowly varying function of R and t, and v(t) such a function of t. Then (5.30) yields
~~
+
A,t
+ 2kArt
2—
—
k
+
=
o,
(5.37a)
and
v
=
v
0 + v,t = 2(k
(5.37b)
— Art).
Eq. (5.37a) leads to
A
2+
=
(2— ~A2)
k
— A,t
—
2kA~t,
(5.37c)
or, by virtue of (5.32), 2 ( Ar~
-
2R021\ 12A1 ~b~!’
~0 \
“oI
and finally
v
= 2{k —2t(A~~1_2)
~}
= 2{k —
________
This means that the packet will move after some time to the origin being accelerated if the initial condition 2/16 (5.38) R02< S holds. The value S given by (5.38) is less than that of (5.35) by a factor 3. This difference appears because (5.38) does not take into account that portion of the potential energy which makes the packet to be a self-consistent one (the compensation of the plasmon kinetic energy in the packet) and also because of the half-quantitative character of the above analysis. We note that the constant-acceleration motion of spheritons to the origin has been observed in numerical experiments [107].
5,4. The CLWin the dynamical model 1. Let us study the CLW in the so-called dynamical model of two interacting fields (2.24). We rewrite it in a more convenient form, introducing a low-frequency potential u by analogy with (l.28b) = ‘F +
l42l~= ‘F + lVsfrl2
(5.39a)
we then have
V2u
=
‘F,
V2(is/s, + V2s/i) = div(’FVs/i). For these variables we easily obtain from eqs. (1.11) and (1.28a)
(5.39b) (5.39c)
80
V.G. Makhankov, Dynamics of classical .colitons
S2 =
f
3r ~
d
=
f {IV2c~l2
— ~IV
s/il4 + ~(u~+ (Vu)2)} d3r,
or
J
2s/il2 +‘FlVs/il2 + .42 + ~(Vu)2}d3r. (5.40) {lV It is evident that besides S 2 also S and S~are conserved because of the gauge and translational invariance of the Lagrangian (1.11) (see section 1); moreover, S is determined by (5.25a). Let us have initially some wavepacket for which S2<0; then from (5.40) there follows the inequality* S
2
=
J ‘F~Vs/iI2
d~r~ >
and by the theorem of the mean we have mm
maxl’Fl> IS
2ISI
(5.41)
(the minimum is with respect to time and the maximum with respect to the coordinates). From (5.41) the statement follows: if in plasma a cavity with a reduced density (‘F <0) appears such that S2<0, it cannot disappear, i.e., in any case, the value of ‘Fl does not decrease during the evolution of the initial packet. 2. Let us consider some general properties of stationary soliton-type solutions of eqs. (5.39). We note that for solitons at rest these solutions coincide exactly with the ones of eq. (5.23). Let ,/i(r, t) be of the form2t
s/J(r, t) = e —IM~(r)
ei~ ~(r). (5.42) Here we take into account the fact that soliton-type solutions which are localized in space (42(r) —*0 as r—*cx) are possible only in the case when A = ..~2
div(IVxl2Vx) = 0.
The solutions of this equation, ~(r, formation:
~),
(5.43) which depend on the parameter
JL,
allow a scale trans-
~(r, j.~)= Therefore in the one-dimensional case (i) S(~.t)=
J lV71(~r)l2
dx
=
~C( 1);
(5.44a)
in the two-dimensional case (ii) S(j.~)= C(2); *
This treatment is due to yE. Zakharov [110],as far as we know. This conclusion can be rigorously proved by the consideration of the asymptotic behaviour of eq. (5.39) of r—~.
(5.44b)
V.0. Makhankov, Dynamics of classical solitons
81
and in the three-dimensional case (iii)
S(j~)= C~31/~.
(5.44c)
This means that in a one-dimensional soliton the number of constituents S increases with the constituent binding energy ~ (i.e., with a decrease in the soliton dimension). In a two-dimensional soliton it is constant, and for a three-dimensional soliton it decreases. Let us multiply (5.43) by (r, VX*), add the result to the complex conjugate one and integrate over volume; we then get for soliton-type solutions [110] 2 d3r+ (4—n) lV2xl2d3r— (2—in) lVxl2d3r = 0 (5.45a) n)
~2(2_
whence
S~=
f
f lVxl
f
~s2S*,
~
(5.45b)
where n is the dimensionality of space (i.e., the number of those spatial dimensions in which the soliton solution under consideration in the usual three-dimensional space is bounded). Therefore in what follows we use n as the dimensionality of the soliton in this sense. We thus have from (5.45b) for the plane (n = 1) soliton* (i) S~=
—
~J.L2S*;
(5.46a)
for the “cylindrical” one (ii) S~= 0;
(5.46b)
and for the three-dimensional soliton (iii) S~
(5.47)
2~
This means that only the plane solitons have negative energy while the energy of two-dimensional and three-dimensional ones is equal to or greater than zero.** Using eqs. (5.44) one can eliminate the parameter ~t from (5.46a) and (5.47) and obtain the relationships between the integrals S and S 2 for the stationary solutions:
(i)
3/3C~,
S~=
_(S*)
(ii)
S~= 0,
(iii)
S~
C~/S*,
~ = S~/C
1
~sis arbitrary,
(5.48a) (5.48b)
C3/S*.
(5.48c)
=
The particular case of a stationary three-dimensional soliton is the approximate spherically symmetric solution (5.31c) obtained in the preceding section. For this solution it follows from (5.32), (5.33a), (5.37c) that =
—A =~((S*)2/16_2R02)=
=
~
(5.49)
or 1aR~= I (see eq. (5.35)). 2/2 = —S2/16 (see (2.31)). This formula coincides with (2.34) with v = 0, ii one takes into account the equality ~ 0, =A Besides strictly stationary solitons (solitons at rest), eqs. (5.39) and (5.23) allow solutions in the form of moving solitons ~(r) exp( — iAt — ikr) (see, e.g., subsection 2.4 for the plane case). For those A is greater than for the solitons at rest. * **
~‘
=
V.0. Makhankou, Dynamics of classical solitons
82
Inserting this expression into (5.41c) we get the relation (5.36), S2 = 48/ S* which we have found above from the condition of extremum of the spheriton energy, dS2/dR0 = 0. This extremum, as 2S follows from fig. 5.3 and because of the sign of the second derivative d 2fdR02, is a maximum. As we saw above when S2 < S~the packet collapses and for S2> S~it spreads. 3. The evolution of various spherically symmetric initial packets was investigated in refs. [108— 110]. In the papers [109,1101 it was found that under the condition S2 < 0 (in fact S2 < S~)a two-scale wavepacket after becoming self-consistent, which gives rise to the soliton emitting ion-acoustic waves and taking it on the quasi-planar soliton shape (5.1k), starts to move to the centre, i.e., it collapses, However, the rate of the packet collapse in the dynamical system cannot exceed v’s’ = I/VS. as follows from the difference of the integrals S2 in the quasi-static and dynamical models. In fact, in the latter model instead of (5.33b) we have
s2=~(2+R02.~_~j(l_sv2)). From the condition S2 V~Eesh=
—
<
S~we get a limiting value for the packet collapse rate: R~a~ 1,
~
(5.50)
where a = S2/S and K is a numerical factor of order of unity. The evolution for various initial data is represented in fig. 5.4 (see [54—55]). Analogous results for the packet collapse have been obtained for a cylindrically symmetric geometry [44]. Finally note, that eq. (5.43) in axially and spherically symmetric cases is reduced to 2 —~
n—l (p+Vrr(p~~2~(p+l(pl 420. 2
4./p
2
~ ~
\
t~6O.
‘~ t~59.
~E’~::~ ~frts;~
:~ ~ *‘
~
~‘
:~/~tTstJL ?29
~)~)‘
t’~g5 ~
r
~
t-1jo.
t—’t3.
‘.1
~541’
i
2f
(a)
—~m36
-
(b)
Fig. 5.4. SS-collapse of two different initial f-wave packets.
V.G. Makhankov, Dynamics of classical solitons
83
This equation has an enumerable set of solutions which are bounded at the origin and vanish at infinity. Since for these solutions 42(r = 0) = 0 the maximum is situated at r R0 l/~.This fact leads to that, for the same dimension of the solitons, CS and SS solutions have an integral S which is greater than S for the solutions with a smaller degree of symmetry for which the maximum of ç may be at the centre. Just the latter solutions are the most probable ones (at any rate these solutions are more favourable from the energy point of view). 5.5. Dipole CLW
The simplest solution has been investigated in a[110].For this solution theplane density of the oscillating 2s/i has a distribution in the form of dipole which lies in the perpendicular to the charge p = V dipole moment. In the centre of such a dipole the field ~pis a maximum. Before a description of the results [1101obtained by numerical calculations, let us consider some properties of the solutions of eqs. (5.23) and (5.39) following from self-similar substitutions [110,103]. 1. For the complete system of eqs. (5.39) one can find the self-similar substitutions in two limiting cases; in the quasi-static one when it is reduced to eq. (5.23) and in a “supersonic” limit when in the right-hand side of eq. (5.39a) one can neglect the quantity ‘F.* In the first case, i.e., for (5.23), the self-similar substitution is [13] s/i
=
exp{—i~s2ln(t 0
—
t)}X(~),
~ = r/\/t0—t,
(5.51)
with the following equation for x(~) 2( ,..t2x + ~ + V2~)+ div(lVxl2Vx) = 0. (5.52) V In his early paper [13] Zakharov obtained a spherically symmetric solution of (5.52) in the region of —
1/it where x satisfies the equation i~V~ = 2,A2X,
~‘
so that xl~l
Xo’
This solution, as was noted in [131,leads to a non-physical divergence of the integral S; moreover, it follows from (5.51) that S 2(t) = S2(0)/V~, which leads to S2 = 0 as dS2/dt = 0. The first difficulty may mean that the self-similar approximation is valid only in a restricted region of space but also can indicate the absence of a region of applicability. The second one is an essential restriction on possible initial states.t 2(t)s/i (which corresponds to the In the “supersonic” limit the substitution [1101is/i1 —p. adiabatic approximation) andwe wemake get from (5.39) —*
2
2
2
V (—p. (t)~+V~)—div(’FV~)=0, =
(5.53a) (5.53b)
En the system of eqs. (2.24) which is more translucent from a physical point of view the “supersonic” limit corresponds to the neglect of the V211 in the wave equation. t The paper [105]also notes that this substitution is erroneous and gives the formula ~ = r(t 2t5 for n = 3. 0 — tY
* term
V2lVxl2.
84
V.0. Makhankov, Dynamics of classical solitons
This system permits among other things a substitution of the form [1101* 2,.
p
2
~p.0
~
fl(~)
—
—
x
—
—
—
~(~)
,h
#~l—2fn,
—
i~ii~ I)
~, I~i~
-2/n
—
\4Tfl ,
—
r(t~—t)
,
(5.54)
H
where n is the dimensionality of the solution as before. Details of the study of the solutions of (5.54) can be found in the original papers [103,1101. For what follows we need the following properties of the solutions of (5.54): (i) in all cases (5.54) leads again to S2 2’F — (t 2 (t V 0 t) 0 r~ , —
=
0;
—
(5.55)
—
2 lVx(0, t)l2 = f(t) = (t
(iii)
(42(0,
t)l
(S.56a)
0—~~’ 413.
(5.56b)
(iii) l~(O,t)l = 420(t0 t) From (5.54) it follows that as t t 2’Ff’F 0 for a plane soliton V 2’F/’F 1, ~ collapse 2’F/’F —*0.for Thea two-dimensional condition S V 1, const., and finally in the three-dimensional case V 2 = 0 which is necessary for the self-similar solution leads to the following restriction of possible initial states in the three-dimensional case** —
—~
—*
—*
0~
(5.57)
The quantity S~for the spherically symmetric collapse was found by us above, formula (5.36). Finally, formulae (5.56) show that, if valid, the self-similar substitution (5.54) leads collapse. to a linear 314 during the three-dimensional time-dependence of the quantities 42(0, t)(~ and (‘F(0, t)( 2. These conclusions have been verified by means of a computer investigation in a series of papers [103,110]. The initial states were chosen in three-dimensional space (in an r, z coordinate system) as follows: —
~p 2s/i=
.~
.
rrz
0Vwsin—~-
w>0,
(5.58a)
p=V w= I—~(r2+z2).
For the system (5.39) we also gave the values ‘F(r, z, 0)
=
—
vs/il2,
‘F 1(r, z, 0)
=
0.
(5.58b)
The distribution (5.58) describes at t = 0 a dipole-type charge directed along the z-axis. The results of the computations are shown in figs. 5.5—5.7. One can see from them that for some
time the collapse behaviour is in fact satisfactorily described by the self-similar substitution (5.54). Given 5, the collapsing soliton has a disk shape, independent practically of the initial data. The deviation from self-similarity is observed at about t **
in all the cases studied; this may be
Note that the substitution (5.54) in the two-dimensional case (n = 2) is valid for the complete system (5.39).
The domain of the initial values (5.57) instead of the point S, = 0 exists, allowed which can lead to a reduction of S, from ~ to zero. **
t0
in fact,
only for the exact system (5.39) where the emission of c1 waves is
V.0. Makhankov, Dynamics of classical t
Fig.
5.5a.
=0.0
~
-0.12
t
Collapse of a disk shaped initial packet of f-waves (see (5.58a)) at Po =
too
85
solitons
t=0.17
13
in the S3 model.
Ipi
t~O.14
100
.
N I
0.1
2 as function of the radius
Fig. 5.5b. ~j coordinate z.
r
~
0.2
and of the longitudinal Fig. 5.5c. Self-similarity curve
t0.0
as function oft.
t=0.30 t=O!iO
IhL
_
t’~O.5O
_
Fig. 5.6a. Collapse of a disk shaped initial packet of f-waves for p 0= 13 in the dynamical model. 2 lP~
_____________________
100
~
0 —iso
——
t
I’P15
‘O.4~
(00
002c
0 —100
0.1
“ ,/ Fig. 5.6b. jg2 and V as functions of r and z.
0.2
Fig. 5.6c. Self-similarity curves g~ and ~ correspond to cI~and the filled dots to it”4.
The open dots
86
V.0, Makhankov, Dynamics of classical solitons
: 0.025
Fig. 5.7. Self-similarity
curves for
~
—
t
2it is neglected in the sound equation (supersound collapse) for p~=
the collapse when V
13.
associated either with computational algorithm defects (this explanation is preferable to the authors) or with the properties of the solution itself. This question is at the moment still unclear.
5.6. Spheriton collapse in the system of coupled Schrodinger and Boussinesq equations We shall now discuss the effect of the non-linear and dispersion terms in the low-frequency equation on the collapse of a spherical wavepacket [60].In this case we have jçt+V2rr 42_~42
/
i9
(5.59)
=‘F~,
2
2\
~Vrr~VrrVrr)’FVrr(’F
2
2
2
2
)Vrr(lçl)
= 4me/3m, is a small parameter). The exact solution of this system is not so far available so we can give only a qualitative picture, which may throw light on the phenomenon under consideration. If the amplitude of the initial packet is sufficiently small, 142m1 ~ 11, then the picture of the formation of a self-consistent spheriton and its initial movement to the centre coincides with that described above in subsection 5.3. The spheriton velocity reaches its limiting value v” = l/’~/~, the spheriton amplitude then increases with decreasing radius according to (5.32), viz.,
and
A/A1
2, =
(5.60)
(R0~/R0)
R 0—R01-\/~t.
After a time t0 R01/V~the packet in the (5.39) model should be collapsed and its amplitude should have become singular. In the model (5.59) considered here there arises a connection between the velocity and amplitude, of the soliton, if the amplitude is sufficiently large. This effect has for the plane case been discussed at length in subsection 2.5 (see formulae (2.40) and (2.41)). The collapse of the spheriton (5.60) is depicted in fig. 5.8 on the A, v-plane as a motion of a point in parametric space along the line II. When this point approaches the curve I the solution must be
V.G. Makhankov, Dynamics of classical solitons
87
17
Fig. 5.8. Spheriton collapse.
reconstructed turning from a one-hump into a two-humped one like the (2.40) one. Hereupon the representing point either goes along the curve I or the spheriton becoming unstable and breaks up. Moving along the curve I we have first
A =2/R~—~A2<0, due to the collapse condition A2> 4/R~.Then the second term in A grows faster than the first one so that
and
(Al ~ A2 ~ (R 4, 0Y ~ A2 ~ (R 8’3, 0)
for the system (5.39),
for the system (5.59).
The collapse results in A( increasing and hence the contraction velocity v decreasing (it is clear the mass of the soliton does not change due to the decrease in R 0).
Since 2OS~=
S
R
( 27)V4
2 3’2 = R 0A
coust.,
we have 2
R2A312— —
A312
0V~i
or
A/AL = (R
413
and
A/A, = (R
0~/R0)
instead of (5.32).
3,
(5.61)
01/R0)”
Finally, when A approaches the value Acr~~ 1
~‘,
(5.62)
the spheriton should cease to contract or becoming unstable it breaks into pieces. The packet width is then about i~r~ 7de. Note that the numerical coefficient in formulas (5.62) and ~r can alter by several times, if one considers instead of the Bq equation its modified variant (see [60] for details).
88
V.0. Makhankov, Dynamics of classical .solitons
5.7. One more example of an SS-packet collapse The equation for the scalar Higgs field in SS-geometry has the form 2
X
3
1VrrXX+X
=0,
(5.63)
and the energy is E=J~YCdr.
~
(5.64)
1)2}.
Hence we have for the quasi-planar bubble (see (3.2)) x(r, t)
=
tanh[~5~(r_R0— Vt)],
(5.65)
and E is proportional to R~. On the other hand the relativistic invariance of eq. (5.63) gives E = Mlr_o x y so that 2. (5.66) E o~R~/V1— V An initial packet in the form (5.65) with v = 0, which tends to decrease its potential energy U =~Jr2dr(x2_1)2,
will collapse. The energy conservation law, dE/dt = 0, then leads to the following equation of bubble motion for R 0~’1 [1121 (5.67) with the solution R0=R~~cn~—~-—,5j \L~0
(5.68)
/
2 = ~).Formula (5.68) describes well the bubble (here cn(x, ~)(extension) is the elliptical cosine k is very thin, R contraction stage as long with as themodulus bubble wall 0 1. When R0 = 1, (5.68) is no longer valid. We shall discuss pulsating solutions of equations of the (5.63) type with a degenerate vacuum in section 7. ~‘
6.
Stationary solitons
6.1. The stability of more-dimensional solitons We discuss some general properties of more-dimensional stationary solutions of some non-linear field equations which follow from their integral characteristics. 1. We start with the simplest case of a real (uncharged) scalar field described by the non-linear
V.0. Makhankov, Dynamics of classical solitons
89
wave equation (6.1) (here III
=
2/3t2
8
—
V2, F’( 42) = dF/d42) and by the Lagrangian
F((p). The Hamiltonian of the stationary field 42(x) is = ~[(42r)~ —
(42x)2]
—
E=J[~(42~)2+F(42)]dx_=K+V. A single integration of (6.1) with 8/at potential part of the energy V: =
(6.2)
(6.3) =
0 leads to the relation between the kinetic part K and the (6.4)
F(42).
The constant of integration is chosen so that for ~ = ~± at X—*±c,2the function F(42) vanishes. Otherwise, the value of V, together with the energy E, is infinite, i.e., it does not satisfy the soliton definition. Here the relation K=V
(6.5)
is the virial theorem. The result stated below was apparently first obtained by Derrick [1131and by Hobart [115],it has a somewhat different form in the paper by Goldstone and Jackiw [114].The result is that soliton-like stationary solutions of (6.1), if they exist, are stable only in the plane geometry. Consider the scale transformation 42~= q(ax). Then the solution of (6.4) will be 42a for a = 1, but this solution gives an extremum of the Hamiltonian (6.3). By transforming the variables in (6.3) we get E[42a]=aK+’ V,
(6.6)
or da
=K-V=0,
i.e., we have the theorem (6.5) again. From eq. (6.6) it also follows, that the value a = 1 gives a minimum of E[42], because K >0 and =2K>0.
da
Similarly, a consideration of the system of interacting scalar fields in an n-dimensional space with the Hamiltonian E=
~f
~ (V42j~d~x+
f
F(421) d~x,
(6.7)
gives after the scale transformation E[coa]a
2—n
K+a —n V,
so that we have instead of (6.5)
(6.8)
90
V.0. Makhankov, Dynamics of classical solitons
~
(6.9)
By calculating a2E2[42a]/da2 for a d2E —~
=
I and taking into account eq. (6.9) we have
=—2(n—2)K,
da
(6.10)
which gives rise to a maximum of E[
42~] for n > 2 and to a point of inflection for n = 2. It means that the solutions under consideration are unstable with respect to a~oscillations*. At once we make a reservation that the statement proved above is valid only for scalar real fields with the charge, Q=if(42*42t_42~42)d3x,
(6.11)
vanishing. The presence of the additional conserved quantity Q may, as we shall see Later, qualitatively change the situation (see also [10]and [117]). 2. As a simple model of a charged field we consider the so-called co theory. Its Lagrangian is 2 ~ m2l4212 ±g2(ço (~, (6.12) = 42t(~ V42( and the Euler equation of motion is —
2—
2
2
L11142±mç±2gj42(42=O. (6.13) We restrict ourselves to solutions depending on time in the formt (we call them below quasi-
stationary solutions) =
~3=
i/i(x) e’~.
(6.14)
Then we have instead of (6.l2)—(6.l3) QpJs/2d3x~p.s1~fr1 = ~{p.2s/i2 —
(6.15)
(Vs/i)2 ~ m2s/i2 ±~g2s/s4}+ const.,
(6.16)
V2s/i + (p.2 ~ m2)s/i ±g2s/i3 = 0.
(6.17)
The energy density corresponding to (6.16) is = ~{(Vs/i)2
+
(p.2 ±m2)s/’2 ~ ~g2s/’4}+C.
(6.18)
The value of the constant C in (6.18) depends on the asymptotic behaviour as x --* ~ of the solution s/i(x). Here we consider only so-called non-topological solitons for which 42(x) as x ~ goes to the constant vacuum value The necessary condition for the existence of topological solitons is the degeneracy of the vacuum. For such solitons the boundary conditions at infinity are different from the boundary conditions for the
~.
*
by
—~
Note that this proof is easily generalized to the case
of vector
fields for which the operator V2 is replaced in the spherically symmetric case
D=V2—21r2. These solutions have
a minimum energy for
Q 0 (see
section
3).
V.0. Makhankov, Dynamics of classical solitons
91
physical vacuum state. In the one-dimensional case examples of topological solitons are the kinks of the equationfor the Higgs field and of the sine-Gordonequation. In three-dimensional space such an example is the magnetic monopole solution of ‘t Hooft [87]and Polyakov [85].More recent results on topological more-dimensional solitons are summarized in [118]. For what follows we rewrite (6.18) in the form 2)(s/i2 s/i~)r ~g2(s/’4 — = ~{(Vs/i)~+ ±m —
(~~2
One of the possible vacuum values of (6.17) is s/i,,~= 0, i.e., we get the theory based on the Klein—Gordon equation and havingsoliton solutions for m2> p2 (e.g. see [115]).The second value of sfi is obtained if one drops the first term V2s/i in (6.17), so that s/i~= (p2+ m2)fg2 which leads to the solutions (including non-topological ones), for which, however, the vacuum value depends on the charge. The physical meaning of this is not quite clear. Therefore we consider the equation V2s/i + (p2
m2)s/1 + g2s/? = 0,
—
(6.19)
which may be obtained by the variation of the Hamiltonian
f
E = ~ {(V~)2+ (p2 + m2)s/? — ~g2çfr4}d3x,
(6.20)
keeping Q fixed. To do this one has to express p as a functional of p
=
s/i:
(6.21)
Q/S[s/j],
and vary E, keeping in mind (6.21). Let us write E in the form E
=
2 + p.2)S — S
S 1[s/i]
2[s/i],
(m
+
2f
~
d3x. Using an a-scale transformation we get
(6.22)
where S1[s/i] coincides with K while S2 = ~g E[s/in] = S 2)S/a3 S 3, 1/a + (p.~+ m 2/a and —
2
26
2
26
p.~—Qa/S
p.a,
or 2
3
2
—3
—3
E[sfi,,]—S 1/a+p.aS+tna whence
S—a S2,
2— ni2)S+ 3S
dE/dalai or
=
—S1
+
3(p.
2
0,
2—m2)S=~S (p.
1—S2.
. (6.23) 3x we get a second relation between the functionals S, S
Multiplying (6.19) by s/i and integrating over d 2 m2)S = S (p. 1 2S2,
1, S2:
—
—
or
(6.24)
2—p.2)S=~5 S2=~S1,
(m
1.
(6.25)
92
V.0. Makhankov, Dynamics of classical solitons 2
2
.
2
~.
Finding 8 E = (d E/da ~ and using (6.25) we see that 6 E > 0 for p > ~m or p > Thus the solitons are stable against a-oscillations (the scale transformation). Now we consider the second variation of the Hamiltonian (6.20) for arbitrary perturbations. Let = s/’~+ 5i/i, then ~
=
J
2
[Js/’~
&/iI~’5s/Id3x +
p.~
os/i
2
1
2
d3x].
(6.26)
We get (6.26) using (6.21), from which it follows that 2
5(p.2S)
=
/
S[s/i~] ~—2
J
s/’~6s/id3x
J_(Os/i)2
[J
d3x
s/,~s/id3x]
S[~ 5]
+2
- ~
In (6.26)* the operator .F~is a test Hamiltonian and is given by formula 2+m2—p.2—3g2s/i~. (6.27) I~I—V It is easily seen that in subsection 4.2 we had a similar operator during the investigation of the planar solition stability. As before we are confronted with an eigenfunction (EF) and an eigenvalue (EV) problem for the operator =
A.y,.
(6.28)
Because of the translational symmetry equation(6.19) has the solution s/’ 5(x + E),** as wellas the solution s/i5(x). For small translations e the function = V (k = 1,2, 3) is also a solution of eq. (6.28) with A = 0. (One easily verifies this by differentiation of eq. (6:19).) This means that the translational mode is a three-fold degenerate state of the operator H with energy zero. Hence, min [119,1161, a spherically andsymmetric there is at geometry the translational mode forms a p-state with! = 1, s/i~=(0s/i5/8r)Y1 least one s-state (1 = 0) with a lower energy, A <0. tFollowing [10]we show that if 1) there is not more than one negative eigenvalue of the operator H, and 2) the value d In Q/dp. is negative, then the soliton solution of eq. (6.19) is stable (at least with respect to perturbations which do not change the symmetry of the solution “longitudinal” stability; see subsection 4.2). Differentiating (6.19) with respect to p and introducing the notation s/i,. = ös/i 5/3p we get s//k
ks//S
—
Ifs/i,.
=
2p.s/i5.
(6.29)
Let us expand the functions s/i,~and s//5 in the orthonormal system of the eigenfunctions of the operator iTt (which we assume to be real) s/i = ~ s/t,.1y1,
(6.30a)
~ s/i~,y1,
(6.30b)
=
and
J
3x
=
61k.
yy5 d
* Note, that as the second term of (6.26) is positive, ,,3/Q>0, the presence of a charge leads to a stabilizing effect.
‘°‘This means
that a translation of the soliton centre from the origin does not change its shape. In general, there would be two s states with zero and unit radial quantum number, respectively.
V.0. Makhankov. Dynamics of classical solitons
93
Inserting (6.30a) into (6.29) and using (6.30c) we have = 2
p.s/i51.
(6.31)
Hence s/’~= 0 for A, = 0. Moreover, we note that 3x = ~ s/i,~A s/i,.iTt~/~,. d 1,
f
but from (6.29) we get 3x = p. s/~,jTts/i,.d
f
f
~—
s/.i 3x 5~d
=
p. ~
(6.32)
~.
Therefore s/I~
~‘
1A,=
or
~/A1
p.~-.Q, =~
(6.32a) (6.32b)
The prime in (6.32b) means that during the summation we omit the terms with A, arbitrary perturbation Os/i as a power series in the eigenfunctions y~, Os/i =
=
0. Expanding an
~
we have from (6.26) 2E =
~
~‘
O~A
3 1+
I5
~‘
s/s~,s/’~O
p.
151 ~~ 51T1183,
(6.33)
where the test matrix T~is defined by 3s/i T1, = A130 +~ p. 51s/i51.
(6.34) 2E it is necessary to find the eigenvalues of the matrix I~,i.e., to
For of the sign of 5 solvethethedetermination equation ~ T 083—flO,.
(6.35)
Inserting (6.34) in (6.35) we get 3s/’
(11— A1)~ p. 51
~‘
s/i~~6~ = 8~.
Multiplying both parts of this equality by ~ and summing over i, we have 3 ~ —1=0. (6.36) g(tI) p. Let us see whether eq. (6.36) has negative roots fl < 0. Suppose the operator If has only one negative ~‘
94
V.G. Makhankov, Dynamics of classical solitons
Ill
ft1 ~\ ) ,\
I
______
Fig. 6.1. The function g(fl) in the case of one negative eigenvalue A in the KG3 model.
eigenvalue A1
=
-
Fig. 6.2. The function g(fI) with A~and A2 both negative.
—A then for Cl<0 —1.
g(fl)=_.4_p.3~I1115~fr~
If the function g(fl) crosses the abscissa axis for fl <0, it means that eq. (6.36) has a negative root. The approximateform of the function g(fl) is shown in fig. 6.1. It is seen that a negative root(1 <0 exists only. if the function g(uI) is negative at the point ~1= 0 (curve II in fig. 6.1). From (6.32b) it is easily seen that (0~=—--~(zI~./A.~— 1= (63~ g / p. ~ ‘.~rSi L’ d ‘ ‘~‘
~
and hence the negative root together with the instability occurs for -~~>0. Qdp.
(6.38)
The existence of two negative eigenvalues of the operator iTt leads inevitably to a negative eigenvalue of the test matrix T11* and hence to soliton instability (see fig. 6.2). The result of this theorem can be formulated also in a negative meaning: it is sufficient to have two negative eigenvalues of the test operator H in an arbitrary geometry (n = 1, 2, 3) for the instability of a stationary charged soliton; in the three-dimensional SS case the sufficient condition for the instability is inequality (6.38) and there exists always a negative EV of the operator H (s-state). The stability theorem for a potential of a rather general form is given in the Appendix. We check the known soliton solutions from this point of view. In the ~ theory (plane geometry) the equations permit the transformation s/15(x,
p.) = p.
whence
2~const. p. The translational mode of the Higgs kink soliton with zero energy, as we saw above, is the lowest one (nodeless), and, hence, all A. ~ 0, i.e., the soliton is stable. For the bell soliton the same mode is not the lowest one (one node), therefore there is one lower EV, A <0 as can be seen in figs. 6.3a and 6.3b. The
Q = p.
*
—
This root is situated between
A~and
A 2.
V.0. Makhankov, Dynamics of classical solitons
95
4~
_____
_______
-~
(a)
~,
(b)
Fig. 6.3. Field function and (a) its derivative (translational mode) for the LGH kink-soliton: (b) its derivative for the KG3 bell-soliton.
second condition leads to I dQ p.2 constdp. vm —p. —
/
2
V 2
2
22
2=
p.
m
m
2
p..
Hence, in the complete accordance with the study of section 5, the bell soliton is longitudinally stable for p.2>~m2. For the SS-solitons we have
Q=
~
2~const.,
Vm-p. and dQ/d~>0, which is in agreement with the result by Anderson and Derrick [116] for the absence of stable time-independent SS-solitons for the ~ (Klein—Gordon) theory.* Given a non-linearity of the form 422 in the equation ofmotion it is easy to check that all the solutions starting with n ~ 1, if such exist, are unstable. The equations with the non-linearities of a more general form demand an additional investigation of the properties of its stationary solutions, in particular, finding the behaviour of Q(p.). In what follows we give some examples. 3. We generalize the theorem for the case of a Schrödinger-like equation 2
2
2
(6.39) Eq. (6.39) is obtained by the variation of the action with a Lagrangian ~
I.
*
*
421)—(V42( —-2gjço(. 2
12
Changing again to the real field function s/i = stationary solutions: —p.~s/i+ V2sfi + g2s/i3
*
(6.40) 2t}, we have the following equation for
4
~
exp{ — ip.
= 0,
It is interesting to note that the same solutions for ~s> m12 turn out to be stable with respect to the scaling mode.
(6.41)
96
V.0. Makhankov, Dynamics of classical 301 itonS
and a Lagrangian L=
—
J
d3x. ((Vs/i)2 + p.2sft2 ~g2i/I4} —
(6.42)
Eq. (6.39) has the first integrals* S E
=
J1~I~ d3x,
=J {(
(6.43a)
2I~(4} d3x.
V
42 ~— ~g
(6.43b)
Eq. (6.41) can be derived by the following variational principles: (1)
OLE s/i]/Os/i = 0,
(6.44a)
given that the variations Os/i are arbitrary; SE[s/i]/&/i
(2)
~s/,,
=
(6.44b)
given that the additional condition OS[s/i] = 0 holds (this condition may be considered as a constraint, and p.2 is a Lagrangian multiplier). Here, to first order, the variation Os/i is orthogonal to the solution
f
ifr
3x =0; 58s/,d (3) The variational principle can be formulated by introducing some function F(S) of the functional (6.43), H = E[i/i] + F(S), into the Hamiltonian (6.43b). Equating the first variation oH/Os/i to the variation OLfOs/i, we get eq. (6.41) and we determine the function F(S) dF/dS =
(6.45)
2
In this relation p. is a function of 5, and the variations Os/i, are arbitrary as in the case (1). For the study of the stability of possible stationary (soliton) solutions of eq. (6.41) it is necessary to find the second variation of the Hamiltonian, 52E in the case (2) and 82H in the case (3): 02E =
f
OI/iIfOs/I d3x,
(6.46a)
given that the additional condition holds
J
{2i/i~Os/i+ (Osfr)2} d3x = 0,
and ITt =
—V2 + p.2
—
3g2s/i~.
(6.47)
Hence it follows that if there is even one eigenfunction Yt of the problem Ify 1 = A1y1 with negative A. <0 satisfying the condition (6.46b), the solution s/i5 is unstable. The presence of two negative EVs A1 and A2 means that the soliton is unstable as the condition (6.46b) is satisfied by the corresponding selection of the constants a and b in the linear combination Os/i = as/i1 + bsfr2. *
The soliton momentum integral in the rest frame is not interesting.
V.0. Makhankov, Dynamics of classical solitons
97
Let us consider instead of eq. (6.39) an equation of a more general form to include so called “saturable” non-linearities (e.g., see [120]) i42,
+
2 V 42 + QF(~42r)= 0.
(6.48)
2)~M as IQt2~~x. Eq. (6.48) realizes the extremum of
A non-linearity the Lagrangian is called a saturable one if F(~q,~ ~~i(ço,*~ç — 42*42,)
=
—
~V
where
J
2U(~
—
42(
2),
(6.49)
42I
l~l2
U(l
2)= 42~l
—
F(x) d3x.
(6.50)
Eq. (4.80) has the integral S (eq. (6.43a)) and the integral 2)}d3x.
E=J{1V
(6.51)
4212+ U(j42~
Changing to the real function s/s = ~ exp{—ip.2t} leads to V2~ p.2~+ sfrF(s/i2) =0. —
(6.52)
This equation is obtained by the variational principle (6.44b). Calculating the second variation of E by analogy with (6.46) we get (6.46a), where
If =
—V2 + p.2 — F(s/i~)— 2s/i~dF(s/i3/d(s/i~).
(6.53)
As in the Klein—Gordon equation case considered above, the translational symmetry leads to the presence of a three-fold degenerate p-state with energy zero (the lowest non-symmetrical state). Hence there is at least one symmetrical s-state with A <0. Applying the method of Lagrangian multipliers we write the problem of the minimization of the functional 02E in the form
Ifo
42 = fl042 + as/is,
(6.54a)
and
J
3x = 0. Oq,s/i5 d We expand 042 and ~ in the complete orthonormal system of EF of the operator (6.53), .FTty, = 842
=
~ CjY~,
s/i 5 = ~ s/i5~y~,
and we substitute these expansions into (6.54); then we have 842
=
a~AQ
Yt.
From (6.54b) we have
a ~‘ As—fl = ag1ffl) = 0.
(6.55)
98
V.G. Makhankov, Dynamics of classical solitons
-_______ ____________ Fig. 6.4. The function g(f)) when there is only one A
1 <0 in the non-linear Schrodinger model.
For a = 0 the function 842 is the same as one of the y~and Cl the same as one of the A1. As we saw above, a negative EV A, exists only for the spherically symmetrical s-state. If such a state is the only one, it is the ground state and if it has no zeros, it cannot be orthogonal to the nodeless solution s/i~~ so that a 0 (in any case for the study of the nodeless soliton stability). Hence g(Cl) = 0.
(6.56)
Differentiating eq. (6.52) with respect to p. and using the above mentioned method (see eqs. (6.30)—(6.32)) it is easy to obtain g1~,1—~A,
4p.dp.~
6
The function g1(fl) when there is only one A1 <0 is represented in fig. 6.4, where we see that the negative root of eq. (6.56) occurs for g(0)<0, or dS/dp.>0.
(6.58)
Hence we may formulate the theorem on the stability of solitons of eq. (6.52) in a negative aspect: the presence of one negative EV of the test Hamiltonian H (and this exists always) and the satisfying of the condition (6.58) are sufficient for the instability of the soliton 42~o~ We note that taking into account the second-order term in the perturbation (042 = y and ~~ I) in the orthogonality condition (6.54b) leads to
iTty = uly +-~as/i5+~y, and hence we have instead of (6.56) g1(Cl) = 0—
The roots of this equation are the points where the function g(fl) intersects some line g~which lies under the abscissa axis. Therefore the soliton instability condition for infinitesimal is dS/dp.~0.
(6.59)
This analysis allows a simple and clear geometrical interpretation. The extreme ~ is a saddle point in
99
V.0. Makhankov, Dynamics of classical solitons
a functional space, and the orthogonality condition determines a surface on which the variations 842 are allowed, given that these variations cross the saddle along some curve. Depending on the sign of dS/dp., the saddle point in this curve is either a maximum or a minimum. Hence the value of dS/dp. is connected with the angle between the variation surface and the plane containing the steepest descent line. Therefore, although the saddle point is unstable, the presence of the additional constraint (6.54) can lead to a stabilization of the solution. Finally we note that the stability theorem for the Klein—Gordon equation (6.19) can be easily generalized for equations of the form V2s/i—(1—p.2)~+s/IF(s/i2)0. 6.2. Some properties of spherically symmetrical solitons Eqs. (6.59) and (6.52) will be identical if one introduces the notation 2_li~p., 2, K ~ p. In that case one has
for for (6.59), (6.52).
—
V2sfr
—
1(21/, +
sfrF(s/I2)
Let us show that as [761*
K —*0
=
0.
(6.60)
the SS solutions of (6.60) are the same as the solutions A~(r)of the equation
1 d / 2dA~\ —~--~~r -~--j_A~+A~_0
(6.61)
(n gives the number of nodes of the function A(r)). Introducing the new variables 42
=
s/i/ic
and
(6.62)
p = Kr,
instead of s/i and r, we have 1 22 V Q—(p+42---2F(I( ~ )0. If one looks for the solutions in a class of bounded functions ç ~ N, we have for KN 4 1 an expansiont 2
x
2, if we redefine 42 by
i.e., we have eq. (6.61) with the accuracy of a zeroth approximation in K 42
2dF(x)
2
=A. 2) be upper bounded (a “saturable” non-linearity) as Let the function F(ifr *
uX
x~’O
~
then one can write
We studied in detail an analogous equation in section 2 (see (2.59)).
fit is natural tosuppose that for small in the definition of ,e.
~‘
the expansion of F(~’2)starts with a term ~
i.e.. F(0) = 0. Otherwise the term F(O)
0 could be included
100
V.0. Makhankov. Dynamics
of classical so! itons
U.
—
,1~
Fig. 6.5. Approximate form of a poteniial in which a fictitious particle moves within the framework of an equation with a saturable non-linearity,
Fig. 6.6. Approximate plot of S as function of , equation with a saturable non-linearity.
= (I
2)112 for an
—
w
its asymptotic behaviour as k=
Fi(~2)=M_(C~)2k,
1,2,...,
(6.63a)
if there is a power series representation, or as F
2) M — C~exp{—C 2} 2(s/i 2s/i for an exponential non-linearity (C
(6.63b) 1 and C2 are some constants which depend on the form of the
function F). Inserting the expansion (6.63a) into eq. (6.60) one can easily see that the extremum points of the potential u
F(x) dx
= _~1(2~2 +
determining the trajectory of the solution in the phase plane* are ~=
*(K2+F(~2))(K2+M_~)0
Hence, besides the point 1/’ = 0, we have also 1
C
~1m~(M_’K2)
112k
.
(6.64)
Therefore 2
ln
(6.65)
C
(6.66)
1 2m
— ~2.
Eqs. (6.64) and (6.66) are valid as *
K2~~*
M.
An approximate form of the potential a is represented in fig. 6.5.
V.0. Mak)iankov, Dvna,nics of classical solitons
101
For the simplest models of saturable non-linearities F1 = [5//2/(1+ ~~2)]q (see, e.g., [120]) and 2}) (see, e.g., [64,68]) we get from (6.64) and (6.65) exact formula F2 = 1/C(1 — exp{— l/q Cs/i ~/iim =
±(1
(6.67a)
2fa)1/2~
1
1
(-~ In ~ — CK2)
5/i2~~ = ±
1/2
.
(6.67b)
Using these results we can study the behaviour of the integral S[s/i] as a function of K. For small follows from the substitution (6.62) that s
=
f
~2
d3x
=
-i--f
K
it
2d3p =
increases as K—*0. As K2—* M —0 the behaviour of the integral S will be determined by the extremum points they determine the scale of the solution as a function of the parameter K. Hence
5/Im
as
S ~ s/’~-~.
It follows from this formula that for equations with a saturable non-linearity there are solutions for which S increases as K2 —~M. The approximate form of the dependence of S on ic is represented in fig. 6.6. As S must be a continuous function of the parameter K, it has a minimum at some ~cr~ It is natural that the derivative dS/dic at Kcr changes its sign. Hence for the class of equations considered here with bounded F(s/i2) there is a domain K> Kcr, where dS/dK > 0. Finally we show that soliton solutions can exist for which the test Hamiltonian H(s/~’ 5)has only one solution with a negative eigenvalue. For this, following the paper [120],we note that as ic—*0 we arrive at the operator of a cubic 3y. = —V2y 2y, equation 3ic21/r~y If 1 + ic 1= A1y1. —
By a numerical investigation [120] it was shown that among the spherically symmetric solutions of this equation there is only one solution with A1 <0. All the remaining EV of SS-solutions are greater than zero. The energy levels of the operator (6.53) are shifted when K increases. Let us show that no one of them passes into the negative region. It is sufficient to show for this that H has no bounded SS solutions with zero energy for 0< ic2< M. Then, for increasing K none of the EV, A1(K), corresponding to a symmetric function, vanishes or, a fortiori, becomes negative. As we saw in subsection 2.7 any trajectory in the phase plane starting from the axis s/It = 0, with increasing r ends at 5/s = 0 or at the points 5/’m (it can also be seen in fig. 2.24). Only for definite “initial” values of the function 5/’(r)Iro do we obtain trajectories finishing at zero as r—*co. The slightest variation of the initial value s/i(0) leads to a trajectory which will finish not at zero, but at a point ~/~m (the well). The solution s/i. with the asymptotic behaviour s/si —*0 as r is unstable with respect to a variation -of initial data (Liapunov instability; see the paper by Zhidkov and Shirikov [76]). IF s~.is a solution of (6.60), which finishes at the point ~/‘m and has boundary conditions 5/i~(0)= 5/i~(0)+ e, ds/i5/dr = 0 and sfr5(r) is a solution of the same equation but with zero asymptotic behaviour, we have —*
e-*O lim5)~=8~1) E t95f’5(O)~°°
102
V.0. Makhankov. Dynamics of classical solitons
Hence, the function x~ as/i~(r)f3s/s5(0)is unbounded as r—* differentiation of (6.60)) an equation =
and satisfies (as one can easily see by
0.
(6.68)
This means that the operator If has no symmetric EF with zero energy, as we wanted to prove. Hence, for any values of sc in the interval (0, Vp.) the test operator H (see eq. (6.53)) has only one 2) spherically symmetric solution with negative energy. means that eq. (6.60) with function which is bounded as s/i—*~has stable solutions for Itsome values of sc> K~r andthe at any rate F(s/s when K2~* M. Note that in the case of a relativistic equation the charge Q is conserved, but not S, so that the sign of the derivative dQ/dp. determines the region of stable solutions. Here it is necessary to bear in mind also that K —*0 corresponds now to p. 1. The second asymptotic behaviour exists only for the case K2_* M or m2(l — w2)—* M so that it is necessary to have M < m2. The above treatment holds only for this case. The asymptote for Q for small w is the straight line co = (1 M1m2)”2, but not the ordinate axis. The decreasing branch of Q corresponds to a stable solution. —*
—
Examples of stable SS-solitons 1. One example of equations having stationary stable SS-soliton solutions was given by Anderson
6.3.
[1171 2~+ g242J42~2— a4214214 = 0. 42 ~- ~,, — m The corresponding Lagrangian and Hamiltonian are
V2
2=142,1 —1V 2
2
2
2
142! ~‘=I42~l +1V421 +m 1421 421
2
—m
2
2
4
12
12
6
—~aI42l,
+-2g 42! 2
1~
(6.69)
4
I~
6
—~g1421 +~aI42I Changing to a real function s/s = ~ e’~’making the usual substitution of variables, co 2 gs/i(1—co) 2)’12 a = &m2(1 w2)/g4, p = mr(1 — co we are led to a one-parameter equation:
(6.70)
.
=
p.1m,
=
—1/2
—
U’~P
-~~ bU’4’
3
5
—~+—---=~—q~ +a~. dp pdp The method of investigating this equation is analogous to the method of subsection 2.7. Rewrite (6.71) in the (anharmonic oscillator) form: —‘(a.
____
\2+i( 4)2~i(h4
d
—
‘
~2’.p/
2J~~
2
aq)6
—
3
—
+
U.
The function u is shown in figs. 6.7a and 6.7b for two characteristic values of the parameter a defined by the extremum points, du/d~,and the zeroes of the function u: 2 _________ F ii~Vi—4a\” 12=~ and Pl.2_~\ 2 ) 2a 3a —
From the boundary conditions ‘I~= 0 at p = 0 and t1(p) —*0 as p —~~ and from fig. 6.7a it follows that for a given a in the interval (0, ~) there are one or several solutions of eq. (6.71) (one solution is
103
V.0. Makhankov. Dynamics of classical solitons U. U-
/~
~1~77~2
(b ) 6) non-linearity model for two values of a: (a) a
(a) Fig. 6.7. Potential in the ~ — ag
(b) a>
<~,
nodeless). For a > ~ there are no values of 4(O) when the solution has the asymptotic behaviour as p—*ca~ (this is connected with the sign of d/dp<0). Anderson [117] gives nodeless solutions of eq. (6.71) and studies numerically their stability as a function of co for different values of the parameter a. In fig. 6.8 [117] we give the soliton instability growth rate as a function of co. This shows that the instability region is reduced with an increase of a. For a ~ the soliton is stable in the interval (0.18< co < 1) (nevertheless the region of small co <0.18 is unstable). The presence of a stable solution for this model, in contrast to the 424 model, can be clearly illustrated by means of the potential part V 5 of the soliton energy density for both modest (see figs. 6.9a and 6.9b). It is seen that in the 424 model the only extremum of V5 (besides a trivial one) is the maximum, hence the soliton “rolls down” either to s/i—*0 (“dissipative mode” of instability in the terminology of [116]) or to 5/,-÷cc(“singular mode”). In the model under consideration the absolute maximum of the function V5 arises at a fixed value of s/i = F2. The soliton has two possibilities; either to roll down from the hump to s/i —*0 (region I) (dissipative mode as above) or to roll down to the well s/i—*F2 (the singular mode disappears!). However, for Q w = 0 this does not yet mean soliton stability. In fact, it can conserve the value s/s = F2 when it increases its dimensions. This gives a negative contribution to its energy. The soliton can compensate this negative contribution by means of increasing the gradient ~/‘r(see fig. 6.10). A kind of a shock wave arises and moves to large r with increasing steepness of its front. The of the charge Q = bounded: const. can depress 3x =presence const. = of Q, the the integral soliton dimension is upper this instability. Because of co f ~2 d R<1~’3 ~ O~~rn~417.wF2(1_co2)) Tm (~)
:
Fig. 6.8. Growth rate of the Q-soliton instability in the
(~4 ag6) model as a function of —
w for various a.
* As in subsection 2.7, nodeless and oscillatory solutions with non-zero asymptotic behaviour are possible.
104
V.G. Makhankov, Dynamics of classical solitons
(a)
(b) 4 model; (b) in the (g’ — ag6) model.
Fig. 6.9. Soliton potential energy density: (a) in the g
For the appearance of the “dissipative mode” the soliton energy perturbation must be of the order of its initial energy E 0 which leads to a soliton climbing to the potential hump at the point F1. The instability modes have been both observed in the numerical experiments of Anderson [117] for the appropriate selection of parameters. Anderson describes a behavior of the soliton in a stable region of the parameters for sufficiently small perturbations of the soliton energy (for details see ref. [117]).The important fact is that the presence of a charge can stabilize the soliton (see also section 3). 2. The second example is the soliton of Friedberg, Lee and Sirlin (FLS) [10].*They find a solution to a system of two (charged and neutral) interacting scalar fields (see eqs. (1.14)). The first equation is an analogue of the Klein—Gordon equation (1.5) and the second one is an analogue of the Higgs field equation. Extracting as usually the time dependence of the complex field, 42(r, t) =
s/i(r) e~’,
3=
(6.72)
we get from (1.14) the system of equations 22 V2 5/i—ax sfr+co 2 sfr=0, 2
22
1
(6.73a)
2
the soliton charge,
Q=
wJ
3x
~2
d
wS[5/’],
Fig. 6.10. Dissipative instability mode in the (~4_~ a~6)model. The soliton ‘sits” in the potential well (see fig. 6.9b). *
An analogous but more complex (spinor field) SLAC bag soliton has been considered by Bardeen et al. fl22}.
(6.74)
V.0. Makhankov. Dynamics of classical solitons
105
and its energy E=
f
~
d3x,
(6.75a)
where =
~{(V~)2+ (Vs/i)2 + ~(w2
+
a2x2)5/’2 + ~(x2
—
(6.75b)
1)2),
and a = rn/p. is the ratio of the charged ~pmeson mass to the neutral x meson mass. Eqs. (6.75) can be derived by a variational principle in the following three formulations: (i)
=
or
= 0,
=
0,
(6.76)
where L
=f2 d3x,
2
=
—~{(V~)2 + (Vs/c)2 +
The variations Os/s and O~are arbitrary and w (ii)
~
=
=
0,
or OEJQ
~(a2~2 —
=
w2)çb2 + ~(x2 l)2}.
(6.77)
—
const.;
(6.78)
= 0.
The variations Os/i and O~are also arbitrary but now co must be considered as a functional of s/i by using (6.74) =
(6.79)
Q/S[s/i].
The charge Q is fixed. From (6.75) it follows that (6.80) Finally, introducing a “reduced” energy ~w=~ü
and
E=f~rd3x~
(6.81)
we get as in the case (6.44b) -.
(6.82)
which means that OE/0~=0,
(6.83a)
OE/8s/i = w2çt’.
(6.83b)
and
Here as earlier w2 plays the role of a Lagrangian multiplier. From (6.83b) we have
~ E(s) = ~w2,
(6.84)
106
V.0. Makhankov, Dynamics of classical solitons
whence (6.85)
E=E—~wQ=—L+~wQ.
The variational principles (6.76), (6.78) and (6.82) are useful not only for soliton-type solutions but also for plane wave solutions (“free mesons”) if one considers the latter in a large but finite volume and sets periodic boundary conditions (see section 3 and [119]). Using the variational principle (6.76) we get the virial theorem by the scale a-transformation
f
~
(6.86)
where 2 + (Vsf,)2}, = ~{(V~k’) 1 2 22 2 1 2 = ~{(w — x )s/’ — ~(x —
a
(6.87a) 2
1)
}
From (6.86) and (6.87b) it follows that for co
(6.87b) = 0
the system (6.73) has no soliton solutions since
3. By using the method described in subsection 6.2 we can study the behaviour of soliton solutions as 1(2 = (1 w2/a2)—*0. Putting p = aicr and s/i = (K/a)42 we have from (6.73b) with an accuracy of —
(x2— t)x0.
The solution x = 0 is rejected since in that case s/i does not vanish at infinity as p solution x = I ± cy and have for it y = ±1(2422 or
—*
co•
We take the
22
X’~42~
(6.88)
Now the first equation of (6.73a) is also reduced to (see (6.19)) 2
3
V~~ 42—42+42=0. The behaviour of theUsing charge Q near zerothat K isthe determined by the 2 (see (6.63)). (6.25) we find soliton mass is relation Q ~ co/K, i.e., it diverges as a E = Qm(1 ~ O(ic~))=rQm + C/Q+~~~. (6.89) —~
We emphasize that as ic —*0 the soliton mass is greater than Qm. This value, as we shall see later, is the mass of “free mesons” in a box with total charge Q. In fact, let us place the system into a sufficiently large volume V. We suppose that the fields have periodic boundary conditions. Then from (6.73) it follows that one possible solution is a constant: ~=±~‘
~=_~(i_~).
Introducing the notation w = a cos E and using the normalization condition Q *
same.
(6.90) =
co5/2 V we get*
It is interesting to note that there are analogous solutions for “free mesons” also in the model (6.69) as V—*c~and formula (6.91) is exactly the
V.0. Makhankov, Dynamics of classical solitons
xcose,
~b(Q/coV)
5~fl~
1/2 ~
v2a E = Qm(sec )(1 — ~ sin2 ).
(sin2 ) cos = 2Qa/ V, For fixed Q we have in the limit V
a,
=
107
(6.91)
—* ~
E = Qrn.
(6.92)
A similar result was obtained by us also in the plane geometry for another models. In this (and only in this) sense the limit ic —*0 for sufficiently large Q can be considered as approaching the “free” field. In all the models considered above it reduces to the 424 theory. We estimate now the soliton mass for small to. For this purpose we use trial functions of the form x
—
—
1~—exp[—(r—
R)//1,
f(5/’~sin
r ~ R, r>—R.
wr)/r,
r~
9 (6. 3) 94 (6.
)
The form of these functions follows from the expansion (6.88) and formulae (6.90). The quantities I and R are parameters with the dimensions of length. Their values are taken from the condition of the minimum of the soliton energy. The condition* R = nIT/co
(6.95)
must be also satisfied. Here n is the mode number (n = I corresponds to a nodeless solution). From the normalization condition we have s// 0 (1/ir)V~j~. Inserting (6.93) and (6.94) into eq. (6.75) for large R and Q we get 3 + 0(R2). (6.96) Emin ~
+
R
Minimizing this expression we have Emin
~ ~1T2”4Q3”4,
(6.97)
which occurs for R
(2Q)”4.
(6.98)
The soliton mass is less than the mass of the “free” mesons in the box, if Emin < Qm,
or 14
Q>Q
4
(6.99)
5=m(s(irla)).
Hence for Q> Q5 the soliton is stable, at any rate with respect to a decay into free mesons. Finally 2). noteMoreover, that formulae are valid forand to =(6.99) (irIR)were ~ 1, when one under may drop (6.96) the since(6.96)—(6.99) the expressions (6.97) obtained the in assumption termsQ 0(R that ~ I, they are a very good approximation as a -40. In the paper [10] it was found that in the * The function ~ is then the exact solution of (6.73a), and
x has the correct asymptotic behaviour as
r-÷s~.
108
V.0. Makhankov, Dynamics of classical solitons
Fig. 6.11. Energy of soliton in the FLS model as function of the charge.
opposite limit, a
—*~,
Fig. 6.12. Soliton charge Q(w)
one has for Q5 the following formula
3,
Q5—= 111.8a i.e., Q 5 —*0 as a
as function of w.
(6.100)
we
~. More than that, for a > 5 have Q. < 1 and the quantum soliton for which Q must be an integer is stable for any Q 0.” The results obtained enable us now to answer the question about the stability of the soliton considered. As we saw above if there is one negative EV, A, <0, of the test Hamiltonian If the stability region of the solution coincides with the region dQ/dco <0. The existence of A, follows from the limiting case K—*0 when the problem is reduced to eq. (6.61) for which the test Hamiltonian (6.27) has one spherically symmetric state with negative energy [116] A, = —I5.6,c. The same reasons as in subsection 6.2 permit us to infer that when K increases, up to unity, the second negative eigenvalue of H does not appear (the same conclusion can be obtained by studying the conditional minimum of the functional E[5/i, xD. The behaviour of the function Q(co) as to —*0 and to known from (6.63), (6.98): as co—*0, Q(co) cx Q(w) ~ (a2—co2)”2, as co—*a. —*
—*
a is
4,
~
From the continuity of the function Q(w) it follows that for the whole of its descending branch the soliton is stable. The approximate behaviour of the functions E(Q), Q(w) and E(S) 110] are shown in figs. 6,11—6.13. In these figures the soliton stability and instability regions are also shown. The expressions for the second variation of the soliton energy (52E)( 0 and the test Hamiltonian H will be considered in the appendix. The details of a proof of the stability of the soliton under consideration (which is somewhat different from our proof) can be found in [10]. In this paper it is shown that with increasing node number (the mode number) the solitons become more unstable. By the method of ref. [18] the quantization of the soliton is carried out and a quantum correction to its mass is found. This correction has with an accuracy of 0(g°)the standard form (6.101) Here
~N
are the normal modes of the operator H2,
‘N
are the occupation numbers, the last term is the
vacuum energy. In this term the summation is extended over all the frequencies ~ of the solution in Here we may remain in the framework of the weak coupling approximation since the quantity a in (6.100) is determined by a relation *
between three constants: the masses of the charged and neutral mesons and the coupling constant.
109
V.0. Makhankov, Dynamics of classical solitons
Fig. 6.13. Plot of reduced energy E as function of S.
the form of “free mesons” (for the plane case see also [85]).The quantum mechanical corrections lead also to a situation when in the region Q~< Q < Q5, where the soliton energy is greater than Qm, the soliton is metastable because of acan barrier penetration since the latter isseeproportional 2) the soliton lifetime be quite long if g2effect. ~ I (gHowever is the coupling constant, subsection to exp(—1/g 3.6). Finally we note that the model permits an evident generalization. Let 2
=
—
8 ~*~9
—
~(D~x)2 —
a2X2!42!2
—
U(~),
(6.102)
where U(~)is an arbitrary polynomial of degree not higher than four in x~so that the theory remains renormalizable. Then if U has an absolute minimum at the point x = Xvac U(~)~ U(Xvac) 0,
0, putting
we have for the charged 42-meson mass m
=
and for the soliton energy density = ~{(Vx)2
+
(Vs/i)2 + (co2 + a2~2)+ 2’W(x)},
(6.103)
where W(x)
=
Hence Emin
~ IT~+ ~ir
U(0)R3 + O(R2g2),
and
g2=x~. Then we can consider two cases (i) U(0) 0, (ii) U(0) = 0,
Emin ~
Em,n~~ir(6Q/g)2’3,
where we have put U(~)= ~g2x2(xXvac)2 —
-
(6.104) (6.105)
110
V.0. Makhankov, Dynamics of classical solitons
and x
= (I +
e’~y’
as a trial function. For both cases the behaviour of the function Q(w) is analogous to fig. 6.12. The conditions of the stability theorem remain also useful, so that the left branch of the function Q(co) corresponds to a stable (in the classical sense) soliton. Finally we note that authors of [10] describe a numerical calculation for finding solutions of eqs. (6.73) and give the functions E(Q) and Q(w) obtained in that way.* For the case of a = I they have Q~= 3.03, and the point C corresponds to to 0.96. For a = 5 the value Q. is 0.21. 4. In concluding this subsection we give one more example of a stable localized spherically symmetric solution of the Higgs field equation [85]. In the simplest case when a uncharged field is an isovector and its direction coincides with radius-vector in any point of space (“hedgehog”) Q
5 = 3.47,
Xa
we have the equation 5’q(r) + i~(r) ~3(r) = 0, i5 = V~—21r2. .1 This equation has a solution with the asymptotic behaviour
(6.106)
—
? 7(r)=C/r,
asr—*0, as r-+~.
(6.107)
This solution is a bubble inside which the Higgs field vanishes. Because of the abnormal boundary2 conditions infinity which energy preventand the hindering bubble from expanding the rterm playing the at role of an (~(r)—* effective1) kinetic it from collapseand forof small such—277(r)1r a bubble is stable. This fact has been confirmed by a direct numerical calculation of Rastorguev at Dubna. We emphasize that the stability of the bubble does not contradict the Derrick—Hobart theorem (6.10) since the bubble is not the soliton of our definition since because of the boundary conditions at infinity and the term —(2/r2)’q(r) the total energy of the bubble is linearly divergent. The same considerations help to understand qualitatively the appearance of the stability region in the example of the two-field charged soliton in the Friedberg—Lee—Sirlin model: the Higgs field ~(r) prevents the soliton from spreading (the “dissipative” instability mode), the presence of the charge Q hinders its collapse (the “singular” mode) because of dQ/dco <0 which for small to leads to dQ/dR > 0, i.e., if Q = Q 0 is given the soliton radius cannot be less than some value R0 (see, e.g.. (6.98)). Hence the Higgs field realizes a kind of a surface tension and the charged field leads to an effective spatial repulsion and the stable soliton solution arises as a result of the balance of these forces. Finally we note that the Langmuir wave collapse considered in = section example of the an 2= — to2 1 —(1—5 is an = 2y~~~O when unstable “hedgehog” in a “non-relativistic” approximation it equation of vector 424 theory is valid. However, in contrast to (6.106) here we have ~)2
I.s/i(r)— 5/i(r)+ s/s~(r) 0, *
The fourth-order Runge—Kutta method was used.
(6.108)
V.0. Makhankoi. Dynamics of classical solitons
111
with normal boundary conditions at infinity, s/i(r)—*0 as r—*~and just this fact leads to the “hedgehog” instability. As we saw above, for scalar fields the soliton stability appears for sufficiently large it, i.e., for small to (“ultra-relativistic” approximation). Just in that region the saturation of the non-linearity and the presence of the “charge” S can lead to the appearance of stable (including the case of non-symmetric perturbations) soliton solutions of the plasma turbulence equations. Thus, for instance, for a model with a non-linearity in the form (6.63b), (6.109) 155/i(r) p.25/’ + s/s(l — e~2)= 0, —
it is easy to find from the virial theorem that —
p.2 ~ ~
J r2 dr(1
—
e~2)~ 0,
that is s/~,as~ 1,
and R 0 ~ const.
Numerical investigations [64] show that the constant is 4V2de. Let us show now that the presence of the charge and the saturation of the non-linearity lead in any case to a stopping of the collapse (i.e. to a suppression of the “singular” instability mode).* 2) M C/s/i2 as s/i ~, and then For this purpose we write as before (see (6.63a)) F(sft E = J r2 dr{~42rI2+ ~ ~ M~ço~2 + tn 42~2~} —*
—
—*
If the quantity S 3 = E + MS which is conserved in time is bounded for t = 0, the packet collapse is stopped. In fact the presence of the collapse means that for any in the interval 0< < S and arbitrary R we can find such an instant of time r starting with which the inequalities
Jr2
drI42(r,
)(2>
are satisfied, i.e., in any arbitrarily small spatial volume practically all the plasmons, which were contained in the initial packet, collect as a result of the collapse. Since S3=j
12 1 2 2 r dr~J42rI+~42I +IlnI42H 2
I
2
is the sum of only positive terms we have the estimate
~f
2 drI
r
2 ~~
42!
-~ ~
dr ~
~—
S 3,
2) =
whence it follows that as R
—~
0 the integral S3 cannot be bounded. For the rion-linearities F(1421
~For eq. (6.109) this result has been obtained in 1641.
112
V.0 5fakI~.~
Dynamics of classical solitons
ko~I(1+ I42I2),~and(6.109) one can obtain exact expressions for S3: S3 =
f
2
dT{142r12 +~
IøI~+ ln(1 + 14212)},
r
and s 3= Jr2dr{l42rl2+~I42l2+(l_e~2)}.
However, from this discussion the longitudinal stability of the “hedgehog” with zero asymptotic behaviour at infinity does not yet follow and even less the “transverse” stability. Nevertheless by an analogy with scalar solitons [1161one can expect that in any case the ground state (a nodeless one if one does not consider the origin) may be stable with respect to non-symmetric perturbations. The unformed two-scaled soliton is unstable with respect to the non-symmetric mode, as follows, for example, from section 4 and numerical calculations [92].
7. Long-lived pulsating solitons We have considered so far non-stationary solutions of non-linear wave equations. There were bound solitons of the sine-Gordon equation—bions, and quasi-stationary bound solitons of the Klein—Gordon and Higgs equations. All of these one-space dimensional solutions we call bions. The last two equations are not completely integrable and the life-time of their bions may be very long, though finite. Therefore the bions of the KG3 and LGH equations are only approximate solutions. There naturally arises a question whether there exist in the real three-space dimensional world certain analogues of the bions. We consider below non-stationary spherically symmetric solutions in the frame-work of the KG3, the sine-Gordon and the GLH equations and show there really exist long-lived pulsating solitons both of large and small amplitudes, which are something like three-space dimensional bions. The chief results have been here obtained by a computer and only in the small-amplitude approximation was it possible to construct a perturbation series for the solutions in a sufficiently consistent way and thus to find analytical solutions in analogy with the one-space dimensional problems (see section 3). 7.1. Meson bubble life-time Let us first consider as an example the evolution of a quasi-plane initial packet x = tanh[(r—R0)/V2], R0>> 1 in the frame-work of the Higgs equation [111,112]. Such a bubble may be regarded as a gluon-type meson without quark—antiquark pairs (for the physical sense of this model see for example [122, 112]. As we saw earlier (see subsection 5.7) the bubble starts to collapse to the centre.* Near the centre a reflection takes place as a result of which the compression is replaced by an *
When at the initial time we have a packet of sufficiently general form
x(r,
R0), but R5’> 1,
with the necessary boundary conditions at the
origin and at infinity, at first the self-consistency of the packet occurs as in the case of the CLW spheriton and then the packet takes the form of a quasi-planar soliton.
V.0. Makhankor. Dynamics of classical solitons
113
130 L
160
~IL ~
10
-
~ t~24
20
~‘III
~T~1C
70
-0.21
,‘
t~22B
~
20
Fig. 7.1. Evolution of a “meson” bubble in the LGF1 model for
R0= 10.
expansion. Because of the law of conservation of energy the expansion can happen until the radius R1 ~ R0, then compression starts again. The picture resembles strongly that of a conventional oscillator. It is clear that the lifetime of such a quasi-planar bubble is completely determined by the radiation of energy to infinity. In the plane world considered above the analogous oscillations of the field correspond to the bions which have a rather long lifetime. Is there the same situation in the real world? Unfortunately one cannot estimate analytically the energy emitted and the bubble lifetime which is determined by this energy. For this purpose Bogolubsky and the author have performed.a series of computer experiments [126]. 2XtXr emitted through a sphere of radius WeR have calculated the total flux of energy Q(t) = ~T rm ~‘ 0 and found the function E(t) and the distribution ~(r, t) (see (5.64)): E(t)= J~rdr~
~
l)2}.
It turns out that for different values of the initial radius the bubble emits over half of its initial energy after only 1 to 3 pulsations. In this process there is both the emission in the form of small-amplitude waves near the vacuum as defined by the boundary conditions and also a fractionation of the field energy into separate spherical layers, some of which move to the boundary rb and leave the domain r> ~,.The computations have been performed for Rc = 5, 7, 8, 10 and 15. In fig. 7.1 we show a typical picture of the bubble evolution (R0 = 10). Fig. 7.2 gives the dependence E(t) for several values of R0. From these curves one can see that the bubble lifetime is not greater than a few (3 to 5) times its radius R0. However, in this figure one can note that as a result of evolution of two bubbles with sufficiently large energies (R0 = 8, 10), identical long-lived one-scale states appear for which the
114
V.0. Makhankov. Dynamics of classical solitons
E 100
lao
200
300
Fig. 7.2. Bubble energy as a function of t for various initial radii R
0.
thickness of the transition layer is of the order of their radius. Both the energy and the field function are concentrated near the centre; moreover, in fact the energy distribution does not change with time and the field function oscillates around the vacuum fixed by the boundary conditions. Naturally we call such solutions “pulsons” (do not confuse with pulsars!). Below we discuss in more detail the pulson properties. Such studies have been performed in order to investigate the bubble evolution in the sine-Gordon model Xrt~rX+s11lX0.
(7.1)
The initial packets were chosen in the form of a quasi-planar solution x
=
4 arctan[exp(r
—
R0)],
R0 ~ 1.
(7.2)
Eq. (7.1) is completely integrable, does not permit stochastization or emission when solitons interact, and has bions as exact solutions in the plane world. Sometimes the hopes are expressed that these unique properties of completely integrable equations must somehow arise in more real world of two and three space dimensions [801.The authors of the paper [126] set their mind on testing whether these properties might possibly occur in the evolution of the bubble (7.2). In particular, one might assume the existence of SS analogues of the bions and, as a result, of infinite oscillations of the bubble. The computation for R0 12 shows a powerful emission already after the first reflection of the bubble from the centre. The picture of the evolution coincides qualitatively with one depicted in fig. 7.1 up to the formation of long-lived pulsons. Thus in this computation we do not observe any unusual properties of the sine-Gordon equation in the physical world. In the next two sections we study the properties of the pulsons of the KG3, the sine-Gordon and the Higgs equation on the basis of papers by Bogolubsky and the author [127—128]. 7.2.
Pulsons of the KG3 equation
Let us limit ourselves to the real scalar field 3=o. xo—V~x+x—x
(7.3)
115
V.0. Makhankov. Dynamics of classical solitons
As we saw earlier this equation has non-trivial stationary solutions, the plane and the spherically symmetric solitons. However, both those are unstable. On the other hand, eq. (7.3) in the plane geometry has rather stable (in any case in the longitudinal direction) localized oscillating solutions, the plane pulsons [82]. Using the method of [82] for small amplitudes and a computer we study the SS pulsons of eq. (7.3). Let ~(r, t) be given by x(r, t)
=
a(r) cos tot + b(r) cos 3wt ~
.
b ~ a.
(7.4)
Inserting (7.4) into (7.3) we get the well-known eigenvalues problem arr+~ar_K2a+~a30,
a~(0)=0,
K2= 1—to2,
a(oo)=0,
or, if A2 = ~it2a2, we have (7.5)
Arr+~ArA+A~0.
The solutions A~(r)of (7.5) are numbered as before so that A~has (n
—
I) nodes, thus
A 1(0)
4.34< A2(0)
14.10< A3(0)
29.13<~~ . <~4(~) <...
In the case of small amplitudes xm we have
2t)xrn A~(a~
x~(r,t) = V~x0A~(a~0r) cos (Vi~~
0r) cos(\/1 as a solution of eq. (7.3). The function b(r) in the same limit is b(r)=-
~X~A~(XOT).
2t), —
a
=
1,
(7.6)
X0
(7.7)
The function (7.6) for the first three modes (n = 1,2,3) has been investigated by a computer for the amplitudes x m = 0.2, 0.4 and 0.7. At xm ~ 0.4 the expression (7.6) is the solution of eq. (7.3) with high accuracy (errors are less than 1%). We note that for sufficiently small amplitudes x~‘~1 the radiation by the pulson is very slight and its lifetime r—~~ as xm—*ø. If in eq. (7.6) xm = 0.7, then the pulson amplitude Xm(t) slowly decreases to Xm(t) = 0.63 (at t = 80), and the characteristic radius grows. The stability of the solution (7.6) is tested by the variation of the parameter a. Values of a < I correspond to a broader initial field bunch with the same amplitude and frequency and values of a> 1 correspond to a narrower one. In the first case, the initial bunch has a greater potential energy than the “equilibrium” one (a = 1), in the second case it has a smaller potential energy. It is natural that the initial evolution of both bunches is determined by this deficit of the potential energy. The positive deficit leads to a compression of the bunch, the negative one to an expansion of it. This fact is confirmed by computer calculations; however, the evolution does not stop at the equilibrium value of a I and goes further. Moreover the first bunch is unstable with respect to a singular mode and the second one with respect to a dissipative mode. These instability modes are both inherent to the charged Q-soliton of eq. (7.3) [116]and were described in subsection 6.3.* Compare the growth rate of -~
* Note that although the solution (7.6) is the real part of the solution obtained by Anderson and Derrick 1116] for the charged bunch, there is a qualitative difference between them. The second one is gauge invariant and conserves strongly the charge Q.
116
V.G. Makhankov. Dynamics of classical solitons
i..L(0,t~
0.2~
(.0-)
~
20
t
~
~7
~
Fig. 7.3. Time-dependence of the envelope of the oscillation amplitude of the field function in the origin (K03 model). (a) a < l (dissipative mode), (b) a> I (singular mode).
Fig. 7.4. Singular mode of the Q-soliton instability in AndersonDerrick’s theory. is the soliton energy density.
both the instability modes for the pulson (7.6) [127](see fig. 7.3) and the Anderson—Derrick Q-soliton (see figs. 11, 12 of ref. [116]).It is easy to do it for the2 =singular mode. From figs. 7.3b and 7.4 we find 4.34 Vi —0.64 2.6. Bearing in mind that the that tp 40, Xp(O, 0) = 0.4; tQ 2.7, XQ = 4.34VI w’ instability growth rate (1/t) cx x2 we have tQ/tp = x~/x~ 0.024. Numerical calculations give tQ/tp 0.067, i.e., these values are of the same order of magnitude. The difference of a factor 2 is either connected with the rough estimate, or it indicates the larger stability of the Q-soliton as compared to the pulson stability. Analogous estimates can be obtained for the dissipative mode. We find the pulson energy —
J
E = 4’z~ r2 dr =
4irJ ~‘dr
= 2irJ
(x~+x~+x2—~x4)r2 dr
(7.8)
for small x
1. We note that because of the scale transformation x(r, Xo) t’°(x~)I~(x~) + I~(x~) = x~(I~’° I~)+ 5(n) + Xo = I~”~(x~) + S m~
—
~
—
=
~0y(~0r) we have
—
where j(n) =
J (~)2r2
2~
dr,
J y~r2dr,
I~ =
S
J
2~ y~r2dr,
I~=2~Jx~r2dr, and
n
is
the ~~1S°’1
mode +
number.
J
i~”~2~ ~
In
the
limit
0)(cos2 tot +
to2
as
Xo~°
sin2 wt)r2 dr
we get
117
V.6. Makhankov, Dynamics of classical solitons
I
Fig. 7.5. The field function
2
x~and
the energy of a spherical layer ~‘,as functions of the radius for the first three modes, n
with an accuracy of O(~~). Hence because E1~~2~J ~
3
&2 =
= 1,2,3.
1 — x2~,we see that
0)r2 dr,
(7.9)
and that the energy density LW,, is independent of time. The functions ~,, and x~for n = 1,2, 3 are represented in fig. 7.5. It is interesting to note that for the same values of x m ~1:2:3:4:9: 1 the pulson masses with the number n are and 5 approximately in the ratios proportional to S~/A~,° 7.3. Pulsons of the sine-Gordon and Landau—Ginzburg—Higgs (LGH) equations 1. Let us consider the properties of one-scale field bunches which are formed as a result of the evolution of “mesonic” bubbles (see section 7.1) in the framework of eq. (7.1), Xtt—~rX+smnX0. Three characteristic stages of the evolution of the pulsons of this equation are observed in computer experiments for the initial data in the form of the quasi-planar bion 1K coswtl ~(t0)=4arctan~— ~, t.w cosh KTJ
/
w= vl—ic
2
(7.10)
,
for
K/to = 10 or to In the first stage (t 0 to 200) the formation of a one-scale bunch with a bell-shaped form of the oscillating function x(r, t) occurs. Then, more than half of the initial bunch energy, E = J 4irr2 dr(r)
=
21T
J (x~+ x~± 2(1
—
cos x))r2 dr,
radiates away (see fig. 7.6 and compare with the bubble (7.2)). Practically from the onset formation process we see a quasi-periodic behaviour of the function x(r, t) with a period T
(7.11)
of 1
after that the period
the 7.5;
slowly increases together with a decrease of the pulsation amplitude C(t) (C(t) is
the envelope of the function Ix(0, t)I). In the second stage (t = 200 to 700) a weakly radiating pulson with an amplitude C(t) modulated by a period T2 10 arises, the amplitude slowly decreases from C(t) 2ir to C(t) ir (see fig. 7.7).
118
V.6. Makhankov, Dynamics of classical so/lions
~00
of Fig.t. 7.6.
Pulson
1000
energy for the sine-Gordon equation as
—
,~nction -
0t)I inEnvelope SCln~) WO]nt-i) Fig. y(0, 7.7. the sine-Gordon p~ricd of the oscillation o~er model r’.”O,l,S-,3... pulson. amplitude of the field function
The field oscillations are practically symmetric with respect to the zero vacuum x~= small change in C(t) in the period T2.
0
because of the
One of the very important facts is that the evolution of such a pulson from the second stage really does not depend on the choice of the initial data ~(R, 0). The fact of the formation of a long-lived heavy pulson of eq. (7.1) from various field bunches indicates its stability for i~~ C(t) ~ 2~.On the other hand, the same results permit to state with a large degree of certainty that even within the framework of the sine-Gordon equation, which is completely integrable in the plane geometry, in the spherically symmetric case there are no exact solutions that are analogous to the plane multi-soliton ones (in particular to the bions). When the amplitude C(t) is equal to C2 ~ir (t = 630) the third stage of the pulson evolution starts. A relatively rapid (~t 80) decrease of C(t) down to C3 4 1 takes place accompanied by a “swelling” of the field bunch. Then again a state with a weak radiation arises. An analytical description small amplitude pulsonsection. C(t)4 The I is principal obtainedterm by the userequired of the 3and of the amethod of the preceding of the expansion sin x x ~x solution can be expressed by the formula —
—
~
t)= \/~~ 0A~(~0r)cos(\/l —X~t),
(7.12)
where A~is the set of solutions of eq. (7.5). 2. The dynamics of the pulsons eq. (5.63) 3 =0, xtt — V~~xx + x for the Higgs field turns out to be very similar to the above one. We note only characteristic differences. Firstly, the oscillations are non-symmetric with respect to the vacuum x~= —l selected for the calculations. The form of the quasi-periodic function x(O~t) (its period gradually decreases from T 5.3 to T V2 1r) even in the second regular stage (t 100 to 1350) differs from a sinusoidal one. At once after the formation of a weakly radiating heavy pulson of the Higgs equation, the modulation amplitude of the function C~0(t)(the upper envelope of the field function x(O~t); because of a non-symmetricity of x it is worth-while to follow its upper and lower envelopes C~0(t)and Cd(t)) is relatively large (see fig. 7.8). When C~0(t)approaches to zero the modulation amplitude tends to zero and the modulation period monotonically increases from TM = 8T at t 10 to TM 17T at 1200. In the third stage, as in the case of the sine-Gordon equation, the heavy pulson becomes unstable. A rather rapid decrease of its amplitude C~~(t) from zero at t 1360 to C~0(t) —0.8 —
119
V.0. Makhanko r. Dynamics of classical solitons
1DOr~
~
~
~O,1,2,3.
jx(O, t)~in
Fig. 7.8. Upper envelope of the oscillation amplitude of field function
the LGH model pulson.
(t
1650) takes place. At this time the oscillations become symmetrical with respect to the vacuum —1 selected for the calculations. Thus the lifetime of the heavy pulson of the Higgs scalar field is more than two times larger than the analogous sine-Gordon equation pulson lifetime. Finally, in the last stage one can describe a Higgs SS-pulson with a small amplitude of oscillations around the vacuum x~= —1 by writing the field function ~(r, t)= u(r, t)— 1, u 41 in the form of power series [85]: =
u(r, t) =
2’~’f
2g 1(r) +
~[e
2”~2g 2~~1(r) sin[(2n
+
l)wtj +
2~~2(r) cos[(2n
+
2)wtlJ,
(7.13)
and making the substitution r = \/~t/(I +
~2)I/2
~ = \[2r/(I
+
~2)l/2
As a result we have for the function A(~)= V~7~f1(~) again eq. (7.5). By making the appropriate substitutions we get the algebraic relations 32
g1=—-4ft,
g2—4f~
(7.14)
With an accuracy of O(E2) we have 2A~(~)(l + cos 2wt),
u(r, t) = ~= A~(~) sin tot —
to2 = 2/(1
+ ~2)
(7.15)
It is easy to check that the masses of the pulsons (7.15), M~ En with different values of n and with the same amplitudes Urn = (2/V3)A~(0),are in the limit Urn 4 1 in the same ratios as S~/A~(0) (~1:2:3:4:9:...). The distribution ~‘(r) having the sense of energy of a unit spherical layer of radius r and the energy density e(r) for small amplitude (x m 4 1, Urn 4 1) pulsons are both stationary; this directly follows from the solutions (7.6) and (7.15) (see also subsection 7.2).
120
V.0. Makhankov, Dynamics of classical solitons
V 0,927 t~370
ft
_____
C~J
~‘
‘7.
Fig. 7.9. Distribution of energy density in a heavy poIson of the LGH model at two times. The distribution is seen to be non-stationary and to resemble the energy distribution in a plane bion.
~
Fig. 7.10. Picture showing qualitatively the evolution of a pulson in the LGH model. The ball dimension is associated with its mass. U~ is the potential energy of the poIson as a function of the field amplitude (and hence mass). The vacuum q~= — I is fixed by a boundary condition.
In this context, heavy pulsons with large amplitudes differ qualitatively from light ones by the essential non-stationarity of ~(r, t) and (r, t) (fig. 7.9). The lifetime of the (7.16) and (7.15) solitons increases with decreasing oscillation amplitude. We also note that solutions in the form of light pulsons for all three types of equations correspond to the so-called near free-field approximation it2 = m2 — w2-÷0. As in the case of the FLS stationary Q-solitons [10] it is natural to expect in the region of large amplitudes the existence, besides the nodeless pulsons, of long-lived heavy pulsons with nodes of the field function both for the sine-Gordon and for the Higgs equations (proceeding to the limit of the functions A~for small oscillation amplitudes). In the meanwhile an analytical description of the heavy pulsons has not been found, and the solutions (7.16) and (7.15) are obviously unsatisfactory for large amplitudes (see figs. 7.7—7,9). Finally we note an interesting feature of the behaviour of the heavy pulsons. It is remarkable that in the framework of the essentially different eqs. (7.1) and (5.65) the massive pulsons are “stable” for amplitudes close to the upper vacuum x,~ (which is distant from the “main” vacuum Xo selected by the boundary conditions). These are IT ~ C~ 21T for the sine-Gordon equation and 0 ~ C~~(t) ~ I 0(t)E of the amplitudes, [0, ir] and [—1,0],respectively, for the Higgs equation. In the intermediate domains the heavy pulsons “lose” their stability and rapidly “fall down” to the region of small oscillation amplitudes near the main vacuum ,v~.This effect can be apparently explained by a prevailing influence (if one wants by the “attraction”) of one of the vacuums. It is schematically illustrated in fig. 7.10 (the dimension of the ball is related to the oscillation amplitude and thus to the pulson mass). It is seen that in the “stability” region the pulson, because of its radiation, must overcome a potential barrier and this considerably increases its lifetime. More work on an elucidation of stability problems for both heavy and light pulsons is carried out now. For the present it can certainly be asserted that the Higgs equation pulsons turn out to be stable with respect to non-symmetrical excitations (the results of these and some other two-dimensional computations performed at Dubna by Bogolubsky, Shvachka and the author will be published elsewhere). Conclusions and acknowledgements Possible development of theory 1. Plasma. Construction of theory of Langmuir turbulence on the basis of “collapsons” as a phase
V.G. Makhankov, Dynamics of classical solitons
121
transition in the plasmon gas near a phase equilibrium point (the liquid phase consists of the collapsed plasmon “droplets”). In this respect it is necessary to study: a) the stability of Langmuir “hedgehogs” for equations with a saturable non-linearity, b) elementary processes of their interaction, even if only for cylindrical geometry. For the first part some arguments raising hopes was given in the text. In the work on the second problem numerical experiments are the most important. It seems to us that efforts in this direction will not perish to no purpose (remember the achievements accomplished in this way in the physics of large dimension excitons [132] and, in particular, the observation of exciton droplets). 2. In particle theory the soliton use for physical applications just begins. It is apparently connected with the fact that, firstly, even in the study of “plane” solitons a new still enigmatic world of a great number of phenomena is revealed, which is interesting by itself and attracts investigators. Secondly, so far one does not know enough about solitons and their behaviour in the real four-dimensional world. The most interesting problem, namely, the soliton interaction still remains terra incognita. However, an interest in this problem arising recently gives grounds for optimism. The first that comes to mind is the soliton form-factors and their comparison with experiment for the extraction of the models with most perspective [19,133]. The second one is the unique properties of QS as coherent states consisting in a strong suppression of a decay channel connected with the total dissociation of a QS into constituents as compared with the transition soliton—~soliton + constituents. Remember the behaviour of psions which can be apparently explained by soliton models without charm [134,135]. The third one is attempts to explain the quark confinement by soliton bags [122,123] and other models [18]. Finally, why can black holes not be solitons [136]and why do laser beams in plasmas not show the wave-particle duality [137]. In fact the transparency of active materials increases by five orders of magnitude when solitons propagate through them [131] and the first soliton had been observed experimentally [2]. The author is extremely grateful to his colleagues for valuable discussions and criticism. Special thanks are due to Prof. I.S. Shapiro for helpful discussions concerning the application of soliton theory to particle physics and for actual initiating our work on three-dimensional pulsons and to Prof. D.V. Shirkov for discussions stimulating our work in the direction of the study of models which are transient between q,” and sine-Gordon theories. The author also wishes to thank Professors K. Nishikawa and H. Flaschka and Doctors B. Fried and R. Shukla for sending many papers which were very useful in writing this review. We would like to express our appreciation to Professors M.G. Meshcheryakov and E.P. Zhidkov for their encouragement, to Dr. I.L. Bogolubsky for reading the manuscript and some remarks and to Prof. V.K. Fedyanin for many valuable discussions and suggestions. Much help by G.A. Makhankova in preparing the manuscript and by A.F. Rubtsova and Yu. V. Katyshev in the translation is also appreciated. Finally, the author would like to express his sincere thanks to Prof. D. ter Haar, since due to correspondence with him the idea of writing this paper was born. We wish to thankfully acknowledge Prof. R.G. Salukvadze and the Sukhumi Technological Institute (where a large part of the review was written) for their hospitality.
122
V.0. Makhankov, Dynamics of classical so)lions
Appendix 1 Let us look for a soliton-type solution of the system i~,+
xc°
~ —
0,
i3~(x2)~~
— 132Xxxft
in the form =
çfi(x
x = x(x
exp[i(~vx —
—
Vt)
—
Vt).
Introducing A
= ci
v2/4,
—
~= x
—
and (3
vt
=
v2(3 2,
we get
2X + f3~~2 = ~2 (A.1.1) 7 Let further x = —A sech2(a~)(the solution of the uniform Bq equation), then the first equation (A.1.1) is +
+
(A + A sech2 a~,)t/i=
Introducing the variable
~=
0.
tanh ~, we find (A.1.2)
+
where Aa2 (n
=
= 0, 1,2,3,. =
$~(l— ~2)A2+2$a2(1 — 3~2)A=
—(1-- ~
(A.l.3)
s(s + 1), r— V~/a. Solutions çl’(~)bounded at ~= ±1 (~= ±~) are found for s — r = n . .), and eq. (A.l.3) is satisfied if n =4~~s = 2. In this case from (A.1.2) we get
B~(i
—
B = const.
~2)(s~~J)/2
Eq. (A.1.3) gives the condition imposed on the constants A and B: (f3~+ /0A2 = B2,
— 2A 7
+ /3
2+~/3A2=
0,
1A
or B2
(13~+
~A2,
A
2(13
+~
(A.l,4)
Finally, =
B(tanh a~)sech a~,
~
x
=
—A sech2 a~,
A=Af6.
(A.1.5)
(A.I.6)
Analogous formulae can be obtained for the case of WS-solitons (the plus sign of the term which is proportional to x in the first equation of (A.1.1)). The solution exists for B2
((3k
—
f3)A2,
A
=
2),
~ = A sech2 a~
(A.1.7)
V.G. ‘lakhankot. Dynamics
the other formulae are the same. Inserting (3
= /3
of classical
and ~
=
so/thins
123
into (A.l.4) and (A.l.6) we have the relation
(2.41): A
=
31 —— 20 y
A model with the IBq instead of the Bq equation leads to
s=’
A=_~(10y2_1)_1.
by (B2/ôt2)~2we get Replacing (8~/~x2)~2
A
—~-(l+9v2)~ty2.
=
Finally, for (a2/ot2)x2 and (i94/9t28X2)~we have A
=
31 — ~—
—~--~.
It follows from these formulae that Acr is very sensitive to the model when
~2=
1
Appendix 2 We generalize the theorem on the stability of a charged stationary soliton to the case of arbitrary potential U(2~(2). 2
2
1
2
~II —~V~ —~U(2l~(), ~r=112+ V~2+~U(2(~(2). “~‘
For i/i e”°’
we have Q =
wS[~i]
~
f
~/,2
d3x,
(A.2.1)
E = ~ J{to2~~2 + (V~)2+ U(~i2)}d3x.
(A.2.2)
The second variation 62E for fixed (A.2.i) is easily obtained (see the main text) 52E =
d3x +
31(II~’Ô~/I
~J-(f
1frSÔ~/Id3x)}~
(A.2.3)
where =
and
i/i.
—v2— ~2 + 8(*2)!+
2~ 8~tp~2l
is the soliton solution of the equation, +
—
~,fr~F(t,fr~) =0.
(A.2.4)
124
V.G. Makhankov. Dynamics of classical solitons
Differentiating (A.2.4) it is easy to check that ~/i~ = ~ is the eigenfunction of the operator H with A = 0 (the translational mode). It means that there is a p-state i/ri = (3~~/’~)Y,m(1 = 1) with zero energy and hence there is at least one spherically symmetric s-state with A <0. Differentiating (A.2.4) with respect to to and proceeding as in the main text, we find V ~ 2fA
~j
—
I
+
.~
d
Q
and =
A.8 11
~
Therefore t = 0; g(fl) w~~ t~41(fl— A1) hence as before we have the soliton instability condition dQ/dw>0, independent of the form of the potential U. For the determination of the stability region it is necessary to investigate the behaviour of the solutions ~fr. as functions of to in order to make sure in the uniqueness of the s-state with negative energy.
Appendix 3 The second variation of the FLS soliton energy can be easily obtained if one puts x the column functions
=
x~+ &~and
= ~/i~ + ö~i. Introducing
‘=(~)
~=(~)~
(A.3.l)
we get (52E)
3x +
~{J~
0
d
~k~j-~ [J ~ d~x]}~
(A.3.2)
where —
(10
°)v2 + 1
(a2c1i~ ~ 2aX
—
1)
2a2~s~frs)
5lJJ. and ~3 is the transpose of i~. Then as in the main text, = 3,,,i~,
=
A1y1,
q= ~
~
~‘ ~/A1 =
.F~’i~ =
~—
2w~,
(Q/to),
(A.3.3)
V.0. Makhankor, Dynamics of classical solions
125
whence T0
=
A.811 + ~J— ~,
(A.3.4)
and g(fl)
~J—~(ci ~‘
—
A1)’
—
I
=
0.
(A.3.5)
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DYNAMICS OF CLASSICAL SOLITONS (IN NON-INTEGRABLE SYSTEMS) V.G. MAKHANKOV Joint Institute for Nuclear Research. Moscow, U.S.S.R
Received February 1977
Contents: Introduction I. General properties of solitons 1.1. Variational principle 1.2. Equations having soliton solutions 1.3. Quasi-particle interpretation of HF solitons 1.4. Solitons and the Fermi—Pasta—Ulam problem 2. Solitons in plasma physics 2.1. Equations describing non-linear phenomena in isotropic and magnetized plasmas and their soliton type solutions 2.2. Boussinesq equation and the stochastization problem 2.3. KdV equation and its ‘improved” version IKdV 2.4. Schrodinger equation with a potential satisfying a non-linear wave equation 2.5. Coupled Langmuir and ion-sound waves in the nearsonic region. 2.6. Schrodinger equation with a saturating non-linearity. 2.7. Soliton-like solutions of equations describing excitons in one-dimensional molecular crystals 3. Solitons in particle physics 3.1. General remarks 3.2. Solitons in a g~field theory 3.3. Properties of plane solitons due to their charge 3.4. Bound states of solitons 3.5. Approximate analytical description of bound solitonstates in non-integrable systems 3.6. Quantization of solitons
3 5 5 5 8 10 11
II 13 17 19 29 33 35 38 38 39 44 47 52 54
4. Stability of one-dimensional solitons 4.1. Some general remarks 4.2. The longitudinal stability of plane solitons 4.3. Stability of solitons in the direction perpendicular to their motion (the transverse stability) 4.4. A qualitative discussion of soliton stability 5. More-dimensional solitons 5.1. One exactly soluble two-space-dimensional problem 5.2. More-dimensional solitons in the framework of the S3 equation 5.3. Collapse of Langmuir waves (CLW) 5.4. The CLW in the dynamical model 5.5. Dipole CLW 5.6. Spheriton collapse in the system of coupled Schrodinger and Boussinesq equations 5.7. One more example of an SS-packet collapse 6. Stationary solitons 6.1. The stability of more-dimensional solitons 6.2. Some properties of spherically symmetrical solitons 6.3. Examples of stable SS-solitons 7. Long-lived pulsating solitons 7.1. Meson bubble life-time 7.2. Pulsons of the KG3 equation 7.3. Pulsons of the sine-Gordon and Landau—Ginzburg— Higgs (LGH) equations Conclusions and acknowledgements Appendices References
56 57 59 68 70 70 74 75 _19 83 86 88 88 88 99 102 112 112 14
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DYNAMICS OF CLASSICAL SOLITONS (IN NON-INTEGRABLE SYSTEMS)
V.G. MAKHANKOV Joint Institute for Nuclear Research, Moscow, U.S.S.R.
NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM
V.G. Makhankov, Dynamics of classical solitons
3
Abstract: A survey of the properties of soliton type solutions to non-linear wave equations appearing in various fields of physics is given. For the most part the description deals with non-integrable systems and remains at the classical level. Some general features associated with ~solitons— stationary states, bound states of solitons, stability, the Fermi—Pasta—Ulam problem, and so on are discussed for both one- and more-dimensional worlds. The results of computer experiments on the dynamics of the formation and interaction (in one-space-dimensional geometry) of soliton-type objects are presented at length. As an application of the general theory we speculate on solitons in plasmas and particle physics. 4 and sine-Gordon theories. Finally, new, spherically symmetric, oscillating solutions (“pulsons”) are examined in the frame-work of (p
Introduction A whole class of non-linear differential equations admits solutions in the form of so-called solitary waves. Although such solutions were obtained fairly long ago (in hydrodynamics more than a hundred years ago [1,2]) the interest in them began to arise only in the middle of our century. This fact is apparently connected with the circumstance that the solitary waves being some very peculiar class of particular solutions to non-linear differential equations were supposed to be a rare phenomenon. Moreover, the non-linearity which is responsible for the formation of the solitary waves, on the other hand, looked like automatically leading to their destruction after an interaction. In fact, there was a hypothesis that even the presence of weak non-linearities in a system always leads to an irreversible redistribution of energy among the degrees of freedom (the ergodicity hypothesis). This hypothesis was checked in a numerical experiment in the 50’s with an example of a system approximating a piece of non-linear string [31.The above-mentioned ergodization was not observed. An analogous result was obtained [4] for the Korteweg—de Vries (KdV) equation. In the problems considered non-linear oscillations arose consisting of the redistribution of energy among several harmonics near the excited one. Then the system periodically returns to the practically initial state (fig. 1.1). This result, being on the face of it paradoxical, or more precisely the attempt to understand it, is called the Fermi—Pasta—Ulam (FPU) problem. The existence by itself of the FPU problem suggests that non-linear interactions are not sure to lead to the destruction (randomization) of oscillations and therefore the solitary waves can be conserved after the interaction. Just such solitary waves are called “solitons” in the American literature [4,5]. However we do not adhere to such a terminology and understand by a soliton a solitary w~iv~having some “safety factor”, i.e., weakly changing in the interaction (see details below). In the review paper [5] sufficiently clear definitions u~ sunning and solitary waves and a class of wave equations having such solutions are given. Also various analytical rnethoc1~for the study of the behaviour of these solutions after their interaction are discussed and an imposing bibliography is given. Moreover, the main attention is paid to equations describing completely integrabie i~miltonian systems. The whole variety of physical phenomena which lead to equations permitting soliton solutions cali be divided into two different classes. They are distinguished both by the problems and by the interpretation of the results, although they come together at the classical level. The problems of the investigation of non-linear wave phenomena in real continuous media (hydrodynamics, solids, and so on) belong to the first class and field theoretic problems and particle physics belong to the second class.
4
V.G. Makhankor, Dynamics of classical so!itons
Fig. 1.1. Non-linear oscillations of the energy distribution over the spectrum in the FPU problem.
In the first case the study usually goes through the following stages: it starts at the classical or quantum discrete level then with some degree of rigour the investigators proceed to the classical or quasi-classical continuum level which determines the form of the resulting equations. The final result of such a transformation is precisely the classical soliton solution (to which through a reverse transformation one can in principle give a quantum meaning). In the second case the models under investigation are constructed on the basis of a Lagrangian formalism and some requirements following from general physical laws (relativistic invariance, symmetries, and so on). In this case classical wave equations appear in the initial stage and are the basis for the construction of real quantum objects. Moreover, unlike conventional atomic physics, the quantum soliton properties are conditioned by classical solutions as h-+0. All this determines the form of the second class of equations. These directions come together in the intermediate stage of the investigation of the soliton solutions of the classical wave equations. Therefore it is interesting to study the general properties of the classical solitons (CS), to some extent forgetting the physical interpretation. As we shall see below the existence of stable solitons in a four-dimensional one-field model with a so-called saturable non-linearity is possible. Although undoubtedly of interest within the first class, they might be rejected for the second class theories because of renormalizability conditions. Below we consider non-linear phenomena in systems allowing stochastization, i.e., in non-integrable systems. The behavior of such systems can be determined by the degree of their proximity to some completely integrable analogies; moreover, different properties are determined by the proximity to different integrable models. In this sense the integrable models can be considered as a zeroth (non-linear!) approximation to the description of real physical systems and further investigations can be performed as a perturbation series in this small deviation. Note that at present only completely integrable models can be strictly analytically investigated by various methods (we have in mind the Cauchy problem solution). As a rule, analytical methods are practically helpless (at least at the present) in the study of the evolution of non-integrable systems. Therefore, with rare exception, all the results on the evolution of ergodic (even one-dimensional) and more-dimensional systems were obtained by computer experiments.
V.G. Makhankov, Dynamics of classical solitons
5
All this makes purposeless to discuss here the ideas and results described in the review paper by Scott et al. [5].If we mention them, it is only when this is necessary for a clearer understanding of the phenomena under consideration. This review is addressed not to the experts but may be thought of as a basis to proceed further. Therefore we try to give a qualitative picture rather than to develop rigorous approaches.
1. General properties of solitons LI. Variational principle The overwhelming majority of equations having siliton solutions can be obtained as a solution of some extremum problem in field theory. Let ~(x, t) be the field looked for. Introducing the Lagrangian density ~ p~, ~), determining, as usual, the action S as the functional* ~
(1.1)
and putting the first variation of S equal to zero, (1.2) we obtain the equation for
~.
From (1.2) we have
ôSa~f a a..~’ a a..~’0 aço at aço, ax ap~ —
13
( )
—
.
The analogous procedure can be made for a system of interacting fields (for a more detailed discussion see e.g. ref. [6]). For some kinds of Lagrangians one can obtain from Noether’s theorem [7] conserved (in time) dynamical quantities such as the energy-momentum, the charge, and so on. The appearance of such integrals of motion is the consequence of the invariance of the Lagrangian with respect to some transformations of the x, t coordinates and of the field function q~(which is in turn the consequence of the appropriate symmetry of the system). However, the aforesaid conserved quantities as well as the conservation laws can be directly obtained from the equations of motion (1.3).
1.2. Equations having soliton solutions The equations possessing soliton solutions can be divided into two types: relativistically invariant ones and relativistically non-invariant ones. 1. The first type of equations is more often found in the papers on elementary particle theory. That part of such an equation which contains time and space derivatives can be (c easily the 2 = p2 + m2 = 1).obtained Passing from from the relativistic dispersion relation between energy and momentum E c-numbers E and p to the operators E ia/at, ~, ia/ax we have -~
(ii~—
v2)~+ F(p)
*
=
0,
—
—~
(1.4)
It should be noted that such a Lagrangian formulation permits a relativistically invariant representation in contrast to the Hamiltonian
formalism.
6
V.G. Makhankov, Dynamics of r’lassica! so)itons
where F, which determines the interaction potential, is some non-linear function of q’. Type (1.4) includes, for example, the non-linear Klein—Gordon equation,
(~+
—
(~
2g2jQI2~= 0
va);
~_
(1.5)
the Higgs field equation,
(~ —
m2)~p+ 2g2~~~2~ = 0
(1.6)
(eqs. (1.5) and (1.6) are attributed to the so-called classical q” field theory); the sine-Gordon equation LI~+sin~=0,
(1.7)
and many others (see e.g., refs. [8, 181). Eqs. (l.5)—(l.7) bring about the extremum of the action (1.1) with the following Lagrangians, respectively =
—
leH~~
~‘SG=
a~*a~rn2I~I2+ g2j~4, 3~(,O+mI~l—g IQI ,
(1.8) (1.9) (1.10)
cos ~,
~
where x~ {x, it} and the summation over dummy indices is carried out as usual. By using the normal procedure [6] for these equations one can obtain the Hamiltonians, which are the time components of the 4-vector of the field energy-momentum, and the momentum vector ale
(1.11)
ç
~‘40t
1 /32’
ale
‘\
(1,lla)
Sometimes systems of interacting relativistic fields are considered. E.g. in ref. [9] the stationary solutions of the system 2~=0, L1~+ ~ g flx+m2xg2xI~I20 (1.12) —
in two-dimensional space-time have been found. The Lagrangian for this system is —
2
2
2
2
2
2
I
2
2
22
c°~ —m ~ +g~~ +5(x~—x~—~L.x). The system of interacting scalar fields in three-dimensional space, L1~+
a2~2ço 0 2 21
2 Ex+axI~I+~x(x —1)=0,
(1.13)
114
with the Lagrangian le=_3*a_~(a~x)2_a2x2I~I2_~(xS_l)2,
(l.1j
has been proposed and investigated in ref. [10]. We discuss the soliton solutions of this system in section 6. Here we only note, that all the above-mentioned Lagrangians possess symmetry, i.e., they
V.G. Makhankov, Dynamics of classical solitons
7
are invariant with respect to gauge transformation -~
e’°~,
which involves the current conservation law (1.16)
apj~=O,
where Ik
=
—
j(~p*3
—
(1.17a)
coak,o*),
p~jo=i(q’*p1_~q,~~),
(1.17b)
from which it follows, that Q=fpd3x
(1.18)
is an integral of motion. Eq. (1.16) is easily obtained from the variational principle (1.2) in which the quantity ~S/e5Ois equated with zero.* 2. Non-relativistic equations having soliton solutions have been known for a long time. Among these equations the Korteweg—de Vries equation (KdV for short), [11] 2)~ + ep~= 0, (1.19) c~+ ~a(~ and its various modifications with a changed non-linear term are apparently the most studied ones. Earlier the Boussinesq (Bq) equation (sometimes it is called the non-linear string equation), [1] (ç2) = 0, (1.20) —
—
had been obtained. Finally, recently theorists have become interested in the Schrodinger equation with a cubic non-linearity (S3), [12] i~,+ p~±g2I~I2~ = 0.
(1.21)
In the theory of Langmuir plasma turbulence the Schrodinger equation with a self-consistent potential (like the Hartree—Fock equation in the theory of atoms)
~
=0
(1.22)
has been studied intensely. The self-consistent potential 1 obeys: (i) a non-uniform wave equation [13] E111 = (I~I2) —
or
(1.23)
(ii) a non-uniform Bq equation [14] —
or
(~2)
—
‘T’xxxx =
(I~I2)x~,
(1.24)
(iii) a non-uniform KdV equation [15] ~I, + I~+ *
I(~2)
+
~
=
—
(JpI2)~.
(1.25)
This conservation law is the consequence of the invariance of the Lagrangian with respect to the rotation in isotopic space (see ref. [6]).
8
V.0. Makhankov, Dynamics of classical solitons
Eqs. (1.19), (1.21) and the set (1.22)—(1.23) can be obtained from the variational principle (1.2), in which2’KdV the corresponding Lagrangians are given by = ~1~x1~t +~ai~+ 1~~~l/J~ ~ where
i~ = ~
=
=
1. * leS351(~t~ (see e.g. ref.
*
2
w
2
4
5)Hw~I ±g ~
(1.27)
,
[5]);
‘. * le~=~i(~ ~
*
2
—u5I~2 ~ I
where instead of 1 the potential u =
(1.26)
~
u,
= t~+
is
4
1 2 2 +s(u—u~),
(1.28a)
introduced by the formulae [16]
~I2.
(1.28b)
The Bq equation and systems (1.22), (1.24) and (1.22), (1.25) (we denote them SBq and SKdV, respectively) may also be obtained from a variational principle, but the form of their Lagrangian is unknown to the author. Concluding this subsection we give one more equation describing the exciton excitations in one-dimensional crystal [17] which has quasi-soliton solutions: (1.29) Henceforth the soliton solutions of relativistically invariant equations and various modifications of the Schrödinger equation are called high-frequency (HF) solutions, and the solutions of KdV and Bq type equations are called low-frequency (LF) ones. (Physical arguments for this will become clear below.) 1.3. Quasi-particle interpretation of HF solitons There are two types of soliton solutions in two-dimensional space-time: a solitary wave with the same asymptotic value at x = oo and x = + ~ and a wave with different asymptotic values. The first one is called a “bell” soliton, the second one a “kink” soliton (for details see ref. [5]). We consider some soliton solution properties which follow from the conservation laws for the case of the Klein—Gordon equation. In two-dimensional space-time we have instead of (1.5) —
(~_~+m2)~_2g2~l2=o.
(1.30)
This equation is that for a scalar field with self-interaction. The quanta of such a field are usually called constituents. Indeed, consider the field ~ obeying the equation 2)~=g2F (1.30a)
(El—m
Here as usually g2 is the coupling constant and F is the source function which can depend on an another field i/i or on q’. In conventional field theory the solutions of (1.30a) are looked for by the method of an expansion in g2, when =
c~o+g2fll2F(~,~o),
V.G. Makhankov, Dynamics of classical solitons
9
where c is the integration constant and q~is the solution of the equation for the free field (Li m2)p0 = 0. The quanta of this free field are just the constituents. Sometimes we call them the 2 is “components”. In our case the function describes the self-interaction of the constituents if g not small the aforesaid procedure does not allow one to obtain the solutions sought for. and Moreover, even when g2 ~ 1 equation (1.30) has, besides the solutions which are power series of g2 and tend to the free field as g2 0, also solutions which are singular in g2. Below we shall see that these solutions are just the solitons. To find a particular stationary solution of eq. (1.30) let us pass to the moving frame ~ = x vt and transform the function —
—~
—
=
i(w
~/i(~) exp{ —
0t Then
~fi obeys
~
—
—
k0x)}.
(1.31)
the equation 2g2i4r3 0,
At/i +
where
(1.32)
y
A = m2y2—w~, k 2. (1.33) 0= w0v, ~2 1—v We can consider to 0 and k0 as the energy and the momentum p respectively, “component” 2y2of ~2the> 0. In fact, the quantum. Equation (1.32)gives has a solution in the form of a bell soliton for A = m first integration of (1.32) —
—
A*2 + ~y2g21/14 =
const.
Supposing cit and ~ vanish as ± we have const = 0. From the condition ~ = 0 we find the dispersion relation for the component ~
A = m2y2—e2=~y2g2l/J,~,, whence it follows that the component energy 2
2
I
2
2
2
1/2
=(m y —~yg cfrm)
—
(1.34)
~2)I/2
exp{ i(t —
—
(1.35) Ef =
my is the energy of the free component quantum.
vx)} sech{(m2y2
—
2)U2(x
—
vt)}
(1.36)
of eq. (1.30) describes a bound state of, in principle, an infinite number of component quanta. In the laboratory frame this state is infinitely degenerate: the energy of each component is the same. The momentum distribution of the quanta can be obtained by the Fourier integral expansion of (1.36). In the rest frame it repeats (1.36) ~‘s(P) ~ sech(~irp/p 0), Po
A.
(1.37)
From eqs. (1.36) and (1.37) we conclude that the soliton width is entirely determined by the interaction energy of the components and decreases with increasing energy. From (1.37) it follows that the increase of the interaction energy (the potential energy) leads to the result that the components with greater and greater momenta are captured by the soliton (sometimes such a phenomenon is called condensation [19] by analogy with a superfluid). At the same time we note that the solution (1.36) is
10
V.G. Makhankov, Dynamics of classical solitons
singular with respect to the coupling constant g2 and does not reduce to the free field ~ as g2—*0. Now we consider the dispersion relation of the soliton itself as one for a quasi-particle consisting of the infinite set of components. Its energy and momentum are round (see (1.8) and (1.11)) from the formulae
E=
J
~dx
f{~~I2+ I~r±m2I~2—g2~4}dx,
=
(1.38) (1.39)
Putting q~’ =
~s(X,
t)
in these formulae and evaluating the inteerals we have
E=~(l —A2)~2(1±2A2)y, P
M
=
E
~~(l
—
A2)~2(l+2A2)yv,
(in the rest frame)
=
(1
(1.40)
—
A2)~2(1+ 2A2),
A~(/my)~1. Here we see that there is the usual relativistic relation between the energy and the momeptum of the soliton quasi-particle: E2=P2±M2.
~1.41)
A similar relation for the case of the real field = this paper can be obtained from (1.40) if one puts ~
E’ =
EIA...o,
P’
=
PIA...O, M’
~
= i/i
has been found in ref. [20].The formulas of
=
The example considered is somewhat more interesting because the soliton as classical object behaves simultaneously both as a wave ~ exp{i(vx t)} and as an extended particle with Mass M, momentum P and energy E, and during the motion this particle is effected by the Lorentz contrac~tin~ far as its width E~xcc l/y. Similar quasi-particle properties are also inherent in many other solitons, in particular the solitons of the S3 and Higgs field equations. —
1.4. Solitons and the Fermi—Pasta—Ulam problem The FPU problem was briefly considered in the Introduction. It consists in that, despite of what people expected for a long time, sometimes even non-linearities which are not “small” do not lead to the stochasticity in a system (i.e., to an equipartition of energy between the degrees of freedom), that is the system does not go to an ergodic state. First this was tested by the example of a segment of a non-linear string. Its oscillations can be approximately described by the Boussinesq equation [3]. Peculiar non-linear oscillations of the energy distribution over the spectrum arise in the system: sometimes the distribution over the spectrum expands (i.e., the energy is redistributed between harmonics which are close to the basic one), then it narrows again almost to the initial state [3] (see
V.G. Makhankov, Dynamics of classical solitons
II
fig. 1.1), i.e., the trend to thermalization is absent. A similar result was obtained by Zahusky and Kruskal [4] for the KdV equation with periodic boundary conditions. Only after numerous studies of non-linear equations did it become clear that some of them are complete integrable, i.e., they have a numerable set of explicit integrals of motion and the corresponding Cauchy problem can be solved by the inverse scattering method (ISM). Reference [5]reviews mainly such equations and the study of its solutions. From all the results of that review we need only the fact that the solitons of complete integrable equations emerge after interaction keeping their initial form and speed, and only a soliton phase shift appears. It means that such equations describe the elastic interaction (scattering) of solitons. This result was confirmed later by a number of numerical experiments [21,22]. Looking ahead somewhat we note that the success of the ISM method in the integration of some non-linear equations* gives rise to the hope (or even an optimism) that all or practically all the systems having soliton solutions are completely integrable. For example, in the review by Scott Ct al. [5] there is even the assumption that “for a non-destructive interaction of solitary waves the existence of a Lagrangian is necessary”, and “the only well-founded method for determining whether a solitary wave is really a soliton (in Zabusky’s definition) remains a direct numerical computation (the second assumption by Scott et al. [5])”. Concerning the first assumption we note that at present there at least three non-integrable examples—the Klein—Gordon equation (1.5), the Higgs field equation (1.6) and systems (l.22)+(1.23) and (1.22) + (1.24). Below we shall show that the solitons of these equations interact inelastically up to the formation of bound states, and the existence of a Lagrangian is of practically no use. The second assumption seems to us to be highly reasonable and moreover such a test has been used in many investigations of the FPU problem [22,24—27]. The inelastic interaction of solitons would indicate the ergodicity of the system which is described by the equation studied. In conclusion we note that exotic properties of equations which are connected with their integrability in two-dimensional plane space-time can disappear for a more realistic physical picture, e.g., for spherically symmetric objects [28]. We shall come back to this problem below.
2. Solitons in plasma physics There are now many original works, reviews and books [29—32]dealing with the investigation of non-linear waves in dispersive media. We discuss therefore only the most recent results. 2.1. Equations describing non-linear phenomena in isotropic and magnetized plasmas and their soliton type solutions A rich family of various oscillation modes (collective degrees of freedom) is inherent in plasmas which are specific gaseous media consisting of two or more kinds of charged particles. This large variety appears especially in magnetized and slightly non-uniform plasmas. There are only three oscillation modes in a uniform isotropic plasma, viz., two high-frequency and one low-frequency modes: *
For recent papers see refs. [231.
12
V.G. Makhankov, Dynamics of classical solitons
(i) Transverse oscillations (t) with the dispersion formula t
2
to ‘(k C
2
2 1/2 +tope)
2/me)U2, k and c being, respectively, the Langmuir frequency, the wavevector and the = (4lTne velocity of light); (ii) Longitudinal Langmuir (~)oscillations with the spectrum (tope
322
to =tope(l+~k d~) W~1Ve being the Debye length, Ve = VTe/me and the electron temperature Te is measured in energy units); (iii) A longitudinal low-frequency ion-sound oscillation mode
(de =
= kv~(l+ k2d~Y”2, where v~= VTe/mi is the ion-sound velocity. Such oscillations exist only in a non-isothermal plasma with Te5~T 1 (T, being the ion temperature). The dispersion curves are plotted in fig. 2.1. The number of modes in a magnetized plasma is more than ten. We can not discuss here all these oscillation branches the number of which increases in slightly non-uniform plasmas because there is an extensive special literature devoted to these problems [33]. An imposing series of papers appeared at the end of sixties in which the propagation of solitary waves (solitons) in plasmas was investigated. Nearly all the oscillation modes in plasma known by now were found in these works to be described in the non-linear approximation by equations having soliton type solutions. It was usually either KdV equation and equations of a very similar type or the non-linear Schrodinger equation with a self-consistent potential. This fact which is paradoxical at first sight can be simply explained: the specific feature of linear plasma oscillation is manifest in their dispersion, i.e., in the dependence of frequency to and wave group-velocity Vg on the wavevector. On one hand it is well known that wave packets disperse when propagating, on the other hand, hydrodynamics tells us that non-linearity can give rise to a steepening of the wave front, i.e., to making wave packets more narrow. The balance of these two contrasting phenomena can give a non-linear stationary wave (see, e.g., [29, 32]). We present here, as an example, the papers not discussed in [5]:Non-linear drift waves shown in [34]to be described by KdV and Bq equations, Bernstein modes by
Fig. 2.1. Dispersion curves of plasma oscillations.
V.G. Makhankov, Dynamics of classical solitons
13
the sine-Gordon equation [35].The soliton solutions of sech and tanh types have been obtained[36]for the Alfvén mode coupled equations (using the familiar technique developed in non-linear optics [37]).It was suggested [38,39] to study the propagation of cylotron waves along the magnetic field H0 usiiii a Schrödinger equation with a cubic non-linearity (S3). The latter describes whistier Naves propagating along H0 as well. There are several ways to obtain (with a certain degree of rigour) non-linear equations describing wave phenomena in a plasma, viz., a Lagrangian formalism, Karpman—Krushkal’s method [40], the reductive perturbation theory [41], the hydrodynamical approach adopted in [13] and developed in [14, 15], and many others. Some of them allow one to take into account the resonance wave-particle interaction which leads to non-local terms appearing in the corresponding equations. In what follows we shall leave apart these effects and discuss only the qualitative features of the soliton phenomenon. 2.2. Boussinesq equation and the stochastization problem The equation 2)
was
+
‘p~~
cott = (px.v + (q~ derived by Boussinesq
(2.1)
in 1872 to describe shallow-water waves. The two-parameter family of
soliton type solutions 2 =
/T
A sech {K v ~A(x
—
vt
—
x 0)},
(2.2)
where V=±(1+7A)U2,
KI
obeys eq. (2.1). Notice that the velocity v of this soliton is completely determined by its amplitude, so that the
solution (2.2) depends only on two parameters A and x0: the amplitude and the position of soliton at t=0. Equation (2.1) describes a large range of physical phenomena including the propagation of ion-sound (s) waves in a uniform isotropic plasma making allowance for the first non-vanishing terms associated with the non-linearities and the deviations from the linear dispersion law. N-soliton solutions of this equation have been found in [42]. It means that the interaction of solitons does not asymptotically change the shapes, amplitudes and velocities of solitons and consequently, that it is elastic. It is this fact, which motivates us to think that eq. (2.1) may be complete integrable [43] and describes a Hamiltonian system not permitting stochastization. The operators needed to solve eq. (2.1) by the inverse scattering method have been found in [43]. Note, that the Bq-equation can not be solved by a computer by using a sufficiently finegrid alongthe x-axis since the appropriate Cauchy problem turns out to be incorrect [24]. The improved Bq-equation (which we call IBq) can be obtained starting with the exact hydrodynamical set of equations 3V/at+(VV)V= —V~, 3n1/at + div(n, V) = 0, (2.3) 2q~= ee~ v Here, we shall, however, obtain it in a less rigorous but simpler and more descriptive way using the —
14
V.G. Makhankov, Dynamics of classical solitons
known dispersion formula for ion-sound waves: 1 ~k2
2
(2.4)
(k is measured in frequency units and the velocity is taken to be equal to unity), or
Then we have in coordinate representation for the linear part of the operator L3=~j~~(l_V2)_V2,
or in one-space dimension world (“plane” world) ~2
~2
~4
L~_~—~-~—a2at2.
(2.)
We obtain now instead of eq. (2.1) the IBq-equation Lço
—
(~2)
=
0,
(~t~j
in which ~ stands for the plasma density in the case of ion-sound wave propagation. Equation 2 = k~(l k2), (2.6) and its dispersion formula (2.4) approach eq. (2.1) and its dispersion formula, to when k e~~ In the region k ~ 1 (a wave length of the order the Debye length or less) the Bq-equation —
gives rise to an unphysical instability. This is the physical reason why the Cauchy problem of eq. (2.1) becomes incorrect since to2 <0 for k> 1. The equation with the expression (2.5) as a linear term was obtained in [21] to describe waves propagating at right angles to the magnetic field. It is convenient to write eq. (2.6) as an equivalent system using a function V which has the meaning of a “velocity”
3/ t9V
~
3
2
This system has the following constants of motion for the soliton type solution
dS
1/dt
= dS2/dt
=0,
S1
=
J
~ dx,
S2 =
J
V dx.
Notice, that the energy conservation law of the Bq-eq. [43,44] 53=~
J
2+V2+~co3—(co~)2)dx (co
is no longer valid in the above form for eq. (2.6). The Fourier analysis gives [14] (~2)~~as a non-linear term and yields in addition
V.0. Makhankov, Dynamics of classical solitons
0.36
X=O
15
~
x=560
Fig. 2.2. Formation of a soliton train from a symmetrical perturbation at rest (half a picture is shown).
Fig. 2.3. Formation of a soliton and wave train in the framework of eq. (2.5).
(~+~2)dx=0. Equation (2.6) as well as eq. (2.1) has a two-parameter family of solutions of the type (2.2) but with K2 = (1 + ~A~’t2. For the equation Lp (~,2)~ = 0 we have solitons (2.2) with K = 1 and v = (1 —~Ay”2.As should be expected the solitions of the Bq and IBq equations do not practically differ from each other when k~1,i.e., A<<1, (v—1)~l. The dynamics of various initial wave packets was investigated in the framework of eq. (2.5) via a computer by Bogolubsky [24]. The formation of solitons for arbitrary initial perturbations repeated qualitatively the picture described in papers* by Berezin and Karpman [21],except that eq. (2.6) gives a symmetric picture, for example, the formation from a packet at rest of several solitons moving in opposite directions. The results of [24] are shown in figs. 2.2 and 2.3; the evolution of a packet which was at rest initially in fig. 2.2, while the packet is moving with unit velocity along the x-axis in fig. 2.3. The most interesting result however, was obtained when investigating the collision of two solitons moving in opposite directions. The interaction of two small amplitude solitons, A ~ 1, on the basis of eq. (2.6) does not practically differ from that on the basis of eq. (2.1). But the picture alters considerably for large amplitudes of the interacting solitons. In fact, the fraction of short harmonics grows exponentially with decreasing soliton width and as a result eq. (2.6) unlike (2.1) leads to an inelastic interaction of solitons (see figs. 2.4 and 2.5). The coefficient of inelasticity grows with soliton amplitude. A modification of the IBq equation analogous to the MKdV equation yields —
Lp
=
~
(2.7)
which we call the IMBq. This equation together with the MKdV can be used to study the properties of an anharmonic lattice and of non-linear Alfvén waves. The interaction of solitons = *
±A
sechf~4—(x±vt
—
xo)},
v = (1 + ~A2)U2
In this paper the authors have studied by means of a computer the formation of solitons for the KdV equation.
(2.8)
16
V.0. Makhankou, Dynamics of classical solitons /9g
~
~o5
___
x=-30
~=3O
x=-30
Fig. 2.4. Dynamics of the nonelastic interaction of IBq solitons (A 1 = A,= 2).
s=30
Fig. 2.5. Head-on collision of IBq solitons with different amplitudes (A1 = 2. A2= I).
of eq. (2.7) describing the rarefaction and compression waves turns out to be inelastic as well [24]. We would stress here, that the hydrodynamic equations can be reduced in the case of ion-sound waves to the non-linear equation for the hydrodynamic velocity V which in a plane case assumes the form 2) LV= —(V 1~. (2.9) The interaction of its solitons
2)112, (2.10) ~+(1 +~A is also inelastic [24] (see fig. 2.6). This result at first sight is in a contradiction to that which was obtained in [45] via a computer simulation of the exact system (2.3). There, the authors came to the conclusion, that the interaction of s-solitons is elastic when their amplitudes are less than a critical one. As we saw the inelasticity of soliton interaction is fairly displayed only at sufficiently large amplitudes. A rather complicated analysis is needed to show this effect at not so large amplitudes (see, for example, [251). Finally, it should be underlined that the improvement of the Bq-equation has first arisen from a pure mathematical point of view (the correctness of the Cauchy problem) and then this “improvement” V= ±Asech2{~~ xjf(x~ vt—xo)}~
V
V
—
—
—————-——
Fig. 2.6. Interaction of solitons for eq. (2.9).
V.0. Makhankov, Dynamics of classical solitons
17
turned out to be not only natural from physical considerations but also to give a new qualitative result; the inelastic interaction of solitons and the stochastization of the system. We shall below in an analogous way “improve” (or “make worse” as one prefers) the KdV equation. 2.3. KdV equation and its “improved” version IKdV The “improvement” of the KdV equation is no longer so obvious and descriptive as it was for the Bq equation. But from a computational point of view (in a computer sense) such an “improvement” is sensible and has a right to be. First, a few words about KdV equation having in the laboratory frame the form (2.11)
4,t+~,X+(q,2)X+~XXX0,
which has been used to investigate firstly shallow water waves [11] then long ion-sound waves in plasmas [5,46] travelling in the positive direction along the x-axis, and many other linear wave phenomena (see ref. [5]and above). The most frequently used form of the KdV equation is + p,~= 0, (2.12) p, + a(~2)~ which is (2.11) written in a coordinate system moving along the x-axis with the velocity of linear waves for k—*0. Equation (2.12) can be easily obtained by the transformation ço—*p(x—t) and describes a completely integrable Hamiltonian system [47]. Its N-soliton solutions had been derived by the inverse method to give non-destructive (elastic) soliton collisions which result asymptotically only in a phase shift of the interacting solitons. Equation (2.11) has been derived in the long-wave limit with the linear spectrum being to = k(l k2). The basic assumption employed in deriving this equation that the whole perturbation propagates only in a positive direction along x-axis, breaks down when k> 1. Such an incorrect description of short wave behaviour on the basis of eq. (2.11) does not lead to the incorrectness of Cauchy problem as in the case of the Bq equation.* We shall “improve” the KdV equation to obviate this short-coming. At the same time our equation will describe the evolution of long waves at any rate no worse than KdV but will be more convenient from the computer study point of view. We take the dispersion relation in the formt to = k(1 + k2~’to obtain instead of eq. (2.11) —
Pt +
~
+
= 0.
(~2)~ —
(2.13)
This equation has been suggested in [49] to investigate tidal waves. In the coordinate system moving with unit velocity along the x-axis, we have ~ + (q~)x+ 4Dm~
= 0.
(2.14)
The evolution of arbitrary initial perturbations and a process of soliton interaction have been studied on the basis of eq. (2.14) in [25] by using a computer.** *
The dispersion, w = k( I
—
k2) leads to an ‘unpleasant” stability condition for computer studies of the KdV equation, r ~tch3 (where c is a
numerical factor of order of unity, while r and h are the temporal and spatial grid steps). This formula clearly coincides with that for the KdV equation when k 4 1. ** The stability condition for a computer study in the case of eq. (2.14) was shown [25]to be not r < ch3 as above but r ~ h.
V.0. Makhankov, Dynamics of classical solitons
18
We have for eq. (2.14) the constants of motion
~fr =
0,
S~=
J
dx,
S~=
J
(~2
dx,
+ ~
S2 =
J (~ —
and the two-parameter family of soliton solutions
A
1/2
4D=Asech2{(2(2A+3)) (x_vt_xo)},
v=~A
(2.15)
(note, that v = I +~Ain the laboratory frame). It has been pointed out in ref. [25],that the evolution of initial packets studied on the basis of eq. (2.14) is, in general, similar to that for KdV equation which has been simulated in [24,21]. A noticeable difference arises only when one considers sufficiently short (k ~ 1) and powerful (A ~ 1) solitons. The picture when a large amplitude soliton (A1 = 6) overtakes a small one (A2 = 1.5) has been examined resulting in the emission of a rarefaction wave of a very small amplitude (A~ 1.7 x 10~) which lagged behind the solitons. Many various manoeuvers were employed providing the display of the inelasticity of the interaction and the computational errors to be less than 0.014%. The effect of inelasticity was found to be about 0.3% of maximum soliton amplitude with the level of computational errors being less than 10% of the effect. Such an effect for head-on soliton collisions within the framework of the IBq equation reached about 10% at A = 2. The difference may be explained by the fact that the “distance” between the IBq and Bq equations is larger than that between the IKdV and KdV equations*. An analogous situation took place in the framework of an IKdV with a cubic non-linearity instead of quadratic one (IMKdV): 4Dt +
(4D~)~ + 4Dm
=
0,
(2.16)
whereas the MKdV equation ~
0
is completely integrable according to Wadati [48]. Equation (2.16) has soliton solutions 2)”2 (x vt x = A sech{(2 + A 0)}, —
2,
—
v = ~A
(2.17)
and the inelasticity of their interaction grows by more than ten times in comparison with that of eq. (2.15). When defined as AineilAmax the coefficient of the inelasticity Kinei depends on Ama,, as shown in fig. 2.7. Thus, solitons of the IBq, IMBq, IKdV and IMKdV equations can show the inelastic interaction which means that these equations describe dynamical systems in which stochastization and hence irreversible energy exchange between degrees of freedom are possible. It is meaningful now to turn to the definition of a soliton which so far has not been formulated very clearly. We broaden it so as to include localized solutions of the corresponding non-linear equations *
Such an explanation was suggested in ref. [43~.
V.0. Makhankov, Dynamics of classical solitons Xuiet
19
% 2.6g
2.75
2. 1
/ ~ ~ Fig. 2.7. Inelasticity coefficient of soliton interaction versus amplitude for eq. (2.16).
which, however, can inelastically interact and even fuse. It seems, therefore, to be more natural to define a soliton as a solution of a non-linear wave equation (or a set of equations) confined in a space at all times and having a finite energy (or a rest mass) (see subsection 1.3). The second requirement is of use when one considers solitons to be classical models of particles [10, 19,49]. All the equations mentioned in this and preceding sections describe, from the plasma theory point of view, the so-called low-frequency branches of oscillations (for example, ion-sound, Alfvén, and many others). From the point of view of a given quasi-particle interpretation their solitons are made of massless constituents the velocity of which tends to unity. Including the interaction makes the components tachyons (t’> 1). Once the frequency of these oscillations tends to zero together with wavevector (to cc k) it is natural to call them low-frequency ones. In subsequent subsections of this section we shall discuss high-frequency solitons which have components with a rest mass of, for example, that of Langmuir waves. 2.4.
Schrodinger equation with a potential satisfying a non-uniform wave equation
Here we examine solutions of the system (l.22)+(l.23) in the plane geometry: i4D
5 +
4D,,,,
—
~I~4D = 0,
L11 = ~
(14D12),
(2.18)
and compare them with the solutions of the S3 equation i4D1 +
4D~,,+
=
4DJ24D
0.
(2.19)
Both the system (2.18) and eq. (2.19) can be obtained starting from hydrodynamic equations that describe Langmuir plasma turbulence. In this case ~ and 1 are respectively the amplitude of the high-frequency electric field and the density variation (perturbation). However, as above, we obtain eq. (2.18) less rigorously but in a simpler way by using the dispersion relation for Langmuir plasma waves 3
,2
fiSk = Wpe+~topeK
with ~
2
(2.20)
tie,
being the Langmuir frequency, and de =
Ve/tope.
V.G. Makhankov, Dynamics of classical solitons
20
A high-frequency wave propagating in plasma creates a Miller force [63] which expels some of the plasma particles from the region where its amplitude is larger, Fcc_~~~(IEI2). As a result, the local plasma density is decreased in the larger amplitude region and the wave propagates in the perturbed plasma with a density n = n0 + Sn. Upon substituting this relation into eq. (2.20) one obtains
2(d~°~)2 + ~w~]Sn/n
tok =
+
~w~k
0.
(2.21)
When obtaining eq. (2.21) we took the density perturbation into account only to first order in Sn and neglect higher-order terms. Let us write eq. (2.21) in the form (0)
3
SWtokWpe
(0)
~tope(de
(0) 2
1 (0) )k 2 +2Wpe Sn/n0,
then changing to the coordinate representation with the help of the formulae Sw iô/8t, k i3/3x, we have 2/3x2 1)4D = 0: (2.22) (i3/3t + 0 the equation for the high-frequency field envelope of -oscillations. In eq. (2.22) we use the following dimensionless variables -+
—
—*
—
2 —x x=~/,~-,
2
t~I~Wpet,
2
~
~=4I~t 1Sn
‘~
2
Mp~n~T’ 3 E
and = me/mi,
Vs =
(T/m~)”2.
We have now to write the equation which governs t, the plasma density perturbation. In a quiescent plasma it is clear that the wave equation is
(~
=
—
0.
(2.23)
In the presence of the high-frequency field the Miller force appears which in turn leads to a source function on the right-hand side of eq. (2.23). This function can be simply found if we recall that the Miller force expels plasma from the region of the high-frequency field, Sn cc El2 therefore we finally have the second equation of system (2.18). The so-called quasi-static limit of (2.18) is sometimes considered which one can obtain by neglecting the term 32c1/3t2 (the limit of infinite ion mass), then —
2
2
= _~(I4Dl2),
or for soliton-like solutions ~t~_l4DI2, and we come to the S3 equation, formula (2.19). It should be underlined, that the above method for getting eqs. (2.18) is valid only in the plane
21
V.0. Makhankov, Dynamics of classical solitons
Fig. 2.8. Quasi-decay instability growth rate versus wavenumber of perturbation.
geometry. In a more general case a more rigorous approach gives the system (see [13])
(~
V2)cI = V2(IVci’12), (2.24) div( 2iV~/i5 VV2cir + 4Vci’) = 0, where cit is now the amplitude of the high-frequency potential, and not the field (we shall return to these equations in section 5). Both eqs. (2.19) and (2.18) describe the instability first examined in ref. [50] and called there the modulational instability. Its initial stage is satisfactorily depicted by the following dispersion equation [25] (Nk—Nk 1.~)dkl =0, (2.25) —
—
—
~
~
K Ve
~
K meve
totokl+tok,_K+1S
where w5,~is the ion Langmuir frequency, the solution of the linear dispersion equation (2.20), k the wavevector of the Langmuir wave, K the wave number characterizing the modulational amplitude, viz., the wavevector of the low-frequency perturbation and Nk is the 1-quantum number. Equation (2.25) can be obtained from weak plasma turbulence theory [25] and also directly from (2.18), if one takes a slightly modulated 1-wave [13] tok
4D4D0+S(pexp(—iwt+ikx),
4oc
exp(—itot+ikx).
The growth rate determined by eq. (2.25) as a function of wave number is depicted in fig. 2.8. There are many names for this instability in the plasma physics literature the modulational, the modified decay, the quasi-decay, or the two-stream instability, and so on. In fact, eqs. (2.25), (2.19) and (2.18) describe the instability of a gas of Langmuir plasmons in relation to their adhesion similar to what takes place in a gravitating gas. The analytic investigation of the dynamics of this instability is quite complicated, except for the solution of the Cauchy problem for the S3 equation. It is because the S3 as well as the KdV equation describes a complete integrable Hamiltonian system so that the inverse method can be applied [12]to look for the asymptotic behaviour of an arbitrary initial packet.* A whole series of papers have recently appeared, in which the attempts were undertaken to examine the dynamics of solitons in the framework of (2.18) by analytical methods. The most interesting of them, from our point of view, are those by Karpman [52]and by Thornhill and ter Haar [53].In the former one the formation of a soliton from a near-sonic initial packet of small amplitude 1-waves is studied on the basis of (2.18). In the latter, an analysis of the conservation laws for eqs. (2.18) found in [54] —
dS1/dt=0, *
1=0,1,2,
In fact, computer simulations have shown a much more complicated picture [25].
(2.26)
V.G. Makhankov. Dynamics of classical solitons
22
2dx;
s
S
1
=
S2
=
f J
(2.27)
l4DI
0~s=J
{i(4D4D~_4D*4D~)+2~u}dx; 2+~l {l4D~l
(2.28)
2+~2+~u2}dx;
(2.29)
4Dl
with ~I~ +1 u~=
0,
was used to examine the features of the interaction of Langmuir solitons with each other and with ion-sound waves. Computer investigations of the dynamics of the formation and interaction of Langmuir solitons in the framework of both eqs. (2.18) and (2.19) and a comparison of their results was first done in ref. [25] and later in [51, 54]. In ref. [25] a quasi-particle interpretation of these phenomena similar to the one, which was discussed above, in section 1, for the Klein—Gordon equation was also suggested. However, we deal now with non-relativistic constituents (plasmons) and non-relativistic solitons. The rest mass of these solitons is determined mainly by the integral S since the Langmuir frequency is much greater than the dispersion and non-linear terms. We can therefore take the first integral as the total number of Langmuir quanta (plasmons) conserved by virtue of eq. (2.26) in all interactions.* The second integral represents the momentum and the third one the energy of the soliton. Lastly, eqs. (2.18) picture the system of two interacting fields with the second one being a massless one transmitting the interaction between plasmons. To proceed further, let us consider some relations following from eqs. (2.27) and (2.28) for the soliton-type solutions of eqs. (2.18) and (2.19). The first one we supply with index d (dynamic), the second with s (static) =
A sech{~(x
I~~l4Ddl7’
—
vt— xo)} expfi(~x—cL~t
—
y
~d4v2~4Y,
~,)},
(2.30)
=l—v,
where the soliton velocity v is measured in units of the ion-sound velocity v. and coincides with the group velocity of Langmuir waves, = 3koV~/wpeVs, Vg
=
A sech[~(x
—
Vt
—
xo)} exp{i(~x
—
fi,,t
—
~
(2.31)
Both families of solutions (2.30) and (2.31) are four-parameter ones (A, v, x0, in contrast to the low-frequency solitons of the KdV and Bq equations, which are two-parameter families (A, x0). The solutions (2.30), (2.31) are sometimes called the two- and one-parameter families, respectively, i~)
*
This
fact is also a consequence of U(l)
symmetry.
V.0. Makhankov, Dynamics of classical solitons
23
forgetting the parameters x0 and ~ which determine the position of soliton at t = 0 and its phase. It should be noted that for a completely integrable equation such as the KdV, S3, sine-Gordonequations and others, it is important to take these parameters into account since only they alter after the soliton interaction. We shall, hereafter, deal mainly with equations which deviate from completely integrable ones since despite of the elegance and the large number of the latter, they seem to be albinos in a normal population. The integral S2 for eq. (2.19) was obtained in [12] and simply coincides with the Hamiltonian H
S2~=
f
(14DXl2
—
‘I4DI~)dx.
(2.32)
The first integral has the same form in both cases. Upon substituting (2.30) in (2.27) and (2.29), (2.31) in (2.27) and (2.32) we have (see [51,53])
Sd
2V’~m,
=
m
=
Aly,
(2.33a) (2.33b) (2.33c) (2.33d)
2)y6},
S2d = \/~{mv2/2_~m3(l 5v —
S,,Sd~y, S2s =
\r2y{mv2/2—~m3y2},
or ~S,,(v2—j~S~), S 2 j~y6S~(l 5v2)). 2d = ~Sd(v One can see from formulae (2.34) and (2.35) that the integrals S and S
(2.34)
S2s =
—
(2.35)
—
2 for a formed soliton are firmly associated. In the static approximation there is only one collective degree of freedom, hence both S and the high-frequency part of S2, i.e., are conserved and the formation of a soliton from an initial packet takes place only with S and S2 being connected by relation (2.34). Thus, given 5, the value of S2 is uniquely determined; if not, viz. S2(t = 0) S2(s) the initial packet can, generally speaking, oscillate infinitely around its equilibrium position [25]. Figure 2.9 taken from [25] depicts the time-dependence of the amplitude for a packet which is initially at rest. One can see that first it narrows with its amplitude increasing, then it passes through the equilibrium position due to inertia (remember the rest mass of the plasmons), that is, the soliton state, and turns back, and so on, while the amplitude of the packet did practically not alter in the whole computer run. In the framework of the KdV equation an arbitrary initial packet at once starts to break up into a number of solitons, which is determined by the number of bound eigenstates of the Schrödinger equation with a potential which has the form of the packet under consideration (see the inverse method and details in [5]),and a weak background which carries the energy 52,,
:~
excess. The S3 equation shows a long oscillating processaround the equilibrium (soliton) state, except for
Fig. 2.9. Amplitude of a Langmuir wave packet as a
function of time (at the point x
= 0)
in the quasi-static approach.
24
V.0. Makhankov, Dynamics of classical solitons
o.5~ .1
0
Fig. 2.10. Dependence of packet amplitude on time (in the origin) in the frame-work of system (2.18).
x Fig. 2.11. Formation of a soliton from an initial packet.
N-soliton initial packets. This is the main difference in the dynamics of arbitrary initial packets for the completely integrable S3 and KdV equations which is associated apparently with the fact, that the linear spectrum for the KdV equation is of a quasi-decay type, i.e., the decay w~(k1)+ w~(k2)= w~(k1+ k2)
(2.36)
is only just forbidden by virtue of the fact that tos(k) cc k and can become allowed in the presence of a non-linear interaction. As a result, we have small amplitude trains of ion-sound waves when solitons are formed from initial packets. The Langmuir wave spectrum, as can be easily seen from the S3 equation, is of a non-decay type with +
to~’,
2w~
(iS~1+~2
tope
being forbidden strongly. That is why apparently there appears no background of Langmuir waves~(in the case of the Klein—Gordon equation with a cubic non-linearity as we shall see below, the situation may alter). The computer simulation results confirm the theory of the S3 equation: 1-solitons emerge from the collision without changing their shapes and velocities. But we have a completely different picture if we study the evolution of initial packets and the interaction of solitons with each other on the basis of the system (2.18). The soliton now forms from an arbitrary initial packet during the time of a few oscillations with the energy excess carried off by s-waves (1 field). Figure 2.10 shows the plot analogous to that of fig. 2.9. but for the system (2.18). It is also seen from fig. 2.11 that, when oscillating, 1-plasmons emit s-waves until the soliton state is reached. It is very interesting to note, that although the oscillations of 1-plasmons in the self-consistent well of the soliton continue, the emission of s-waves ceases. It means, that in the Classical system considered there arises a prohibition for emission as occurs in a familiar quantum theory of atom, so that a classical object displays not only quasi-particle properties but also certain quantum prohibitions (see also [10]).
*
In principle, even in the S3 model the energy excess may be carried off by very small amplitude solitons but the smaller the amplitude of the
soliton the slower this process goes since r ~
V.0. Makhankov, Dynamics of classical solitons
25
!P12 ~
/fr --
____ 0/
AJL
-<2>
x
/1
Fig. 2.12. Coalescence of two identical Langmuir solitons followed by s-wave radiation.
Fig. 2.13. Interaction of t-solitons with different masses (a light soliton hits a heavy one).
~I2
~
l’~
Fig. 2.14. Interaction of solitons with approximately equal masses.
)~-~
Fig. 2.15. Interaction of solitons of different masses (a heavy soliton carries with it a light one).
26
V.0. Makhankov, Dynamics of classical solitons
With the help of a computer simulation the dynamics of the interaction of -solitons with each other and with ion-sound waves has been studied in [51, 54]. In ref. [54] most attention was paid to a study of “elementary events” of collisions of solitons with various masses and it was found that only solitons of masses very close to one another interact effectively. The greater the difference between the masses of the colliding solitons, the smaller the effect of their interaction which gives rise only to changing the soliton energies contained in the integral S2. The number of solitons in such interactions is conserved, as a rule, but only the shapes and relative velocities of solitons change, except the case when a heavy soliton colliding with a light one can absorb it. Figures 2.12—2.15 [54] give pictures of these processes. In figs. 2.16—2.17 the interactions of ~-solitons with compression and rarefaç~tion sound pulses are shown. In the same paper it was found that when two identical solitons collided head-on they either pass through each other or fuse depending on their amplitudes. The analysis of soliton adhesion conditions was in detail carried out in ref. [511to obtain the region of parameters of solitons (namely, velocities and amplitudes) where two identical solitons adhere. This region is depicted in fig. 2.18 in the m, v-plane above the solid curve; it was determined on the basis of a series of numerical experiments. In the same papers the effect of boundary conditions on the process studied by a computer was shown to be significant, leading, e.g., to the adhesion phenomenon becoming reversible if the system is either placed in a potential well reflecting s-waves [54](boundary conditions with fixed ends t(— L) = c1(L) = 0) or with a feedback (the periodic boundary conditions) [51] (see fig. 2.19). It was also shown in [51] that the interaction (as well as the formation) of e-solitons passed through two stages. The first, the rapid stage, is associated with the attraction of solitons which start to interact and with the fusion of them into one soliton, which is then in an excited state; in plasma theory terms, this stage is usually called the aperiodic instability. The second stage, which is a slower one consists in 2 P1
/
/
/
1,:
A
Fig. 2.16. Interaction of (-soliton and s-pulse of compression.
‘I
/
/
‘
\,i
Fig. 2.17. Interaction between -soliton and s-pulse of rarefaction.
V.G. Makhankov. Dynamics of classical solitons
27
Fig. 2.18. Ares of fusion of two identical solitons in the mu. c-plane.
the emission of the excess energy (mass “defect”) in the form of acoustic waves; if the compound particle does not manage to break up into two, it goes over to the ground state (for details see [51]). Notice, that the analytical results of ref. [53] obtained by studying the constants of motion confirms qualitatively what was found in the computer simulation [51, 54]. At the end of this section we discuss briefly the whistler solitons studied in refs. [31,55, 56]. When propagating along the external magnetic field helicon waves (w) have a spectrum (iS
= (1)HekOC/Wpe,
I’f~
J\~~ 31~ ~4.33I0
(1
ml
II
i~.l0
a
t-O.8
~‘~_
~
I
~
I
I
2 3,4j~
~
31210a
2
!~I~A~ /
I
Fig. 2.19. Reversibility of a process of two soliton fusion under 2 ~‘ periodical boundary conditions. ~/ ~ ‘I ~jIj - S6.,0
28
V.0. Makhankov. Dynamics of classical so/ilons
which implies that
depends strongly on the wavenumber k0 and in that sense, the helicons differ qualitatively from Langmuir plasmons; indeed, they have no rest mass, which constitutes the main part of Langmuir soliton energy. Therefore, as was shown in [551only in the case of packets which are sufficiently narrow in k-space, one can obtain a system resembling (2.18): ~4Dt +
to
4Dxx + ~t~4D = 0,
=
(2.37a)
fr (l~l~)~
(2.37b)
where ~ is the complex envelope of a helicon field amplitude which is defined by the relation h,,
ih~=
—
exp{iw0(k0)t},
~
where 2/W~,e(flo),
wo(ko) =
c
toHe =
eH/mec,
toHek~C
is the velocity of light, and the following notation is introduced: t =
X = (top2eVs/WHeC2)X, =
(Wp2eV~/WHeC2)t,
(w~tek~c4/w~ev~)Sn/no, 14D12 = (k~to~ec4/41rnomiv~we)I4Dl2.
It should be underlined, that the opposite sign before the term of interaction in eq. (2.37a) leads to the fact that the helicon solitons move faster than sound waves, v’> v,,, and have a density hump instead of an 1-soliton density pit. The soliton solution of system (2.37) has a form similar to (2.30), except for the expressions for 1 and v which now are =
14D12/(t)
—
1),
V
v’;Ivs =
2(toHekoC2/to~eVs)>
1.
Moreover, the system (2.37) has just the same constants of motion, except for the sign of the term proportional to 1 in S 2, namely,
J{I4DxII4Dl+}dx,
S2
=
c11
+ u,
and =0.
Then we have 2—~y6S2(v2—5)), ~y2 = v2— 1. S = 2V2A/7, S2 = ~S(v Note, that in this case the energy part determined by S depends on the soliton velocity and vanishes formally as v —*0:
E= WHeCV 5
w~
4
The condition for the validity of the system (2.37), iXk ~ k0, implies that A~v/V2y and
S2>O.
V.0. Makhankov, Dynamics of classical solitons
•28.
29
~_~__~i~
l~I~
~
~ 2.0
II
f-2.06
~
___________________
~/\ ~ ~
t=6.75
t=3.513
~
~
Fig. 2.20. Formation of a supersonic helicon soliton.
Fig. 2.21. Head-on interaction of two identical w-solitons.
The formation and interaction of whistler solitons were studied numerically in ref. [55] (see figs. 2.20, 2.21). There the role of ion-sound waves in such processes was found to be decisive in spite of the w-solitons being hypersonic. Moreover, although decays of the type w —* w’ + s are forbidden within the framework of the weak turbulence theory [56], head-on collisions of solitons and their interaction with ion-sound waves give rise to a new (soliton type) channel for the dissipation of energy contained in the w-mode. We have been purposely dwelled on the results of ref. [55] since the influence of ion-sound oscillations on high-frequency dynamical processes is often ignored. For example, an erroneous statement has been made in [54] that there is a wide region of parameters, I
2.5. Coupled Langmuir and ion-sound waves in the near-sonic region A first look at the solution (2.30) is quite sufficient to see its short-coming (inconsistency): the density of the plasma perturbation 2
*
2
‘~Y 4Di We emphasize that the S3 equation has formally e-soliton solutions in the parameter region mentioned and it is in this sense that the
quasi-static approach is reasonable.
30
V.G. Makhankov. Dynamics of classical solitons
together with the constituent energy, and the inverse soliton width, yA/”/~,tend to infinity as v —* 1 with the soliton mass m approaching zero at the same time. This fact was first noticed in refs. [44,51,58,591. In [14] the inhomogeneous Bq and in [58] the KdV equations were suggested to replace the second equation of the system (2.18). In that case the perturbation of the density obeys, more consistently. the ~d,
equation ~
—
—
/3(t2)~+~
or I~+ ‘1?~+
+
~3(~2)
2)+~~ a4~~5++ = (l4Dl a’I~,,~ = — (14D12)x,
(2.38)
which equations must now be satisfied by the potential in the Schrödinger equation (a and /3 are small parameters proportional to me/mi). Approximate solutions of system /
‘4Dm +
4D~,,—
=
0,
4\
I
2
011 —
~ — ~—‘~
2
(1
2
)=
U ~
2
(l~l),
(2.39)
accurate to the linear terms in = ~me/mi have been found in ref. [14].the exact soliton solutions ~t the above systems were obtained in [58,60,61] by various techniques. It follows from (2.33b) that the faster the soliton the larger the relative contribution of the kinetic energy to total energy. Besides, ~n [51]it was found that fast solitons with S 2>0 interacted like sound pulses loaded with high-frequency field but not vice versa. At the same time in [51] the description of the plasma density perturbation (i.e., the ion-sound waves) was linear, so that as v 1 the soliton interaction dissappeared completelv,* On the other hand, a consistent description of the ion-sound waves using a Bq (or rather an IBq) equation gives a two-parameter (see section 2.4) family of solitons, i.e., there appears as compared to (2.30) an additional connection between the amplitude and the soliton speed — which is larger than the sound speed. It was shown in [58,601 that bound Langmuir-sound(1—s) solitons, which are stationary solutions of the set (2.39) (or of (2.38) plus the Schrodinger equation) form, like the s-solitons, a three-parameter family of solutions of the form —~
—
—
A tanh{B(x — Vt — x0)} sech{B(x 2{B(x — vt — 6A sech V = V—A, A2 == 48A2, A = =
—
Vt
—
~v2,
—
x0)} exp{i(~vx — 1~t—
~)},
(2.40)
(2.40a)
with A the eigenvalue of the Schrodinger equation being determined by the formula (2.41)
which means that the coupling between the amplitude and the velocity of solitons appears again. Note, that the solutions (2.40) as well as (2.30) are subsonic, V2 < 1, giving rise to A <0 and to attraction among the constituents in the soliton.t An elegant method of solving the system analogous to (2.39) (for a near-sonic w-soliton) was developed by Bogolubsky in [611.We give it in the appendix. * The
author has been recently informed by Zakharov, that the system of the Schrodinger equation with (out + O/Ox)4 = —(3/ox)IcoI2 governing the
selfconsistent potential is completely integrable [23]. ~ Actually, from (2.30) it follows that A = [1—~v2 (eigenvalue) is given by the potential energy of the constituents in the soliton.
V.G. Makhankov. Dynamics of classical solitons
31
The system (2.39) has the constants of motion [44,60, 62] S=fl4Dl2dx,
f =f
—
S~=
~{i(4D4D~ 4D*4D~)+ ~u} dx,
S2
{4DXI2+~c2+l
(2.42)
~
which, as before, can be taken as a high-frequency quantum number, momentum and fs-soliton energy. Note, that in contrast to slow 1-solitons, the solution of (2.40) is a density cavern loaded with a high-frequency field the energy density of which has a two-hump shape (“camel” shaped). We have envisaged in previous section the dynamics of the formation and interaction of 1-solitons and in fig. 2.18 we depicted the region of adhesion of two identical 1-solitons moving towards each other in the m, v-plane. Consider how the picture of formation and interaction will change for 1-s solitons (2.40). First, a few general comments. Turn again to the m, v-plane (or, which is the same, the V—A, v-plane). Relation (2.41) corresponds to the line I in fig. 2.22. Points lying below this line give the four-parameter family of 1-solitons. Under but in vicinity of the line I, this 1-soliton family loses its stability (for instance, the point A) for it is found to be in a region of influence of the solution (2.40). Due to the conservation laws the soliton can only decrease its velocity v; hence it will, emitting s-waves, go off the line I along a straight line parallel to Ov towards small v where the effect of the solution (2.40) weakens. When it is given by a point above the line I, say B, the Langmuir wavepacket will move (in parameter space) along this line either emitting s-waves or breaking down into several packets (possibly both simultaneously) depending on the initial data. A similar movement in parameter space will occur due to the interaction of different types of solitons. Following the paper by Thornhill and ter Haar [53], Watanabe [62], starting out from the constants of motion (2.42), has found the selection rules for the processes of interaction of 1, ls-solitons and s-waves: 1+14-*1+ls,
t+s’E-*l+Is.
Somewhat earlier in ref. [611such processes were studied by means of a computer. There the Bq-equation
Fig. 2.22. Dynamics of ts-solitons (packets) in the A, v parameter plane. The curve I corresponds to the relation (2.41). A and B are possible initial points.
32
V.G. Makhankov, Dynamics of classical solitons
was replaced by the IBq equation (LI
3
ax2at2)~
8 (~2)= 8 (14D12)
—
in the system (2.39). Soliton type solutions to such a system coincide with those given by (2.40). The stability of two-humped solitons was verified by numerical experiment for v = 0.95 on the basis of the coupled Schrödinger and IBq equations. The solution (2.40), being conserved with high accuracy, propagates up to distances many times longer than the size of soliton. The head-on collision of two identical ls-solitons (with v = 0.95) turns out to be inelastic. Just after the solitons come in contact with each other there occurs an exchange of energy and momentum among the plasmons; the inside humps lose some of the plasmons and the outside ones gain them; finally there is a strong throwing-out of supersonic 1-plasmons (see fig. 2.23a). The fraction of these plasmons is rather considerable,. about 20 per cent of integral S is carried away by them. The remaining part of 1-plasmon comes together again in two diverging two-humped packets, which seem
~~ x—-z2
~1v~c
(~
\\
l~ \.~ X=—2.2
A
(~
-L2240
~
\\
1
//
ItI
~ X=2.2
“
2
t=e.8
—1 z5.#O~ ~
f—a
~
x~2.2
‘I
_122.102
~
/
IPI~3.6
\‘1/...~...
2.1 -O.82.i05
~
A
~
—2.5IO~
I~
~
~I/’ I
_,.12.,02
I
1
( a)
‘‘
~
—1. 18~i02 ( b)
Fig. 2.23. (a) Head-on collision of Es-solitons: (b) Interaction between an Is-soliton and a compressional s-soliton.
V.0. Makhankot, Dynamics of classical solitons
33
to turn into smaller amplitude Is-solitons. A similar effect of 1-plasmon acceleration has been seen in Is-solitons interacting with s-pulses. Already for an amplitude A,, = 0.4cI~,,an s-soliton knocks about 40% of 1-plasmons out off an Is-soliton, and these free plasmons are again supersonic (fig. 2.23b). A rarefaction pulse interacts with an Is-soliton significantly less strongly. Recall the opposite picture that took place in the interaction between 1-solitons and s-pulses. At the end of this subsection note, that even a small correction proportional to in the low-frequency field equation turns out to be able to rearrange the solution for the high-frequency field. New two-humped Is and Iw-solitons, members of a three parameter family, were shown to be less “conservative” in interactions than their four-parameter 1-soliton and two-parameter s-soliton brethren. We see in this example that a coupling of fields can enlarge the efficiency of soliton interaction. In addition to the dynamical effect mentioned above, the solutions (2.40), (2.41) have one more attribute distinguishing them qualitatively from an 1-soliton (2.30). The latter exist in the whole area of the parameters A, v on the left from the line v = I but Is-solitons exist only under the line I. This means the larger the soliton mass, m cc 5, the less its velocity and as v—*0 we have —
A
(2.43)
i.e., the mass of a soliton at rest (and hence its amplitude) is limited from above. Having used (2.40) we get [62] S = 4A3~/(27)”~, which together with (2.43) gives 4(3~ \5\/~J
m ~ m ~s
=
(3V3
\ 10
\/~\~3/2 me!
There exist no stationary soliton-type solutions to the system (2.39) with a soliton mass exceeding ~ Thus, non-linear and dispersion terms in the low-frequency field equation have given a restriction on the mass (amplitude) of high-frequency solitons.* As we shall see below, an analogous restriction on the soliton mass arises in the framework of the Schrodinger equation with a so-called saturating non-linearity. 2.6. Schrodinger equation with a saturating non-linearity Let us consider here as an example the properties of the stationary solutions of the equation i 4D,
+ 4D~+
‘4D(1
—
exp( — E14D12)) = 0,
(2.44)
which gives the S3 equation when expanded in powers of , if 4D124 1/e to order ~(e’) in . Here, as before, = ~me/mi. Equation (2.44) is more correct than S3 if one examines the behaviour of sufficiently large amplitude packets of I-waves, 14D12 l/e near the stationary state. Such stationary solutions satisfy the equation
— *
(~ —
2}= 0.
— ~‘exp{—~fr
It should be noted here, that the system (2.39) strictly governs coupled Langmuir and Ion-sound solitons only in the region v—si.
(2.45)
V.G. Makhankov. Dynamics of classical solitons
It is convenient to change from A to ep(x, t) = ~/i(x)exp(i~2t).
_~2,
i.e.,
Upon integrating once one gets
c~=(2I)~2l(~2)+ The soliton boundary condition, ~/i—*0 as x—*±~, gives c point, where = 0, we have
1/2 and utilizing that at an extremum
t/,~
/i
I
2
2
2
T(~1m+eXp(~,&m)
Wm
~
Note, that the derivative I =
1
1~2
((2)~2(~2)+)
can vanish only if (2.46) and we come again to a condition like (2.43) which is needed for soliton solutions of eq. (2.45) to exist.* We shall not here discuss the properties of the solutions of the above equations in more details.** We only want to emphasize, that higher order in non-linearities (i) can essentially change the stationary solution for the high-frequency field as well as (ii) lead to an upper bound for the soliton amplitude (mass). This, despite of all the distinctions between system (2.39) and eq. (2.44), displays their common features. As we shall see below conditions of type (2.46) are intrinsic not only to two-dimensional problems but also to many-dimensional ones. We refer readers wanting to become more familiar with the results of studies of equations such as (2.44) to recently published original papers (see, e.g., [64—68,129]). Of most interest from the plasma theory point of view is to find the magnitude of the constant a in the inequality 2
JAJ=~
It turns out to be very susceptible to a change in the model. The transition from the system (2.39) to eq. (2.44) changes a from 3120 to unity; replacing in (2.39) the Bq equation by the IBq equation also changes the magnitude of a (see appendix). On the other hand, its magnitude determines the width i~x of the soliton and hence the characteristic wavenumbers (momenta) of the constituents of the soliton and, finally, the efficiency of linear Landau damping [60]. Various numerical experiments [691 including non-one-dimensional [70] ones show the production of an anomalous high-energy tail in the electron thermal distribution. This may mean that the width of I-wave packets is at least larger than several (5 to 6) times the Debye radius de and a <1. * Equations such as (2.44) or (2.45) were applied by many authors to examine plasma turbulence as well as self-focusing of laser beams ins medium with a saturating non-linearity [64—67]. ~ Not so long ago there has appeared a work [129]in which the authors investigate also solitons in the framework of a model with a low-frequency saturating instability but in a form which is more general than (2.39) and (2.44).
V.0. Makhankov, Dynamics of classical solitons
35
2.7. Soliton-like solutions of equations describing excitons in one-dimensional tnoleCular Crystals A series of papers has appeared recently in which it is shown that various excitations in solids can be treated on the basis of equations like the Schrodinger equation. This refers, in particular, to excitons occurring in one-dimensional molecular crystals [71,73, 74] and in a-spiral biomolecules [72]. In refs. [71,72] the description of exciton chains was reduced to an S3 equation. However, as was pointed out in [73,741, the Heitler—London approximation is usually used to obtain the S3 equation, which implies dropping from the initial Hamiltonian a small, but as we shall see below, very essential term.* The exciton spectrum was found both for a model with rigidly fixed molecules [73] and a model which took into account an exciton—phonon interaction [74]: w(k) = ((~+ 2~cos ka)2 + (2js’ sin ka)2)°2. Equations for the creation and annihilation operators were also obtained; it was shown in [741that the classical analogs of these equations have the form ‘P
~4Dt=
—
(X4D~— ~
4D14D12,
—
(2.47a)
and i4D~=
—
4D* + aço~+ 4D~+
(2.47b)
4Dj4DJ2
We see, that (2.47a) differs from the familiar S3 equation by the term ~ but usually a I (for a definition of a and scale on which ~, x and t change, see [75]). Let us consider some properties of soliton type solutions of the systems (2.47). Multiplying (2.47a) by ~ and subtracting from it (2.47b) times ~ we get the “current” conservation law m9.Ia *2 2 * * ~I4DI~—I~(4D —4D )+(q ~ (2.48) ‘~
2
This yields the integral of motion (exciton number) (2.49) where, as before, S =
f çJ~dx. However now the current j
consists of two parts which we denote by
= ~(a)+ ~
(2.50)
where j
•(a)
=
*2 —I. 2ia(p ), i(4D~4D 4D*4D~). 2
—~
~(r) =
.
(2.51)
(2.52)
—
Relation (2.52) is a well-known current which usually appeared in the Lagrange field theory (see subsection 1.2). The sense of such a splitting up will be clear, if, using the conventional procedure, one tries to derive the energy integral. After some algebra it is easy to see that
2 i~ (ko+1
— ~I~I~) =
fi
(~~I~2 — ~) +
i( 4D~4D~~q~~)+ ~I2j(~} —
—
‘P12 ~
* Moreover, in obtaining the S3 equation one changes from Pauli to Bose operators which is an incorrect procedure and which leads to an uncontrollable error.
36
V.0. Makhankov, Dynamics of classical solitons
which was a constant of motion representing the energy of excitons in the a—*0 limit, now is no longer conserved. Therefore S~ associated only with the energy of the excitons cannot be thought of as the total energy of the system. Remember, that the terms ap~ and a4D~describe a reversible resonance interaction between excitons and molecules which may lead to a broadening of the spectrum line. Proceeding in the same fashion one can obtain i~(4D*4D+ ‘P~’P)= 2a(~+ 4D*2) ~a{
a(4D*4D~+ 4D4D~)+ I’PxI2_~I4DI4
—
Finally, integrating (2.53) over x yields 2~i~ dx, dS~°~ = 2a dS~°> = l4DI
J
J
(tp~+
4D*2)
4D*4Dxx
—
4D~4D}.
dx,
(2.53b) (2.54)
where 5~~) = if (4D*4D~ ~ dx is the momentum of a packet in the a —*0 limit. That is why we have put a superscript (0) on S~and S 2. After changing to the conventional “polar” representation of a complex function —
—
pçlie we have dS~°~/dt = 4a
J
dS~”/dt= ~a —
{(.~/,2
J
cos 2i~ ~
~,282)
sin 28} dx,
—
8X~//’cos 28
dx.
(2.55)
Let us now consider qualitatively possible soliton solutions of the system (2.47). In the a ~ 1 limit, to zeroth order in a, we have for the soliton solution the expression (2.31) in which 2—~A2. fl. 1+~v Employing such a solution one can, in principle, obtain corrections of first, second, order in a (e.g., by the method of successive approximations). Let us, however, try to find some particular solutions of the exact system (2.47). To do this, let 8 = kx tot, ~ = x Vt; instead of (2.47) then we have —
—
+ (to
—
1
—
k2)~/i+
~3)2
—
a2k2~fr2= (a2
—
(v
—
2k)2)iJi~,
2 2 —— a2ki/iI/i~ a k ~/~‘ (v 2k, I/fE + (v 2k)tfr~(t/f~ + Ai ti + Ijf~)~ 2
\2
—
an
2
—
(2.56) 2
57)
—
1
This system has as a particular solution a packet at rest governed by the following equation —
(~O
+
‘P3
(2.58)
= ±a~,
where the upper sign relates to the right-hand side of space (x ~ 0) and the lower one to the x ~ 0 half-space. Due to the x x symmetry of the initial system and hence of (2.58) and because we have ~ in eq. (2.58), the derivative ~ should be continuous and equal to zero at the origin. Bearing this in mind we discuss the equation -+
‘P+~’P+’Pa’P+.
—
(2.59)
V.0. Makhankov, Dynamics of classical sol(tons
37
Fig. 2.24. g,,, g-phase plane for eq. (2.55). 9D
__ ~ Fig. 2.25. Potential in which the fictitious particle moves.
Fig. 2.26. Solution of eq. (2.55) with a = 0.
This equation governs the oscillation of a point particle moving in a non-linear potential 12
U~~4D
14
+~4D
(2.60)
under a frictional force a4D~while ~ is the “position” of the particle and x is the “time”. To put it in other words, one may define I
2
2
14
e(4D)~{q,~—4D +~4D}~T+u as the energy of the “particle” and rewrite (2.60) in the form = (~+~ —
‘P
+
4D~)4D~ =
—
a’P~.
(2.61)
Figure 2.24 taken from [19] shows the contour curves of constant in the ‘P04D~-plane,and fig. 2.25 shows the potential u(4D) as function of the particle position ~p.The points F1 and F2 are those at which the energy has its minima. The point S is a saddle point. The solutions of (2.60) depend on the “initial” values of ~ and hence on e at x = 0. Now let a = 0. Given ‘P(0) such that uI~0= 0, i.e., 4D(0) = V2,* ~. = 0 there is one half of an oscillation from point F. to S corresponding to half a soliton ~This value corresponds to the point F, in fig. 2.24.
V.0. Makhankov, Dynamics of classical solitons
38
Fig. 2.27. Soliton solution spectrum for the case of a non-vanishing dissipation, a 0.
(2.31) with v =0, i.e., A
(see the curve C1 in fig. 2.26). Given initial values 42(0) in the segment [1, V2] we have oscillations of the C2-curve type. Finally, when 4Do(O)> V2 we arrive at oscillations of the type shown by the curve C3. The “dissipation” term ap~changes the picture. There are now, apart from a non-oscillating (nodeless) solution, three families of oscillating solutions (which have one or more nodes). The nodeless solution (curve C1, a normal soliton) starts now from point ç,~higher than 42(0) = V2 and the greater a the higher the initial point will be. Solutions of the first family C~ approach ‘P = 1 as x ~, for the second family ç(co) = I ; lastly, the third family of solutions being of the soliton type tends to zero as x It is quite natural to join the nodeless solution and the third set into one family as they have a similar asymptotic behaviour and to count C’~the nodeless solution belonging to C~family. Then we may arrange solutions from C~in an ascending order ‘P~(x),n = 1,2,3,. , where n 1 stands for the number of nodes. This means that even with 4D(c~)= 0 we have apart from the convex solution of the kind (2.31), the whole family of oscillating solutions ç~(x).The nodeless solution of (2.59) is approximately = ‘Pmax = V~
—
—*
—* ~
..
—
exp{ — ~ax}AV2 sech(Ax). To end this section we note that although we are unable to solve eq. (2.56) in the general case, k 0, one may suppose that such a solution will possess features somewhat similar to those of the C~ family, but being in motion.
3. Solitons in particle physics
3.1. General remarks The quasi-particle properties of extended solutions mentioned earlier have for many years attracted the attention of research workers [76—78]. The intuitive idea of particles as localized field bunches are older and start from the work of Born and Infeld [781.However, only recently the interest in such particle-like solutions of non-linear field theories has grown (one might even say exploded) again with a new force. There are apparently several motives for this explosion of interest and one of them is the *
See curves C~.i = 1.2,3.4 in fig. 2.27.
V.G. Makhankov, Dynamics of classical solitons
39
accumulation a large body of mathematical information about one-space-dimensional classical solitons Step by step there have been appearing the requirements which classical solutions must meet as a basis for a construction of quantum field particle models. The theory is based on the variational principle, which assumes the existence of a Lagrangian and hence other properties of Lagrangian systems. One may consider various combinations of fields as well as a single field with self-action, especially in the one-space-dimensional case. In the three dimensional case rather strong restrictions arise due to the renormalizability requirement. As a result we can have in the Langrangian a not higher than 4th-order polynomial potential. We shall see that, therefore, stable static solitons may occur only in a system of two or more fields. One of them is usually connected with spontaneous symmetry breaking. The natural requirement such a theory should also meet is that there are no field singularities (see [19]and papers cited there). Finally, the theory must be relativistically invariant. To begin with, define a classical soliton as a solution of the equations of such a non-linear theory which (i) has a finite and non-zero rest mass, (ii) exists in a finite region of space, and (iii) is stable (i.e., lives for a sufficiently long time). A final requirement refers mainly to the one-space dimensional theory: we require that (iv) solitons may interact inelastically. The first three requirements coincide almost completely with those given by Lee in [49] (see also [10])except that our definition includes pulsating solitons as well. This definition differs radically from a more narrow one given by Zabusky to describe solitary wave solutions of the KdV equation interacting elastically with each other. Such a narrow definition is of no use in particle physics since, as was reasonably pointed out in [10], “such a highly restrictive definition would automatically exclude all the four-dimensional local field theories that are of interest to particle physicists”. The soliton solutions are sometimes treated as “soliton bags” [9] defining “a potential or a self-field preventing a certain particle or constituent field quanta to tend to its asymptotic free field in remote past or future or as the coupling constant goes to zero”. And although those “bags” have been used to construct dynamical particle models the authors, probably motivated by the success of Gardner, Greene, Kruskal, Miura and Lax’s inverse method associated their theory, in fact, only with completely integrable equations. It is in this sense in which their approachdiffers from the one developed by Finkelstein, Lee, Vinciarelli, Bardeen et al. [122—123],Dashen et al. [83], and many others [80]. .~
3.2. Solitons in a (p4 field theory We have briefly discussed in subsection 1.3 soliton solutions of the Klein—Gordon equation with a cubic nonlinearity (KG3) and seen that they had certain particle-like properties. Here, to display characteristic features of such solutions we consider some one and two-field theories in two (one space and one temporal) dimensions. We restrict ourselves to non-linearities not higher than cubic in the equation (quartic in the Lagrangian) the so-called 424 field theory. 1. Let us begin with Landau—Ginzburg—Higgs (LGH) equation for a real (i.e., neutral) field (3.1) *
Another one is the way in which these solitons may be quantized as well as work on the theory of “topological” solitons (see the review paper by
Rajaraman [80]).
40
V.0. Makhankov, Dynamics of classical solitons
Fig. 3.1. Kink-soliton of the LGH equation. ~ is the energy distribution density in the kink.
its two-parameter soliton solution for sublight (v x(x, t)
m =
—
Im’y tanh~ç,~ (x
—
ut
—
‘1 x0)1,
< C)
constituents,* (3.2)
is of the form of a step (“kink”). This implies different boundary conditions at two infinities x = ± x, = ± m/g, and distinguishes the kink from a bell-shape-soliton of the KG3 equation. It is now convenient to rescale the variables as follows: ~ = (g/m)~,I = mx, t = mt; the results are then invariant with respect to the parameters m and g. The energy density for (3.1) is I
2
2
1
2
2
—1)}. (3.3) defined by (3.2) and (3.3) are plotted in fig. 3.1, where one can see that the kink-soliton energy, as for bell-soliton, is concentrated finite 2 = M2in+aP2 alsoregion holds:of space. It can be easily verified that for the soliton (3.2) the Einstein formula E ~C”~{x1+x~+s(x The functions x and
M=q-~-,
~
EMy,
P=Mvy.
The potential of the LGH system has the form depicted in fig. 3.2a, the corresponding one for the KG3 equation in fig. 3.2b.t We can also see that the kink starts in one of the two minima, goes over the barrier and finishes in the other minimum (track of a traveller passing over a hill). A bell-type soliton starts and finishes at the same level of the relative minimum (the track of a traveller climbing from a valley to the top of a hill seeing on the other side a bottomless pit and hurrying back). These characteristic features of kink and bell solitons determine as we shall see below, their stability. 2. Systems of interacting fields are sometimes considered which have soliton-type solutions. Note here refs. [9, 10, 122, 123] where relativistic equations were studied and [15,51, 60] devoted to non-relativistic systems. As we have seen above from the example of Langmuir solitons a system of interacting fields gives a considerably richer world of phenomena associated with the formation and interaction of solitons. Moreover, this occurs also for equations which are completely integrable in the self-action approximation.** *
~~15 Interesting to note, that constituents of this field have imaginary initial masses that seemed at first paradoxical and repelled theoreticians.
However, after the appearance of papers which initiated theories with spontaneous symmetry breaking, the interest in such equations started to grow (see refs. [130]). t Note, that the potential function a being introduced in subsection 2.7 which governs the motion of a “particle” differs from a real potential V of the system by its sign, i.e., a = — V. **The most striking instance is the interaction of s-solitons because in the self-action approximation both the S3 and Bq equations are complete integrable.
V.G. Makhankov. Dynamics of classical solitons
(a)
41
(b)
Fig. 3.2. Potential energy density in (a) a kink as a function of the field
x: (b) the hell-soliton
as a function of the field versus
~.
Consider as an example soliton solutions of systems with the Lagrangians ~/.3 ~
~SK’P
I
~
I
2’\
2
2
22
x)~
(3.4)
and 2
-~KKH’P1I
—l’P~I —m 2
2
2
421
2
+g
I IcoI X+~(X 1X~/.L x). 2
2
2
22
(3.5)
In the first case we have interacting complex Schrodinger and real Klein—Gordon fields: (3.6a) 2
2
2
LI~+~t~—g~4Dj
The
=0.
Lagrangian gives rise to interacting real and complex Klein—Gordon fields 2ço g2çox = 0,
~KK
—
U4D
+
m
(3.6b) (3.7.a) .(3.7b)
Following [9] we use a parametrization to find solutions to the systems (3.6) and (3.7) ‘P(x, t) =
~I~(h)
exp{iA(~)},
~(x, t) = I/i(f),
(3.8a) (3.8b)
where h=k v
=
f=
1x—w1t, k2x w2t, k3x — to3t, —
with k, and oi1 (i = 1,2,3) being constants. Upon substituting (3.8) and (3.9) into (3.6) one arrives at 2~~I/i = 0, ~ ±2A’) A~I+ g
~
(3.9a) (3.9b) (3.9c)
(3.lOa)
42
V.0. Makhankoi.’, Dynamics of classical solitons
(3.IOb) (3.l0c) Then as in the case of the Schrödinger equation (see subsection 1.3) we search for a particular solution of (3.10) assuming the function exp{iA(v)} to be a plane wave, viz., A’(v) = const.
(3.11)
We then have from (3.lob) (3.12) or 4’ = 0 which is of no interest. Relation (3.12) yields inw~
/to
1
to1w2
A =-~--1--v=ml-i--x—-~—i-—t ~i~2
\“i
t(1K2
which means that k0~p
=
rn(w~/k~), ~
=
rn(w1w2/k1k2).
(3.13)
Using (3.12). (3.lOa), and (3.l0c) one obtains (3.14a) (3. 14b)
2~2= 0. (k~ w~)I/i” In the case where —
—
+
~
g
to
± k3,
3
we obtain the exact solutions to (3.14) putting (3.15)
which gives f = h, or k1 = k3 and to1 = to3. Then eqs. (3.l4a) and (3.l4b) coincide if a compatibility condition is fulfilled leading to the following dispersion relation for the constituents: p
2
2m
2
2
~/2m _p 2/,LK, 1—p/rn 2m
22
2
and 2
12
K
where
2/rn2. —p Note, that in subsection 2.4 we got a formula analogous to (3.16) for a massless second field. -2
~
p2
A2
2
and the non-linearity played effectively the role of mass with ~
=
A2/2.
V.0. Makhankov, Dynamics of classical solitons
43
The solutions of the system (3.14), ‘p= ±~~_?~sech2{~(x — Vt — xo)} exp{i(px
=
—
(3.17a)
t)},
sech2{~(x Vt — xo)J~~
~
(3.17b)
—
are singular with respect to the coupling constant g and can be interpreted as follows. The x field defined by (3.l7b) is a self-consistent potential governing the solutions of the Schrodinger equation.* Its solutions (3.17a) describe a finite (extended) non-relativistic particle consisting of constituents with a binding energy, =
~2/2m,
—
which is determined by the mass ~zof a ,y-”quantum” and may be found from formula (3.16) putting V = p/rn = 0. This formula displays also the inconsistency of the system (3.6), since one of the interacting fields is relativistic but the other one is not. This fact is more clearly seen from the difference between the soliton velocity v and the group velocity, de Vg=•~[~=
/
2
2
V1 1j—----flf
The validity condition for the above system is therefore ~ m, that is, p ~ m, V Vg. The parametrization (3.8), (3.9) and the relation (3.11) applied to the more consistent (in the above 2 = (p2/2)< 1, mentioned sense) system (3.7) gives for V ‘~
‘P(X 2t)
= ±~T
x(x, t)
=
sech2{~(x —
j~rsech2{~(x
—
Vt
Vt
—
—
x0)} exp{i(px
—
t)},
xo)},
(3.l8a) (3.18b)
and the dispersion formula 2
=p +m —p~ 2
2
2
hence — Vg~
2<
The
V
2
__p_
p
I V condition leads now to 2
i.e., the mass of a “quantum”-carrier of the interactions is less than the mass of a “charged” 42-field “quantum”. The binding energy of the latter in a soliton is defined by the formula 2 — 2_ m. =
\/m
An analogous consideration applies for constituents with imaginary masses; we then shall get kink-solitons instead of bell-ones. *
“Quanta” of the x field can be interpreted as carriers of interactions between the “quanta” of the complex g field.
44
V.0. Makhankov, Dynamics of classical solitons
As an example of systems consisting of a complex 4D-field (with the zero boundary conditions at both infinities) interacting with a scalar k-field with a degenerate vacuum, we give the models suggested in [10] and [79]. The first one is described by (1.14) or in the plane geometry (broken symmetry) by 2X2’P0, 319 ‘P11”’P~~+a
x
2xI’Pl2+~x(x2—1)0;
(
1~—x~~+2a the second one by
LIi4D+(m2—1+X2)4D+2423=O, Ux + I + 2142 I2)x + =o. (—
320
~
-
One can easy verify that both systems have as exact particular solutions the following solitons x =
tanh(Kx),
(3.21)
‘P
V2 A exp{
—
i(tot
—
8)} sech(Kx),
where A = (1 4K2)~2/2K, K2 = a2 to2 in the first case, and A = (1 2K2)~2,K2 = m2 to2 in the second one. The solutions (3.21) exist, if to2 = K2 = a2/2 for the (3.19) model, and if max(0, m2 ~)~ to2 ~ rn2 for the model (3.20). This means, that (3.21) is a solution of (3.19) only in a single point to2 = a212 from a total region of the existence of soliton-type solutions 0~to2~a2,and thus the amplitude of this solution is fixed. In the second case we have a range of variation of the frequency to and hence of the amplitude A. This means that such solutions, like those in the case of the complex KG3 equation, have an internal degree of freedom (U(1) symmetry of the 4D-field). The mass of a soliton of eqs. (3.20) is 42 2K )/K), (3.22a) M =charge 2(~K is+ rn(1 and the —
—
—
—
—
2
—
Q = 2to 1—2K2
(3.22b)
This soliton may be considered as a “bag” of the x-field in which charged “mesons” of 42-field are locked in (see analogous models in [79,85, 122, 123]).* 3.3. Properties of plane solitons due to their charge We have early defined the charge of a soliton Q=iJ(’P*’Pt_ 4D~’P)dx as the constant of motion associated with the U(1) symmetry (the global rotation in isospace) called sometimes the internal symmetry [79]. It means that in the Q 0 case we have an added internal degree of freedom as compared to the neutral scalar field and as we see later it effects the stability of solitons. *
Using the Lorentz-invariance of the system (3.20) one can easily get solutions of the type (3.21) for moving solitons.
V.0. Makhankov, Dynamics of classical solitons
45
Following refs. [79,84] we illustrate the main properties of solitons due to their charge. Consider a model with the Lagrangian — a ~p*a(p V(I42l2), where we assume that V~0and V-+rn214212 as =
—
I4Dl—~0.
The equation of motion then is 142
+
=
0,
(3.23)
and the energy of a soliton
E
f
~ = I~I~ + Ico~l2+ V(1
~ dx,
2).
(3.24)
421
1) Varying E with respect to
~,
but keeping ~ and Q fixed we find that
ç(x, 1) = ~I~(x, t) exp{ — i8(x, t)} is the lowest energy solution if 1
8~(x,t) 1(x, 1)
4D(x, t) =
~ç~= 1(x) exp{
—
i(wt
—
=
=
const, and 8~= 0. Therefore
0,
8~)},
(3.25)
and r=~+to2~2+
Q= 1
(3.26)
dx,
12
+
V(q2),
to2~I 21 —
(3.27)
~
V(~42)= 0.
If one regards, using (3.27),
to
(3.28)
as a function of Q and a functional of c~,then eq. (3.28) is easily verified
to be a consequence of the variational principle (ôE/ô~t1)IQ= 0,
(3.29)
from which the relation* dE(Q)IdQ
=
to
(3.30)
follows. 2) The solution of the form (3.25) has a finite energy only if the 4-field tends to zero at both infinities and if to2 < m2. The first statement follows from V ~ 0 and formula (3.26). The second one is obtained by employing (3.28), the first condition, and the fact that V = m21(p12 as ~ 0. Then for large xl =
*
(m2
—
w2)~t
Note, that (3.28) as well as (3.30) may be obtained by varying the functional F = E — wQ with w being a Lagrangian multiplier (for details see [10]
and below).
46
V.0. Makhankov, Dynamics of classical solitons
so that the function exp{— KlXl}
= C
vanishes as lxl—*~,only if to2< rn2. 3) The energy of a soliton is always less than the energy of a plane wave packet with the same charge Q. We can enclose the system in a linear box of length l—*cx. Then, from (3.27) and (3.28), one can find, neglecting terms O(l_3/2), the plane wave solution —‘
(~~) 1/2
‘Ppi
exp{i(kx
=
—
wt)},
to2 =
k2+ in2,
with an energy E
0~= toQ which is a minimum for k
E~ ~ Emin =
=
0, and
Qrn.
2< rn2 (in fact, since both the energy and the Lagrangian include only conjugate From the condition to pairs, ~ and 42*, one may simply assume to ~ 0) and relation (3.30) we get
dE/dQ
= to <
rn.
Consequently, for any Q 0, E(Q)
=
J
to
dQ < Qm.
(3.3 ~)
The following four properties of charged solitons will be formulated here without proof. We shaH discuss them later in more detail. 4) The value of to is in the three-dimensional case always greater than a certain tomjfl which follows from a virial theorem. 5) Though possible values of to lie in the range tomin < to < rn, the charge Q has only a lower bound. As an example, see formula (3.22b) which gives Q ~ as K —*0. 6) In the K—*O case (w—*rn —0) the solutions of eq. (3.28) tend to those of the KG3 equation for which —‘*
E =
Qmfl _~(Qg2)2± O(Qg2)4},
V
m 2~2(l 2g2~2+
Qg2~1,
if —*
—
as —*0. In this sense the frequency range to —~rn 0 may be considered as the one near “free” field-plane waves in a large box. It should, however, be stressed that soliton solutions differ principally from free field ones by virtue of the third property (formula (3.31)). Note also, that in the K—*O limit the size of a soliton increases without bounds. 7) There exists a conserved, so-called topological charge in the theories with a degenerated vacuum (see, e.g., the kink-soliton of the LGH equation) —
Q t0~,=
x(x
= +
~) ~(x —
=
—
cc).
V.G. Makhankov, Dynamics of classical solitons
47
3.4. Bound states of solitons 1. The bound states of solitons have been examined at length in the sine-Gordon theory [80—83]
xt
2 sin
—
x~+ m
x =0.
An exact doublet-soliton solution has been obtained:*
x
=
4 arctan{tan a sech z’ sin(z” + 8)},
(3.32a)
where m’yV’l A2(x = myA(Vx t),
=
—
—
Vt),
(3.32b)
(3.32c)
‘—
2
arcsln[m 2(1 A )], (3.32d) and A = /my (see (1.40) in subsection 1.2). The solutions such as (3.32), representing a kink-antikink bound state, have been given various names; for instance, “mesons” (Perring and Skyrme [77]). “breathers” (Ablowitz et al. [81])and “bions” (Caudrey et al. [81]),and many other ones (see 180,811). The mass of such a bound state which, following Caudrey Ct al. [81], we call a “bion”, varies from zero up to two soliton masses, viz., a =
—
0~Mb~2MS~ l6m,
(3.33)
depending on the internal state. Note that due to the complete integrability of the sine-Gordon equation “bions” can not appear as a result of a collision of solitons (a more detailed discussion of sine-Gordon bions for the interested reader can be found in ref. [80]). 2. We shall now consider bions (hereafter we use bion without quotes) for the Higgs field equation (3.1) which is not completely integrable. This last statement follows, for instance, from the fact that the collision of solitons in the framework of eq. (3.1) can give rise to the creation of bions [26]. Examine the evolution of the initial state
x
1~(x,0)
=
tanh
~3’= (x —
Vt
—
x0)
—
tanh
(x
+
Vt + x0)
—
1,
(3.34)
which is a state consisting of a kink and antikink moving towards each other (see fig. 3.3). If the distance between the solitons is large, so that V2x0~1/y,
(3.35)
then the function (3.34) is a solution of eq. (3.1) with exponential accuracy. The difference
2)x 2~, m 1~ + g can be easily obtained. In the centre-of-mass system =
F(~)~ (Li
I
x+vt+x
3Lt~~
~,,_
=
+tanh *
—
0 —tanh x—vt—x011/ x+vt+x0\/ x—Vt—x0\ j 1yanh ~ )I~,,tanh ~ ~,,_
x+Vt+x0 — —tanh x—Vt—x0 — —l V2 V2
This is probably of immediate concern for the complete integrability of the sine-Gordon equation.
(3.36)
48
V.0. Makhankov. Dynamics of classical solitons
///~/ ~
/
~- I
\~ ~
__
/
__
Fig. 3.4. Bion field function x in the frame-work of the LGH equation at various instants of time.
Fig. 3.3. Initial state for the interaction of kink and antikink solitons.
Since the exact solution of the above initial problem has not so far been obtained, some steps have being undertaken to study this problem by computer by Kudryavtzev [26] and by Bogolubsky, Getmanov and the author. We place the solitons at rest at relatively small distance apart, x0=1/V~ they attract each other and an oscillating state similar to a sine-Gordon bion arises. In fig. 3.4 the function x(x, t) is shown for various times and fig. 3.5 depicts X(x, t) as function of t at the origin. The system is seen to achieve a quasistationary oscillating state in quite a short time, so that the period of oscillations, T = 5\[2/m, does not alter practically for a time of the order of hundreds reversed masses. This means, that radiation at infinity due to these oscillations is very small (some calculations give the energy decrease per period to be 0.5%). An analogous state has been obtained as a result of the head-on collision of two (kink—antikink) solitons moving with equal velocities, v = 0.1, [26]. In order to clarify this phenomenon let us follow ref. [26] and consider how the potential energy of a two-soliton system such as (3.38), +2 ~(x~ 1)2] dx, (3.37) E~=
J
—
~ [(a~x1~)
depends on the distance between the solitons I
2 describing the kinetic 0. Note that the term (öxXjn) energy of an oscillating motion of constituents in the soliton (i.e., a motion of the “quanta” in the =
2x
______
Fig. 3.5. Plot of function function of time,
x(0~t)
in the centre of the bion as a
Fig. 3.6. Potential energy of a kink—antikink system as function of the distance between the solitons.
V.0. Makhankov, Dynamics of classical solitons
49
0
x Fig. 3.7. Field function for solitons passing through one another. It is not even approximately a solution of the Higgs equation.
well), is a fraction of the potential energy of the whole soliton since E~=
EI~
0
(Ek + EP)JVO.
An approximate plot of the function E~(l)is depicted in fig. 3.6. It can be readily understood as follows: when 1 ~ 1 the potential energy is a constant and equal to the sum of the soliton masses:
~
=
/ 4v2m
3
—~-—--—~-
=
2M5
(3.38)
when / —*0 (overlapping solitons) E~tends to zero with I. After solitons begin to diverge E~grows with jl~remaining positive. This becomes obvious if one pays attention to the fact that the function x~is no longer a solution of eq. (3.1) after the solitons have passed through each other, and the longer the distance between them the greater the value of 5~*Therefore from figs. 3.6 and 3.7 it follows that neglecting radiation solitons repel one another and will diverge. Thus in the absence of radiation two types of motion may occur: an infinite one if the energy exceeds the sum of the two soliton masses and a finite motion in a well for smaller energies. Radiation changes the picture. The creation of a bion may take place (and the motion becomes a finite instead of an infinite one), if an excess energy equal to or larger than L~E= E(x!fl)
—
2M~
(3.39)
can be emitted during the period of one oscillation. The radiation power has been evaluated in ref. [26].It tends to zero with ‘rad
tanh
ç~=(i
—
J/J as
tanh 1)
(3.40)
It is this which ensures us that such a bound state is quasi-stable and that the bion is long-lived. The above estimate of the radiation power (0.5% for a period) gives the following lower bound for the bion life-time 3/m. Tb~200T= \/2 l0 The upper bound for the velocities of solitons which can collide to produce a bion state is thus Vlhresh =
*
0.12,
The distance between solitons which have passed through one another is given a negative sign.
50
V.G. Makhankov, Dynamics of classical solitons
in accordance with the value that has been independently obtained in computer experiments carried out at Dubna. 3. Shirkov has recently suggested and examined the whole class of equations of the LGH type with polynomial potentials in a Lagrangian of higher degree than He has noticed that for the potential ~.
U(~,n)=~—fl (1-g~)~/fi lwiwn
(3.41)
(l+y~)~
1wjw~n
one can obtain a quadrature solution (see subsection 2.6):
f
d42
=x—x0. (3.42) J \/2U(ço)+c~ There are 2n + I absolute minima for such a potential (stable equilibrium position; see fig. 3.8) symmetric around the central one ~ = 0: 1, k1,2,...,n. ‘Po=0, ‘P~k=±g~ The integral (3.42) is for c 1 = 0 reduced to 2x,
2
1”k
‘P c e± fl (1 —g~(p2)~ =
The potential
(1—g~u)fJ(1+y1u) 11(1 —g~u)
(3.43) u=g2
-
U( 42,
1) yields
~~2~±2x
2~2~ I four solitons (see fig. 3.9) with equal masses:
which describes for g
M=J(~
42~+U)dx=2JU(42)dx= JV2Ud42.
In the case m
=
0, 2X)h/2~
M =
(344)
~
(I + e± We have then four pairs of solitons at n = 2, m = 0. Two of them “tie” the central vacuum ~ with the first one p~g~ = I the two other ones the first vacuum with the second one p~g~ = 1. In the particular =
—
Fig. 3.8. Potential having n + I stable equilibria.
V.0. Makhankov, Dynamics of classical solitons
SI
P -~
—--
~-=-------—----
Fig. 3.9. Four solitons for the potential U(~,I).
case g1
=
‘~
Fig. 3.10. Eight solitons for the potential U(~,2).
\/~g2 g the masses of these solitons are, respectively,
2. 5/24g and their solutions
M1
~/
=
g~p~ = I
±(1 +
M
2,
2
=
l/24g
2 e~)112.
(3.45)
The general structure of such solutions is depicted in fig. 3.10. These results may be qualitatively generalized as follows: the potential U(
42, n) defines n pairs of four solitons. A soliton of the kth foursome “ties” the (k — 1)th vacuum with the kth one and has a mass Mk
gf
\/2U(42)dço.
k-~I
This class of models is apparently intermediate between the 424 and the sine-Gordon ones. In the n cc limit the potential assumes a periodic structure and solitons become identical. Moreover, one may assume that for U(42, n) there exist a finite set of bound bion-like solutions which is transformed into a continuous spectrum of the (3.33) type in the n —*cc limit. This assumption was verified by computer by Katyshev and the author and we found: (i) that collisions of solitons in the U(42, 1) and U(42, 2) models are inelastic, and (ii) that bions of the type given in fig. 3.4 may arise as in the Higgs model. The inelasticity of the soliton interaction and hence the radiative emission is very small so that the collision of solitons with different masses are like those of billiard balls. An analogous investigation was carried out (by computer by Getmanov) to show the possibility of the creation of a bound two-soliton state in the frame-work of the KG3 equation. Such states were, indeed, found [27]. This fact is no surprise, if one takes into account that we had a picture of non-linear oscillations of an f-wave packet in the non-relativistic limit, i.e., the S3 equation. That picture resembles the bion state (see fig. 2.9). However, due to the complete integrability of the S3 equation the creation of bions as a result of soliton collisions was forbidden. Including the second field ~ led to the dissipation of the non-linear oscillation energy and to the appearance of bound soliton-states becoming in the limit a large mass soliton. In the present case the weak radiation is possibly associated with the quasi-decay type of the small-oscillation spectrum, 2 = k2 + m2, —*
to
since the mass term may be compensated by a non-linearity.
V.0. Makhankov. Dynamics of classical so!itons
52
3.5. Approximate analytic description of bound soliton-states in non-integrable systems Here we discuss a method of approximately solving the equations of the 424 field theory which can be developed since these equations are “near” the integrable sine-Gordon equation. The basic idea of this method has been prompted by computer experiments on (i) the Fermi—Pasta—Ulam problem, (ii) the scattering of solitons in the Higgs, Klein—Gordon, Boussinesq and Korteweg—de Vries models and has been surmised from time to time by various authors. Its outline is the following: the behaviour of a system may be defined by its “vicinity” to a completely integrable model. Such a “vicinity” might be verified, for instance, by means of the scattering of solitons. The smaller the coefficient of inelasticity of the interaction of solitons in a given model, the nearer this model is to the corresponding completely integrable analogue and with the higher accuracy many-solutions of the latter approximate solutions of the former one. This is especially related to the bound soliton-states. This idea was used to find approximate many-soliton solutions in the framework of the ~ field theory in ref. [27].In an earlier paper [82] an analytical approach was developed to investigate non-linear self-localized oscillations of the KG3 equation on the basis of an asymptotic method. However, that technique was valid only if the amplitudes of the oscillations were sufficiently small (in the region near “free” field solution when K 0).* Following [271we consider (3.46) L1~+m2sin~0, -+
with the one-soliton solution
x =4arctanet~,
~
—
rny(x— Vt)+6,
(3.47)
and the equations Li~+ m242(1
0,
(KG3)
(3.48a)
1142—2m242(1—422)0,
(LGH)
(3.48b)
_2422) =
(the coupling constant has been removed by a i~calingg2422—~~ Solitons of eqs. (3.48) are, respectively, = ±sech
z,
(3.49a)
±tanh
z. The transformations
(3.49b)
= ±sin ~,
(3.50a) (3.50b)
‘PH =
42k
‘PH =
±cos
~hange the one-soliton solution (3.47) of eq. (3.46) into the solutions (3.49) of eqs. (3.48). The main assumption is that the transformations (3.50) applied to exact multi-soliton solutions of (3.46) give approximate solutions of corresponding equations (3.48), apart from small inelastic effects. This hypothesis has been verified by computer simulations. We give a few examples. Collisions and bound states of a soliton—antisoliton configuration in the barycentric system for the sine-Gordon model are x=4ar *
11 sinh myVt an~— [V cosh m’yx
.
A similar work has been done by Dashen, Hasslacher and Neven 1831 but for the Higgs field equation.
(3.51)
V.G. Makhankov. Dynamics of classical solitons
53
Using (3.50b) one can get the relation — ~PH—
sinh2(m’yvt) v2 cosh2(myx) sinh2(myVl) + V2 cosh2(rnyx)’
(3 52)
—
which describes an elastic collision of the kink-antikink solitons of (3.49b) quite satisfactorily. As t—*oo, we have ‘PH
—~
±{tanh[my(x vt)+ 8]— tanh[my(x —
+ Vt) —8] ±I},
6
=
In v.
The relation (3.52) pictures well a situation taking place in a numerical experiment by Kudryavtsev for V> V(hresh = 0.2. When, on the contrary V < VthreSh, an energy greater than a sum of the kinetic energies of the colliding solitons is carried off by the radiation and as we have seen a bion is formed. The bion of the sine-Gordon equation may be described by the formula (3.51) if one puts v—*iu:
x
=
4 arctan{i~(x,t, a)},
(3.53)
where ~(x, t, a) = tan a sin(mt cos a tana=l!u,
—
6) sech(mx sin a)
0~aE~ir.
Application of (3.50b) gives the formula (3.54)
which describes the bion of the Higgs equation, if the parameter a is taken as a slowly decreasing function of time. A quite similar operation may be applied to (3.53) by means of (3.50a) to obtain a KG3 equation bion: 2~
(3.S~
42k~2+l.
Such a bion has been found in computer experiments carried out at Dubna. This state turns out to be stable for sin a ~ 0.2. The oscillations (3.55) are followed by a slight radiation, the intensity of which decreases with amplitude. As the result of such radiation large amplitude (heavy) bions subject to eqs. (3.54)~and(3.55) transform into small oscillations around the vacuum fixed by the boundary conditions (~ = 1 for (3.54) and q~= 0 for (3.55)). The spectrum of the “free” small oscillations in the Higgs model is of the form toi~,. = k2 + 4m2. The relation (3.54) shows that the bion frequency lies in a forbidden region of the small oscillation spectrum, viz., tob < and tends to the lower bound of spectrum as a 0. Thus even a very light bion can not decay into free oscillations. Light bions of the KG3 equation have been studied in works by Kosevich and Kovalyev [82] and of the Higgs equation in that by Dashen, Hasslacher and Neveu 183]. But the series obtained by the above authors are asymptotic and hence their results have rather denoted the possibility of the existence of such oscillating solutions. Numerical experiments [26—27]have confirmed the reality of such states. —
0lj~i
—~
54
V.G. Makhankov, Dynamics of classical solitons
3.6. Quantization of solitons From the particle physics point of view the greatest interest is to develop the classical level of a description of particle-like solutions into the quantum one. This problem has now its history. One of the earliest work in this direction was done by N.N. Bogolubov [88]. But it had no followers at that time probably because the classical theory of solitons was at a very initial stage. At the present time a fast flow of papers devoted to these and associated problems has sprung up. The main results of these researches up to the begin of 1975 have been summarized in a good review by Rajaraman [80]printed in this journal. Furthermost development of the theory was reflected in the proceedings of the Conference on “Extended systems in field theory”, Paris, summer 1975 [142], as well as in the rapporteur talk by Faddeev at the Eighteenth International Conference on High Energy Physics, Tbilisi, July 1976 (see also his review paper “Searching for many-dimensional solitons” in the Proceedings of the Fourth International Symposium on non-local, non-linear and non-renormalizable field theories [140]. Therefore without giving technical details we summarize here only a few results (undoubtedly well-known to specialists), as the subject of the present paper is the classical theory of solitons. To begin with, let us note that quantum solitons are descendents of classical ones and in a weak coupling approximation, g2 ~ I, quantum soliton (QS) properties are with good accuracy described on a classical level. The masses and the form-factors (dimension) of QS approach their corresponding classical values and remain finite, as g —*0. This is radically different from conventional quantum objects (bound states), i.e., atoms and molecules which occur only at the quantum level. First some general remarks [49]: 1. It is convenient to construct a theory of QS if all non-linear couplings would be defined by a single constant g2. In that case, for example, let (3~42)~—~ V(g~),
~=
where V has a minimum at ~= ~
0, so that
V(g42)=~m2422+O(g(p3)+O(g2424)+~,
(here ~ may stand for a system of interacting fields). In the g —*0 limit we have the linear equation L1(p2 + m2p = 0 with solutions which are the usual plane waves (free field). 2. The classical soliton solution, as we have seen above, is singular in g therefore we may define ‘Pci =
(pig.
Then
and ,.1f~=
—
~(a,~
— 2 V( 42) 42) will not depend on g and neither will the soliton solution
~,
as long as
h.
In fact, for
g 0. (This fact was repeatedly pointed out earlier.) 3. The quantum action Sq =
Sci
is proportional to ~ or
~
2 plays a role similar to that of
One may expect that g
V.0. Makhankov, Dynamics of classical solitons
55
small g the quantum soliton solution may he written as a series in g2 like that in h. Moreover, the leading term of this series corresponds to the WKB approximation and tends to the classical limit as
g—*0. 4. A method of an expansion of quantum field operators near the classical soliton solution [10, 18, 80] has been applied to quantize stationary solitons and “charged” Q-solitons reducing to them:
{-~
ç(x)
=
+
~ q~(t)~tn(x)} e’~’.
Two types of divergences, the ultraviolet and the infrared ones, naturally appear when calculating the energy of QS. The first one may be treated with a correct procedure of introducing the counter terms when we subtract the vacuum energy [80,85, 86], which is apparently possible for renormalizable theories. The latter being associated with the translational modes may be removed by means of either certain artificial manoeuvres [114] or by the method of collective coordinates [10,84] (see [80, 131] as well). In this way the mass of a kink-solition in the Higgs model with allowance for quantum corrections associated with zero-point oscillations is M_2\1’2m
3m
3g2
—
+
m
V2~r 2\/6~
The first term in the g —*0 limit is the energy of the classical soliton. Taking the translation mode into account gives rise to the correct value of the soliton energy [10,80]:
E~ VM2+P~,
P,, =~2rrn/L,
where P~is the soliton momentum, L is the length of the box. An analogous formula for a kink-soliton in the sine-Gordon model,
is
8m3 m8m M—---7-—-——-—--, g ir y
—
y—
g2/m2 2
I—g/8irrn
The quantization2 ofwith a Q-soliton in a weak coupling theory leads to theofsoliton masscharge to coincide its classical one. However, the eigenvalues the total Q as ina the lowest order in g generalized momentum conjugate to the angle variable i~ (Q = i~8/M) are all integers: 0, ± 1, ±2, ±3 Applying, e.g., the Bohr—Sommerfeld quantization rules gives in fact 5.
E~ E~ —
1
dE/dn
=
whence, using the relation (dE/dQ) =
f dE n=j Eo
to,
we get
Q(E,)
F,,
—~=
I j
dQ=Q(E~)—Q(E0),
Q(Eo)
which proves the suggestion. 6. The mass of a quantum bion in the sine-Gordon model may be obtained with the help of the
56
V.0. Makhankov, Dynamics of classical solitons
WKB method (for arbitrary g2) [80]: Mb(N)=~~~sin(L~—7),N1,2,...<8ir/y. 16 The continuous spectrum of a classical bion becomes a discrete quantum one for 8ir/’y> 1. The quantum features of a bion are very remarkable. In its lowest state, N = 1, the bion corresponds for small values of ‘y/i6 to a free constituent and in the case N > 1 it represents a bound state of N constituents. A more detailed description is beyond the scope of our review so the reader is referred to the paper by Rajaraman mentioned above [80] and to works by Faddeev and coauthors [81]. 7. The formfactors of QS may be obtained on the basis of the classical solution using Fouriertransforms: (p!
0~(O~ 0)Iq) =
where
4.
p
—
~—
J
exp{—i(p
—
q)x}~(x)dx,
q is the momentum transfer
Stability of one-dimensional solitons
4.1. Some general remarks One of the most important and natural requirements for solitons both in plasma theory and in various field models is the condition of stability. We have already mentioned that the soliton must have a sufficiently long lifetime from the point of view of the processes under consideration. In other words, the soliton lifetime must be well above a characteristic soliton interaction time. In some papers the more complex requirement of absolute stability of the classical soliton is put forward; moreover the requirement lies at the basis of the definition [49]of a soliton as “a classical solution of a non-linear local equation which is non-dispersive in space at all times” and, generally speaking, has no singularities [19]. In the case of initial perturbations having the soliton symmetry we call the soliton stability a “longitudinal” one. In the opposite case, when the trial perturbations have a smaller symmetry we have a “transverse” stability. For plane solitons the latter imply a study of the stability with respect to soliton perturbations which are perpendicular to its direction of motion. The first investigations of the existence and number of particle-like solutions were mainly related to the longitudinal stability. Thus, in refs. [76] soliton-like solutions are considered of the equations 42”
=
42(1
—
421X)
with the boundary conditions 42(0) = ‘P(co) = 0. For n> I the absence of Liapunov stability of all possible solutions to this problem was demonstrated. Further work in this direction was devoted to the stability of plane solitons of the sine-Gordon and KdV equations. This problem is discussed in detail in the review paper by Scott et al. [5]. The paper [891 is related to the Klein—Gordon equation. Since the publication of this paper a considerable amount of work on the longitudinal and transverse stability of different types of solitons has appeared. We note merely papers on the transverse stability of solitons of the KdV [90]and Schrodinger [91,92] equations. These papers use both analytical and computational methods. Finally, we must emphasize that one cannot consider the soliton stability problem as a completed (historical) one. Papers on this topic appear with a rising frequency. Therefore our review does not
V.0. Makhankov, Dynamics of classical solitons
57
claim a completeness, but is rather an attempt to demonstrate some methods which are efficient in these problems. It is natural that as before we do not describe the results of the review by Scott et al.
[51. Some stability problems of more-dimensional solitons are discussed in section
5.
4.2. The longitudinal stability of plane so/itons The purpose of this section is to remind the reader the method used for the relativistically invariant Klein—Gordon equation with a cubic non-linearity and the Higgs field equation as examples. 1. We have = 0. (4.1) ‘Pt, V242 + m2p — 2g2~42~242 Let us consider this equation in a one-dimensional geometry. If the field ~ is real, we have —
2
23
L142+tn 42—g42 =0.
(4.2)
This equation has a two-parameter family of solutions ‘P,,(x, t)
m sech{m
=
7(x
—
Vt
—
x0)}.
(4.3)
We say that the soliton solution (4.3) is stable when we write the solution of (4.2) in the form
(p =
42,,+~/,where
II ~II ‘~
at t0
ll~lI~
we have in some metric that I~/’Ilremains bounded for all t from zero to infinity. In ref. [93] it was found that the solution (4.3) is unstable. Let us find analytical solutions describing the evolution of this instability in the soliton rest frame and the instability growth rate 1~.Introducing the dimensionless variables I -~tim, x x/m, 42-4 m’P/g and linearizing eq. (4.2) near the solution (4.3), we get for small perturbations the equation* —*
ITh/i+(1 —3q~)~=0.
51R(x), we have an eigenvalue problem
Separating variables, ~Ji(x,I) = e’ ~ +(E+3ço~)R= 0, E——(1+fl2),
R—*0,
Jxl-*co.
It is easy to see that this problem has a solution related to the translational symmetry 42(x) (see, e.g., [5,86]): R
2(x), 4(x) ~ d42,,/dx, that is, R5(x) =
sinh(x)/cosh
E
1
=
—
1,
—*
p(x
+
/)
fi = 0.
Since this solution has a node, a nodeless solution with E0 < E1 must exist. One can find it by the method summarized in ref. [1191 2(x)’
R0(x) = cosh *
E
0 = —4,
El =
±V3,
Here and below we study the linear stability only, since one can conclude the occurrence of an instability from the linear approximation.
V.0. Makhankov, Dynamics of classical .so(iton~
and show that the problem has no bound solutions except R0(x) and R~(x)(see [119]).This means that the correction to the solution (4.3) can be expressed as 2(x) eSlI, cJ ~ 1, ~4’(x, t) = c sech where the instability growth rate is El = V3. In the laboratory frame the growth rate is smaller: ~iah
=
2. For the Higgs field equation,
x~—V2~—m2~+g2~3=0, we have rn Iym X~=±-~tanh~-~(x--Vt—x 0) Letting again x
=
x. +
~i
we get for q~ithe equation
One of the stationary solutions to this equation is the function 2(x/V2), cosh which has no zeros and k?~= 2. — Separating the variables ~t’= e~’R(y),y = x/\/2, we see from the equation =
=
—
cosh2
~R~ +
y R
(El2 + 2)R = —
E
0R, 2 = 0 is the state with minimum energy. For other states E 2 + 2 < 2, or that El 9> 2 and El Therefore the function ~t~’ is oscillatory: ~/‘~ exp( ±i~flIt). 3. We return now to eq. (4.1). It is convenient to write the solution of (4.1) in the form
112<
0.
—
=
B 0 sech(7B0(x — vt — x0)) exp[
±
—
B~(vx
—
t) +
ii9j.
(4.4)
Here we introduce the amplitude B0 which is related to A0 by
B~=1—Ag, and changes within the interval 0 ~ B0 ~ Bmax = I. As B0 I the soliton solutions (4.4) are reduced to the solutions (4.3) which are unstable with respect to longitudinal perturbations. Let us consider the limit B0—+0. In this case introducing the function 42(x, t) which is a slowly varying function of time, —‘
ç=e
/ x ‘I~~=,t
and neglecting ‘P,~,we get instead of (4.1) the Schrodinger equation with a cubic non-linearity (S3) i~,+ ~
+
I~I~ =0.
Soliton solutions of this equation are stable with respect to longitudinal perturbations. This is confirmed by a direct numerical calculation (see [51]and section 2.4) and follows also from the complete integrability of S3. Hence as B0—*0 the solitons (4.4) are stable too, since in this case
V.G. Makhankov, Dynamics of classical solitons
59
In ref. [891it is proved that the solutions of eq. (4.1) of the form ip = f(x) eAi are stable with respect to small longitudinal perturbations for A2> An analytical proof of the instability of these solutions is given in section 4.3. Numerical experiments [94] on solitons (4.4) for eq. (4.1) which are at rest, performed at Dubna, confirm the instability in the region A2 < Bearing in mind that for the solitons (4.4) A2 = y2(l B~) we get finally that these solitons are unstable for ~.
~.
B
—
2))112,
(4.5)
0> (~(l+ v
i.e., for a sufficiently large velocity close to unity.
V -4
1 one can obtain stable solitons with an amplitude arbitrarily
4.3. Stability of solitons in the direction perpendicular to their ,notion (the transverse stability) Let us consider the transverse stability of solitons for the KdV, Schrödinger and Klein—Gordon equations by varying the action with a Lagrangian integrated over the longitudinal coordinate x, and a class of trial functions restricted to soliton-like functions.* I. Let us illustrate this method by the example of the KdV equation, 42,
+
+
~2)
q~ = 0,
(4.6)
which is derived by the variation of the action S = f ~ dx dt, where .~(4/J, 1) ~ + + t,,i/i~+ and ~ = 4k,,, f/I = The terms arising in the equation at a weakly oblique propagation of solitons, i.e., propagation at a small angle to the x-axis can be easily found from the linear dispersion equation which in the laboratory frame is —
_l
‘~
~2
~2
Itj~+Ity
~i/2 ‘~
to_(1+k2)1I2_~1+k2+k2)
—k Il
ik2
—
2
j2
lit
x+2k2
y
2k2
In a system moving with unit speed in the x-direction we have
to’=w—k~=1(1 —3k~).
(4.7)
Here the value of a = I corresponds to the positive dispersion and a = I to the negative one (see, e.g., [901).Eq. (4.7) is obtained under the assumption that k~~ k~,i.e., the angle i~ = k~/k~ is small. This means that one may consider perturbations across the soliton motion with a wavelength which is well above the soliton width. Note that for a quasi-longitudinal soliton in a magnetoactive plasmas the first term of (4.7) vanishes (see [95]). Using (4.7) we get the desired equation —
42, +
+
‘PXXX
J
+
~
dx’ =
(4.8)
~
which one can sometimes rewrite in another, more convenient form: ~ *
+
~
+
~
+
+
=
0.
For Schrfldinger-type equations with a self-consistent potential such a problem is solved in ref. [92].
(4.8a)
60
V.0. Makhankov, Dynamics of classical solitons
Eq. (4.8) is derived by the variation of the action with the Lagrangian 2= ~{cI~I~,+
+ 24~~b~ + ~,2 +
~a1’~çb~ + ~a[I~
—
tt3~(x=
(4.9)
±cc)]}.
However, when one varies S = f 2’ dx dt dy and gets (4.7), one must bear in mind that only the leading terms of the expansion in the small parameter ~ = k5Ik~are taken into account. We rewrite (4.9) in a more convenient form: 2’ = ~ Here 1
=
+ ~423+
24242~~ + 42~+ 2a424235 + ~aq~I~
+ + ~a~I~42~Y+ ~[I~—
I~(x=
±cc)]}.
(4.9a)
f~pdx’. Moreover, when obtaining (4.9a) we used the relations
= l:I:i~~~ +
+
=
ills = tI~xxy,
in the third, fourth and fifth terms of the right-hand side of eq. (4.9), respectively.* As we saw above the family of soliton solutions to eq. (4.6) is 2{A ~= 12A~sech 0(x M0)}, where M0 = 4A~t. —
(4.10) (4.l0a)
The function (4.10) is also a solution of eq. (4.8). The study of the transverse stability of a plane soliton of the form (4.10) means that we put some perturbations on the function (4.10) and investigate the behaviour of this perturbation in time as t—*cc. If this perturbation remains bounded as t—*cc, the soliton solution (4.10) is called (transversely) stable with respect to the perturbation under consideration. If the perturbation increases unboundedly with time the soliton (4.10) is unstable. Here we restrict ourselves to small perturbations of the soliton amplitude A0 and its phase M0, when the functional dependence is given by (4.10). Suppose that A and M are functions of a transverse coordinate y and time t. Substituting the trial function 2 sech2(A(x — M)) = l2A into (4.9a) and integrating over x from x = —cc to x = +cc we get ~
—~M,A3+~A5— a(~—+4)AA~_ a~A~M+~ (ç_2)A
Varying the action S
= f
A~+2~ A3MY2.
~ dy dt over A and M independently we have the Euler equations
~
AA~=
(411)
-~-A~-a-~-A~M +~-~-~-A~M0. ,9t i9y 5 t9y For the study of the dispersion properties of this system we linearize it near the soliton solution (4.10) putting
A=A 0+6A, *
M=M0+6M.
Also note that eqs. (4.9). (4.9a) and (4.8) make sense only for solitons in our broad conception, i.e., ~ goes sufficiently fast to zero as x-*±x.
V.0. Makhankov, Dynamics of classical .colitons
61
In the zeroth order approximation in ÔA and SM a relationship between A0 and M0 in the form (4. IOa) arises naturally. Linearizing the system (4.11) in SA and SM and expanding the required functions in Fourier integrals we get the dispersion equation ak~—~A~)[8A~+~
=
(1 — 24A~)].
(4.12)
This equation describes transverse instabilities of ion-acoustic type solitons both in isotropic and in magnetized plasmas which were investigated also in the papers by Kadomtsev and Petviashvili [90] and Spatschek, Shukia and Yu [95],respectively. In the first case we have j2
/
to
2
8
221 ak5A0~1+a~-~-~ 0
which differs from the formula of the paper [90] by the second term in the brackets of (4.13). As we see easily, for a = 1 (negative term only a frequency If a = for —1 (just in this 2
(4.14)
For the instability of the ion-acoustic solitons in magnetized plasmas with Te ~ T 1 we have from eq.
(4.12) 4 21 ait ~ (4.15) Here, as in the first case, we get an additional term in the brackets and, moreover, a numerical factor in the formula for the growth rate in comparison with the result of ref. [95].We note at the same time that in this case the instability arises for waves with negative but not for those with positive 2
32
to
.
dispersion.
From (4.15) it follows that (k~)thr>3A~
3k~,
i.e., the instability threshold of (4.15) lies at the boundary of the applicability region, of eq. (4.8). It is interesting that this statement does not refer to the instability of (4.13), since (4.14) holds for A0 < 1, but the condition for the use of (4.8) is A0~l. Note the important fact, that the instability growth rate Im to is proportional to k5 and vanishes together with k3. An approximate dependence of the growth rate on k~is represented in fig. 4.1. This testifies to the longitudinal stability of the ion-acoustic solitons. 2. Using the described formalism we consider the stability of solitons for the Schrodinger equation with a cubic non~linearity* 2 142, + V 42 + 421242 = 0, (4.16) *
Note, that the longitudinal stability of soliton solutions of this equation was easily shown by a direct computer experiment [51].
62
V.0. Makhankov, Dynamics of classical solitons
for which the action reaches an extremum value for the Lagrangian 2+ 2’ = —~i(42427— ~ V! As in the preceding section take a trial function in the soliton-like form
~
(4.17)
—
=
V’2A sech(Ax) ~
(4.18)
and study the soliton stability in its rest frame. Integrating 2’ with the function (4.18) over the longitudinal coordinate we get =
4{A~i, A4~—f3(A~/A)+ ~A3},
(4.19)
—
where
I J
(1_~tanh~\2d
ir2+
cosh~ ) Varying over ~ and A we have 132
\.
36
~
12
11 ~
(4.20)
3y A
0.
A,-2-!-Acby
Putting again
(4.l9a)
~ = ~+
64 and A
=
A 0+ 6A, we get from the system (4.20):
(i)
in the zeroth-order approximation the conventional relation
(ii) in the first approximation in 64 and SA the dispersion equation 2 = —4A~k~(1 — f3k~/A~).
(4.21)
The instability (4.21) resembles surprisingly one for the case of the KdV equation (formula (4.13)) with the exception of the numerical coefficient and the sign of the dispersion. An approximate dependence of the growth rate of (4.21) is the same as the one in fig. 4.1. A linear growth for small k5 becomes a plateau for k~ A0ff3. The instability disappears for perturbations with wavelengths which are estimated from the formula k~>A~If3.
(4.22)
This fact is very interesting and is evidence in favour of Langmuir wave collapse (see [13] and section 5*). Taking into account the interaction of Langmuir and acoustic waves introduce practically no change in the picture of the instability, at any case for its initial stage [22]. 3. The Klein—Gordon equation with a cubic non-linearity for a scalar field with the parameters m and g, 2
2
23
42,,—V42+m ço—g(p =0,
(4.23)
* Note, that although an equation describing more-dimensional oscillations differs from (4.16) nevertheless this fact does not effect the result qualitatively (see, e.g., [92]).
V.G. Makhankov, Dynamics of classical solitons
63
Im (~)
Fig. 4.1. Approximate plot of the growth rate of a KdV soliton with a positive-dispersion instability.
is, as we saw earlier, derived by the variation of the action S÷=~J{’P~_(V’P)2_ m2’P2+~g2424}dxdydt. Becauseof the relativistic invariance of (4.23) it is sufficient to study only the stability of a soliton at rest: ‘P~=V2~sech(mx).
(4.24)
We vary the action with the Lagrangian 2’
~{~—
~
42~m2422+~g2’P},
and take the trial function in the form =
V2A sech Bx.
=
-~-
(4.25)
Then (A~- A~)—
(A,B,
-
A
2B 5B5) +
~
(B~ B~)- ~A
The equations of motion take the form A
2 5B~-A1B, ~ll
~
B~—3B~ A ~AB -
-
2m
-
m2
4~- ~g2A4 +
~.
-
+
~g2
~-
-2
(~-~-~f~)
+
-
~-
~
-
Putting A = A SA and 2. B = B0 + SB, we get in the linear approximation in SA and SB (4.25a) k~ — 2.93m Hence it follows that even for k 5 = 0 the solitons (4.24) and (4.3) are unstable, i.e., a variation of the amplitude and width of the one-dimensional soliton leads to its instability. The instability is stabilized for wavelengths of perturbations ~( = ilk5 which are the order of the soliton longitudinal dimension, 1/rn =~x.The transverse stability of the envelope soliton (4.4) for eq. (4.1) is found by the trial 0+
2
64
V.G. Makhankov, Dynamics of classical solitons
function B
=
sech(Bx) e’4.
(4.26)
In this case we have ~ (~ - ~)B
+ ~ B~- B~- B + ~B3.
Hence by taking the variation we find
+ B2 =
0.
Putting again B = B 0 + SB
and ~
=
~
+
~ we get the soliton relationship between B0 and 4~and the
dispersion equation 2),
(4.27)
k~+ (~± + 4(1 B~)k~f3)iI where z~= I 2B~.Since the condition for the existence of the soliton (4.3) is =
(~2
—
—
0~B~1,
it is advisable to consider the three possibilities (i) ~
(ii) ~
and (iii) ~
In the first case, B~>~, we have an instability when k,,2
< (k~)thresh
—
(B~
(4.28)
~).
As B 0—*
I we get an instability of the (4.25a) type.* If B~= we have
k~±~= k5B0.
=
(4.29)
For k~< B~/f3eq. (4.29) has an “unstable” root (Im to > 0) which is proportional to \/k~for small k5. Finally in the case of B~<~we find from eq. (4.27) that an “unstable” root arises also, if k~
~-1—2B~
(4.30)
we have (4.30a)
2’
Im to *
=
Ic5 (1 — 2B~)”
Note that even the value of the growth rate (4.25a) coincides satisfactorily as k,,
.-
0 with the one obtained in subsection 4.2, ft = V~.
V.G. Makhankov, Dynamics of classical solitons
Tm
65
(.3
(n)
(It)
(I)
_______________ 1’~tk,~.L
~<*
~ ~ ‘K5I~,~L
~
‘Kth~~L 4(~
Fig. 4.2. Growth rate of the KG3 soliton instability as function of the transverse wavenumber.
which because of a factor 2 arising when we change from d2/ 2 to id/dt is the same as (4.21).* The approximate behaviour of the growth rates Im to as functions 19t of k 5 in the regions (i), (ii), and (iii) are represented in fig. 4.2. In these figures one can see a characteristic fact: if the curve of Im w(k,,) starts at the origin there is a “longitudinal” stability of the soliton, at any rate with respect to perturbation of the types considered. In the case (i) when B~>~ the function Im w(k5) is not zero for k5 = 0, which means longitudinal instability of the soliton (4.4). It is interesting that this result coincides with one obtained by Zastavenko [89] in a completely different way. (In the paper [89] only the instability threshold for a soliton of the (4.4) type with respect to a variation of the amplitude B0 was found.) Because of a Lorentz transformation we have formula (4.5). 5. Let us study the “transverse” stability of solitons (of the kink type) for the Higgs field equation. For the real field we have the equation (m = g = I) (4.31)
(~_v2_1)x+x3=o,
which is the zeroth variation of the action with the Lagrangian 2+ l)2}. (4.31a) 2’ = (V~) To begin with we investigate the simplest case when only the soliton shape is perturbed and the vacuum values of the field ~± = ±1 are conserved. Here the trial function has the form (2
—
x~= ±tanh(B(y,t)x). Inserting it into (4.31a) and subtracting the vacuum value of 2=~fdx we get after the variation with respect to B i I
~
(C
2
C
2 \
i
3
—2
),
L) 0\Ui
i.e., B~=
~,
ayi
and for SB = B
(~_~)oB+~SB
—
B0
we have the equation
=0.
It is easy to check that the function = ±A tanh(x/\/2) 213t2 = 1— 2i(a/~t). *
]n fact, changing from g to g e’ we have 8
66
V.0. Makhankov, Dynamics of classical solitons
also leads to stability: (~_~)oA+~SA=0, where SA = A 1. Here we note that the variation of the soliton amplitude gives rise to a simultaneous variation of the vacuum values of the field and hence demands in principle an infinite energy. However, because of the stability of the vacuum perturbations (the dispersion relation is to2 = k2 + 2) we can subtract from (4.3la) the corresponding part of the Lagrangian which is relevant to the vacuum perturbation and vary the remaining finite value of 2,, = 2’— 2’vac where —
i
2
2
2
i
4
sx}I~==.
2vac~{XtXy+X
Now it is also interesting to note that the use of the trial function
x~= A tanh(Ax/V2)
(4.32)
as a soliton-like solution leads to an instability: w2=—~+k~,A
0=l.
This probably non-physical instability arises because of an artificial stiff coupling (resonance!) between oscillations of the soliton and the vacuum introduced into the system by the choice of the trial function x~in the form (4.32). Actually breaking this coupling, i.e. choosing a trial function
x~= A tanh(Bx) leads to the dispersion equation 2 = (1.69±0.l41)+k~,
(4.33)
to
which has no unstable roots. Now consider the general case of the complex Higgs field. The equation, the Lagrangian, and the soliton solution (for V = 0) are* (434 ‘Pu v’P 42+4242—u, a
,,.
2
2’ = 42,,
i
2
2
—
2
I
4
IV’Pl + 1421 H’PI },
(4.34b)
= ±exp{±iA
0t}V1+ A~tanh(\/~T+A~)x),
(4.34c)
respectively. The choice of the trial function is determined by the form of the solution (4.34c) and by the feature noted above (the possibility of resonance). Therefore 4’ A tanh(Bx), (4.35) = e and the variation is carried out with respect to 4,, A and B, independently. The Lagrangian integrated over x is ~=
~
—A
2B —~-+~-}. 5B~)+~~(B~— B~)—~A
*
An equation of the type (4.34a) can describe supersonic Langmuir solitons.
(4.35a)
V.0. Makhankov, Dynamics of classical solitons
67
At first, let all the derivatives with respect to y be zero (“longitudinal” instability), then, varying with respect to A, B and 4, we have, respectively, A,, B,, A 4,, 4 A 8A3 2~ 5AB -2 + ~ -~-+2 —~A -0, 2 A2 2 A4 A,, 4 2 B,, ~çb,yA +~~A~A ~-0, A —
2 -
-
-
2
2
a A24,, at
B
2
—
— -
In the zeroth order of perturbation theory we have =
A 0,
with A~= A~+ 1, B~= ~ In the first approximation in SA = A we get a homogeneous system of equation, from which we find 2 = 3.69A~ —2 ± \/~.97A~ — 11 .95A~ + 4.
—
A0, SB = B — B0,
and 84,
=
4,
—
(4.36)
to
The right-hand side of this equation is always greater than zero since A~= 1
+ A~.As
A 0-*0 the value
of A0 tends to unity and eq. (4.36) coincides with (4.33) with k5 = 0. And finally, if k5 0, the dispersion equation is 4i+ (4 + 12A~ + 8A~ — ~A~k~)to~ + 4A~A~k~ = 0, (4.37) + ~(43+ 94A~)to where w~= k~. For the simplest case of k 5 ~ I it is easy to find three roots for to~.The first two large ones coincide with (4.3), and the third (small) one is —
2i ,~2
to
~2
2 —
‘~
I A0~,A0 +3A~+2A~’
or 2
i.e., we obtain a stable solution again. In the general case it is necessary for an investigation of eq. (4.37) to calculate its discriminant which is always negative (for example, for A0= 1, D = —124+ l24k~—l9.2k~—2.35k~)and eq. (4.37) has no complex roots for to~. Besides a detailed study of eq. 2 ~ 0 (the equality is possible when k (4.37) performed in [94] shows that for its roots to 5 = 0). This means that the solitons of the complex Higgs field are also stable, both in longitudinal and transverse directions, at any rate with respect to the fairly general types of perturbations considered.*
*
Note that this result is in contradiction to Potyakov’s statement [86].
68
V.0. Makhankov, Dynamics of classical solitons
4.4. A qualitative discussion of soliton stability The soliton instabilities considered in the preceding sections have a clear physical interpretation. Let us start with the longitudinal instability of the bell solitons of the Klein—Gordon equation with a cubic non-linearity. As we saw in subsection 4.3 the soliton (4.4) is “longitudinally” unstable for amplitudes B~> Let us consider what this instability means, for example, from the point of view of a quasi-particle interpretation.* A scalar complex field ~ has a charge ~.
Q
=
i
J
~
—
~
dx,
which is conserved together with the energy integral E
=
J {l’P,I2~l’P~l2~
m2I42I2—g2I42j4}dx.
For the soliton (4.4) these quantities are Q=8~~AVl_A2,
E
=
8~-Vi
—
(4.38a)
A2(l —~(1 A2)),
(4.38b)
—
respectively.t The functions Q(A) and E(A) are represented in figs. 4.3a and 4.3b. It is shown that
(a)
(-s)
Fig. 4.3. (a) KG3 soliton charge *
Q as function of the frequency s
(constituent energy). (b) Soliton energy as function of w.
Below we follow ref. [961. Here we have again introduced the quantity (phase) A which is related to the amplitude by A = I — B2.
V.0. Makhankov, Dynamics of classical solitons
69
E esc
0.5 Fig. 4.4. Soliton energy as function of
a Q.
both Q(A) and E(A) have maxima at the point where A = i/V2. In fact, equating dQ/dA and dE/dA to zero we get A~.= ~in both cases. Further, we have ~m
~lA~-~It2~,
EmElASII2~V~, g Q—~-----~Q, E-~---~E. -
The soliton (4.4) is stable in the region where
where the condition E < Qrn is always satisfied (see section 3 and ref. [84]),i.e., the energy of a bound state is smaller than the sum of the energies of its components. Let us consider a possible evolution of the state S~with charge Q1, energy E~and, naturally, soliton phase A1 in the instability region. From eqs. (4.38) one can eliminate A and express E as a function of
Q: E =
(‘ -s-V ~
)
421/2
~i
±
—
Vi —
4~2)}~
(4.39)
This function is shown in fig. 4.4. The lower curve E,,~corresponds to right-hand parts of the curves Q(A) and E(A) starting with A ~ 11V2 and the upper curve to the left-hand ones. One can see from (4.39) that the spectrum E(Q) is not a decay one, i.e., E~(~1)
Qi =
Q2 + Q3,
therefore the decay of the soliton E~Xinto two solitons E~X and E~
2’~
Imto =~(I—2B~)=
31
to a one-soliton stable state which lies on a curve with the same charge and the energy excess is
70
V.0. Makhankou. Dynamics of classical solitons
radiated in the form of a neutral field. Once more note here that an analogous picture arises for the soliton solutions of an equation with a logarithmically diverging potential [961. Similar considerations remain, unfortunately, valid only for longitudinal instabilities when there exists a region of stable solutions. As we saw above (subsection 4.3) the soliton stability region disappears for two-dimensional perturbations of a scale which is in any case greater than the soliton dimension. In this case the interpretation given above is inadequate. Most probably this means that two-scale solitons of the KG equation (one scale determines the soliton dimension and another one its energy localization domain) of a bubble type will be unstable and begin to break into smaller formations as occurs for solitons of the non-linear Schrodinger equation [92].The latter phenomenon is well known in plasma theory as the Langmuir wave collapse (see below and [43]). The problem of the stability of one-scale solitons (including pulsating bion-type ones) remains unresolved in the meanwhile. Apparently just such an instability is responsible for the failure of the self-channelling of a laser beam propagating across plasmas (the dispersion formula is to2 = k2 + to~e). The kink-type solitons exhibit a surprising feature to remain stable even in the case of a complex field when these solitons are in fact, a rotating mixture of soliton and antisoliton states. Therefore, we think that just such formations are the most interesting one from the point of view of constructing extended particle models, provided one succeeds in finding longitudinally stable solitons in threedimensional space. In any case it is necessary to have a Higgs field among the interacting field as a stabilizing factor [10,86].
5. More-dimensional solitons From the title of this section it follows that we shall now be dealing with solitons in the real three-dimensional space. This material is still developed least of all and hence is less ready to be reviewed. Research work in this direction is only slowly become strong both in quantity and in quality. Since there is no general mathematical theory of non-linear partial differential equations various less or more exact soluble models are now being studied. The greatest number of papers are devoted to considering stationary solutions which are sufficiently symmetric to make the problem one-dimensional in a mathematical sense, i.e., spherically and cylindrically symmetric models are being examined, As regards the dynamics of “more-dimensional” solitons, viz, formation and interaction processes, computer experiments are still in the foreground. If in considering stationary states the latter are only auxiliary in nature (like a slide rule) they become a competent tool for theoreticians (together with pen and paper) for studying dynamical problems. All of this, namely, an incompleteness of the material, a variety of models and approaches, an unstable terminology, and a growing flow of publications puts naturally its imprint onto the choice of material as well as the mode of presentation of the following sections. We apologize to those authors whose works are not discussed here. Moreover, some of the problems discussed below will probably be reviewed in separate papers* in which the reader would find a more complete bibliography. 5.1.
One exactly soluble two-space-dimensional problem
We have earlier declared that our main task is discussing non-integrable equations. Here we violate ~We have heard of several such papers which either are in preparation or about to be published [23].
V.0. Makhankov, Dynamics of classical solitons
71
this rule and following Hsing—Hen—Chen [97] consider two-dimensional solitons of the Kadomtsev— Petviashvili equation* 42~± ‘Pyy + ~
+
+ 42xm
(3422)
=
0,
(5.1)
which describes perturbations in a weakly dispersive and a weakly non-linear medium. It seems to be more natural, to begin with such a simple example of an exactly soluble system that the reader can feel some specific features of more-dimensional problems. Let us formulate briefly the statement of the problem following Lax: given two commuting operators (the so-called L—A pair)
Ai =
~i,
0,
(5.2)
which are linear with respect to required function ç, there are two coupled eigenvalue problems LxPr~A~I1, A~Pza’h1r,
(53)
with A and a independent of time. The operator equation (5.2) is then satisfied on a class of functions ~ which are solutions of eq. (5.1) and the spectral problem (5.3) may be employed to obtain a Bäcklund transformation. The operators L and A for eq. (5.1) have been found in [98]. L=
a~+42—ba5,
A=
4id~+ 3i(42a~+ d~42)+ 3ib
(5.4) dx + i(a~+ a,).
(5.5)
Putting b = ±l/V3one gets the upper sign before the second term in (5.1) (i.e., negative dispersion), while b = ±i/V3gives the minus sign and positive dispersion. Examine first transversely stable solitons: ~
+ 42~,,+ 42~+
(3~~)~ + ~
=
0.
(5.6)
One might proceed from (5.6) to the equation with positive dispersion by the substitution y —*iy.t We show that eqs. (5.3) may be considered as a relation between the solutions of (5.6). Putting = In ‘I’ and q?.. ~ we rewrite the system (5.3) as follows: (5.7a) 4i(4,~~~ + 3~i~çli~ + ~)
+
3i(24~~4,~ + I~)±iV34~+ i~+ i~,= a,
(5.7b)
and eliminating s1 we get ~
Thus, for every pair
(5.8) (s/i,
a) satisfying (5.8) there exists a pair
(—
i/i,
—
a) satisfying the same equation
* Eq. (5.1) may be obtained from (4.8a) by rewriting (4.8a) in the laboratory system and dropping the last term which stabilized the transverse instability of solitons in positive dispersion media. 2,, and the transformation y-*iyItleads to k~—~—ik,. makes the result obtained in subsection 4.3 clearer, since the sign in formula (4.13) is determined by that of k
72
V.0. Makhankov, Dynamics of classical solitons
with the lower sign in the fourth term. For this new pair we have a new solution of eq. (5.6); therefore (5.9a) ~
(5.9)
Taking the difference and sum of eqs. (5.7) and (5.9), integrating the results over x and thus eliminating* the constants A and a (for details see [97])we get the Bäcklund transformations (~-
‘F)2~
2(~+ ‘F)~~
J
(~- ‘F),,
dx’
=
0,
(5.10)
~
J(~-’F)tdx=0~
24i~~ =(F—’F)~.
(5.11)
The double sign appearing before the derivatives with respect to y reflects the fact that there may exist waves propagating in both directions along the y-axis. Eq. (5.11) which is an intermediate step in getting (5.10) is needed below. Equation (5.10) allows us to construct a new solution ‘F of (5.6) once we know one solution, ‘F. Since ‘F = 0 is one of those solutions we can start just from eqs. (5.3) which are now (5.l2a) 4s/im
+ 4’~+
s/i,
=
0.
(5.l2b)
The general solution of this system is 2y+ i(4k3 = ~ s/’~exp{ikx ~ V~k Eq. (5.11) then gives for ‘F=0 ~4~=24i~~
k)t} = ~ s/it. e~.
(5.13)
=2(s/i~/s/~r)~,
and hence ~ =2 ~
—
~&kke~)-
(~
~kk:’)(~~ke~)
(5.14)
(~e4) s/’k
is the required solution of eq. (5.6). A cnecial choice of s/i 8k,-iAi +
5kjA’
1 results in special solutions. For instance,
one gets
putting s/’k = = 2A~±) sech2{A(±)[x ±2V~A
1_1y 4(A~±) + 3A~)+ ~)t]}, —
where
A(±)= ~(A1+ A2), *
A(_) = ~(A1 — A2).
The possibility of eliminating these constants implies that they a,e not essential in constructing new solutions.
(5.15)
V.0. Makhankov, Dynamics of classical solitons
73
WaVe ~rorst ~ane
Fig. 5.1. Angle between the wavefront plane of a two-dimensional KdV soliton and the abscissa axis.
The solution (5.15) approaches in the A(..)-+0 limit the conventional one-dimensional soliton of the KdV equation (see (4.10)) if one takes into account that the coefficient before the non-linear term in (5.6) differs from that in (4.8a) by a factor ~, which leads to (5.16)
‘PEq.(5.6) = ~‘PEq.(4.8a)~
The relation A() = 0 implies A(±) = A 1 = A0, and using (5.16), the above statement follows. The soliton (5.15) has a plane wave front obeying the equation x ±2V3A()y = constant. The angle between the wave front plane and that in which x = 0 is determined by (see fig. 5.1) =
arctan2V~A(.),
(5.17)
and lies in the region (5.18) To see this, let for example A1 A ~ A2 then (5.15) may be rewritten as follows 2sech2{~A[x ±\/~Ay— 4(A2 + ~)t]}, 42* ~A so that ~*zzarctan~.J~A~*V3A, since A~1.
(5.19)
(5.20)
This means that i~ is bounded with the soliton amplitude and lies in the range (5.18). Positive dispersion. This case corresponds to the transformation y-*iy. We then have from (5.15) an exact solution of eq. (5.1): (p
=
2A~÷~ sech2{A(±)[x ±i2V’~A()y— 4(A~÷) 3A~...)+ ~)t]}, —
(5.21)
which is singular (recall the linear instability of a soliton in the positive dispersion case) at the points of intersection of the following lines 2V~A(÷)A(..)y = 1T(~+n),
n =0,1,2,...,
(5.22)
and x
= 4(A~) +
3A~)+
This singularity implies apparently that there is no non-linear stabilization of the instability (4.I3),* ~This result is in agreement with that obtained in [99]when studying resonance interactions of wavepackets and contradicts one obtained in [100] for the non-linear dissipation of an unstable KdV soliton.
74
V.0. Makhankov. Dynamics of classical solitons
which resembles very much the collapse of Langmuir waves which we shall discuss in the next section. In the negative dispersion case the multi-soliton formulae may be obtained as well. However, we do not do this here and refer the interested reader to refs. [97, 1001. 5.2. More-dimensional solitons in the framework of the S3 equation Both S3 and KdV equations are completely integrable and therefore Bäcklund transformations exist to construct solutions of the S3 equation. We give now a comparative analysis of the solutions of these equations disregarding their specific features. As we have seen solitons of the KdV as well as of the S3 equation may be unstable but for opposite signs of the ratio ~ Moreover, the dispersion formulas (4.13) and (4.21) are remarkably similar except for the numerical coefficients which are determined by the particular form of the one-dimensional solution and the different degree of the non-linearities. This is because the structure of the function to(k,,) is the same for both equations. This is of no surprise if one sees how those relations are obtained. In fact, after integrating over x the solutions depend only on y and t. This dependence is governed by the approximate equations
±J
42, +(~~
~
dx
=
0,
i42,
2±p,,,, = 0.
+
‘PI421
The first equation resembles the second one if one takes t it and to —ito (which is valid for solitons at rest). This in turn implies that the sign before to2 in (4.13) must be changed to correspond to (4.21), so that the instability of solitons for the KdV and S3 equations occurs for opposite signs of the ~,,,, term. “Positive” dispersion in the S3 equation gives rise to (in the transverse direction) stable solitons* since making the substitution y iy, k,, —ik,, we have from (4.21) to2> 0. By analogy with the KdV equation we may therefore suppose that a stable soliton moving at an angle to the x-axis is described by —~
-*
~=
—*
—*
V8A,A
2 sech{(A1 + A2)x ±(A1
—
A2)y} exp(i4A,A2t),
as can easily be verified by substitution into the S3 equation with “positive” dispersion. This formula is valid if IA1 A21 ~ IAi + A2j. For the normal S3 equation (“negative” dispersion) one arrives at a solution which is singular in the points —
A()y
=
~(n
+~),
x = 0.
Thus the equation
142~+’P~~’P,,~+I42I420 has stable more-dimensional solitons as solutions [101] while the equation
2 i421+42~~+42~~+I42I=042
may have singular solutions. In the following subsections of this section we shall consider the collapse of Langmuir waves in various one- and two-field models. *
See for example ref. [loll. The additional double integration over x of the ç,,, term does not alter our discussion.
V.G. Makhankov, Dynamics of classical solitons
75
5.3. Collapse of Langmuir waves (CLW) Pinching of a wavepacket in non-linear media was first met in a study of the propagation of powerful laser beams through matter and plasmas [102]. An analogous phenomenon has later on appeared in theoretical investigations of the dynamics of Langmuir turbulence in plasmas [13]or more precisely in clarifying the mechanism of dissipation of the “Langmuir condensate”,—the turbulent energy accumulated in the long wavelength region of the spectrum. As we have seen (see [50]) Langmuir turbulence turns out to be unstable with respect to a coalescence of plasmons. This instability resembles the well-known gravitational one. The regions with a higher density of high-frequency energy become the centres of its condensation and of a further collapse of energy bunches (wavepackets). The problem of the Langmuir wave collapse has now its history. Zakharov by analogy with the self-focusing of light beams suggested this idea in 1972 and it called forth a lively discussion going on up to the present. Many papers [70, 103] especially in the Soviet literature are devoted to this problem.* There are both analytic and computational works. It should be emphasized that the problem of collapse has two somewhat different aspects: the mathematical and the physical ones. A basic question, from the point of view of turbulent plasma physics and its application is whether the CLW can provide a transfer of the energy of the Langmuir oscillations (e.g., due to Landau damping) to the bulk of the electrons or whether this phenomenon is not effective enough and will result only in the appearance of high-energy tails in the distribution function. It is expedient to separate the mathematical aspect from the physical one in order not to affect any thermonuclear projects which are at the present time realized or developed. We shall chiefly deal with the mathematical aspect and examine the possibility of CLW in various models described by the appropriate systems of non-linear equations. These models may naturally be improved from the physical point of view. The latter statement does possibly not refer so strongly to the finite size particle method [103], although this method is also not free from its own defects [106]. We start from the simplest model, viz., the Schrödinger equation with a cubic non-linearity. 1. The CLW is described in the 423 approximation by the equation (5.23) V2(is/i, + V2,/i) div(IVs/iJ2Vi/i) = 0, —
where s/i is the high-frequency potential envelope, and not the field envelope as it was in subsection 2.1 (see (2.24)). If the initial packets are spherically (SS) or cylindrically (CS) symmetric, eq. (5.23) can be reduced to 2 n—-l 2
l(pt+VrrcoyçP+Ic0I420,
(5.24)
and n = 2 corresponds to the CS and n = 3 to SS geometries. The field ~ = —Vs/i is subject to the condition 42(0) = 0 at the origin. Eqs. (5.23) and (5.24) have, respectively, the first integrals S =
S
IVs/~I2
—
2 *
f d3r, d3r, f {1V2s/i12 ~Iv~r}
(5.25a) (5.25b)
We know only a few works performed in the USA which are concerned in some way with the collapse of f-waves [64,92, 104,105].
76
V.G. Makhankov, Dynamics of classical solitons
or
S
=
=
J 112r2 J
dr,
(5.26a)
{I(r~I2+ 2142 12_
‘r2I I~}dr
(5.26b)
in the SS case, and
S
=
s 2
=
J
~
J
dr,
(5.27a)
{vV~~I2 +
~
~
dr
(5.27b)
in the CS case. To show some qualitative features of the CLW we discuss it in the SS geometry. In one of the first works towards a numerical study of the CLW an arbitrary two-scale wave packet (the energy containing layer width z~rmuch less than the size of the packet R0, that is, a “bubble”) turns into the quasi-plane soliton which moves to the origin accelerating (see fig. 5.2) and oscillating about the “soliton state”. Such a dynamics is natural for packets, when the instability condition S2~0holds. Packets with S2> 0 disperse. Such a behaviour may be realized with the help of the constants of motion 5, S2 and following the discussion of refs. [13, 1021. Let us consider 2)S D
(r
=
J 14212r4
dr.
Making the Ansatz (5.24), we arrive at =
6S 2 —2
JKr)r12
dr
—~J114r2 dr
<
6S 2,
Fig. 5.2. Initial stage of the SS-collapse of an f-wave packet in the S3 model.
V.G. Makhankov, Dynamics of classical solitons
77
or, upon integrating over t, D<3S2+Ct+C2.
(5.28)
As D> 0 (and hence C2 >0) this inequality puts an upper bound t0 on possible values of t, where t0 depends on the initial conditions, i.e., on the constants C1 and C2: (529)
6JS2j
0
It follows that the solution of the initial value problem exists for only a finite time and gives rise to a singularity as t t0. The value of t0 depends on the sign of C1 which may be thought of as the initial radial packet velocity. If C~> 0, the packet disperses for small t, while in contrast C1 <0 leads to its contraction; t0 enlarges or decreases correspondingly. We have for a packet, initially at rest, —~
Ro/ 2 (C2/3~S2~)~ 42m,
t0
i.e., the contraction time of a packet is defined by its major radius and stored energy (42m being the field amplitude so that 42m 42max). We have already noticed that the evolution of two-scale initial packets starts, as in the planar case, with the formation of a self-consistent quasi-planar spheriton* and that self-consistent packets move to the centre with an appreciable velocity [107, 108]. In this connection it is convenient to change in (5.24) to the function =
r42,
which gives 2~2~0.
(5.30)
j~t,r+
The solution of this equation up to terms of order o(~r/R 0)and phase multipliers is the spheriton =
Ae~’sech{V~R(r_Ro)},
A =~(2_~A2).
(5.31a) (5.31b)
Therefore ~=
where A
A e~’sech{~(r_ R0)}. = Af R0.
or
(5.31c)
The condition i~r~ R0 can now be rewritten as
A>>V2/R0.
One can ascertain from (5.25a) and (5.31a) that the following relations hold for contracting packets: 2 = const AIR02I, AR0 *
Though one-scale packets with z~r R
0 evolve in a more complicated way, the picture resembles qualitatively that discussed above [108,109].
78
V.0. Makhankov, Dynamics of classical solitons
a V
Fig. 5.3. Approximate plots of the potential energy. the effective kinetic energy, and the total energy of a soliton as functions of its radius R~.
and
Ar0 = A1R01
(5.32)
(A, and R0, being, respectively, the initial amplitude and average radius of the packet). The integra)~.S’ and S2 may be evaluated with the same accuracy and we have for the spheriton S~= S/R~, 2 — S2)
S
R~S0I= R022V2A,
S2
R02S~PI+ 2S~~ = ~ (2 + R02
(5.3~. (5.33b
v
In (5.33b) the two first terms are related to the effective kinetic energy, the last term to the poterIiaI energy. The collapse condition, S 2 < 0, for packets initially at rest is 2>96R02, (5.34) S
or
A2> 12fR02. This condition is, indeed, sufficient but not necessary for the collapse to begin. To show this, let us consider how the kinetic and potential energies of a packet at rest with a given S depend on its position R 0. In fig. 5.3 one can see the function S2(R0) which has a single maximum at the point R~ defined by the condition dS2/dR0 = 0, whence 2= S2/48, (5.35) R~
which differs twice from the value given by (5.34). One can see also in fig. 5.3 that if the packet for a given S is on the right of the point R~it will disperse infinitely, and a packet with R 0 < R~will contract. Thus, the collapse condition can be written as* S2< S~~(48fS)>0.
(5.36)
We can examine the initial stage of the packet contraction (dispersion) using the geometric optics *
A similar inequality should exist for arbitrary packets in three dimensional space since in getting (5.28) we dropped negative terms.
V.0. Makhankov, Dynamics of classical solitons
79
technique [109].We take the field ~ in the form = tF(~)exp{i(kr
—
~ = r — v(t)t,
A(R, t))t},
and assume A to be a slowly varying function of R and t, and v(t) such a function of t. Then (5.30) yields
~~
+
A,t
+ 2kArt
2—
—
k
+
=
o,
(5.37a)
and
v
=
v
0 + v,t = 2(k
(5.37b)
— Art).
Eq. (5.37a) leads to
A
2+
=
(2— ~A2)
k
— A,t
—
2kA~t,
(5.37c)
or, by virtue of (5.32), 2 ( Ar~
-
2R021\ 12A1 ~b~!’
~0 \
“oI
and finally
v
= 2{k —2t(A~~1_2)
~}
= 2{k —
________
This means that the packet will move after some time to the origin being accelerated if the initial condition 2/16 (5.38) R02< S holds. The value S given by (5.38) is less than that of (5.35) by a factor 3. This difference appears because (5.38) does not take into account that portion of the potential energy which makes the packet to be a self-consistent one (the compensation of the plasmon kinetic energy in the packet) and also because of the half-quantitative character of the above analysis. We note that the constant-acceleration motion of spheritons to the origin has been observed in numerical experiments [107].
5,4. The CLWin the dynamical model 1. Let us study the CLW in the so-called dynamical model of two interacting fields (2.24). We rewrite it in a more convenient form, introducing a low-frequency potential u by analogy with (l.28b) = ‘F +
l42l~= ‘F + lVsfrl2
(5.39a)
we then have
V2u
=
‘F,
V2(is/s, + V2s/i) = div(’FVs/i). For these variables we easily obtain from eqs. (1.11) and (1.28a)
(5.39b) (5.39c)
80
V.G. Makhankov, Dynamics of classical .colitons
S2 =
f
3r ~
d
=
f {IV2c~l2
— ~IV
s/il4 + ~(u~+ (Vu)2)} d3r,
or
J
2s/il2 +‘FlVs/il2 + .42 + ~(Vu)2}d3r. (5.40) {lV It is evident that besides S 2 also S and S~are conserved because of the gauge and translational invariance of the Lagrangian (1.11) (see section 1); moreover, S is determined by (5.25a). Let us have initially some wavepacket for which S2<0; then from (5.40) there follows the inequality* S
2
=
J ‘F~Vs/iI2
d~r~ >
and by the theorem of the mean we have mm
maxl’Fl> IS
2ISI
(5.41)
(the minimum is with respect to time and the maximum with respect to the coordinates). From (5.41) the statement follows: if in plasma a cavity with a reduced density (‘F <0) appears such that S2<0, it cannot disappear, i.e., in any case, the value of ‘Fl does not decrease during the evolution of the initial packet. 2. Let us consider some general properties of stationary soliton-type solutions of eqs. (5.39). We note that for solitons at rest these solutions coincide exactly with the ones of eq. (5.23). Let ,/i(r, t) be of the form2t
s/J(r, t) = e —IM~(r)
ei~ ~(r). (5.42) Here we take into account the fact that soliton-type solutions which are localized in space (42(r) —*0 as r—*cx) are possible only in the case when A = ..~2
div(IVxl2Vx) = 0.
The solutions of this equation, ~(r, formation:
~),
(5.43) which depend on the parameter
JL,
allow a scale trans-
~(r, j.~)= Therefore in the one-dimensional case (i) S(~.t)=
J lV71(~r)l2
dx
=
~C( 1);
(5.44a)
in the two-dimensional case (ii) S(j.~)= C(2); *
This treatment is due to yE. Zakharov [110],as far as we know. This conclusion can be rigorously proved by the consideration of the asymptotic behaviour of eq. (5.39) of r—~.
(5.44b)
V.0. Makhankov, Dynamics of classical solitons
81
and in the three-dimensional case (iii)
S(j~)= C~31/~.
(5.44c)
This means that in a one-dimensional soliton the number of constituents S increases with the constituent binding energy ~ (i.e., with a decrease in the soliton dimension). In a two-dimensional soliton it is constant, and for a three-dimensional soliton it decreases. Let us multiply (5.43) by (r, VX*), add the result to the complex conjugate one and integrate over volume; we then get for soliton-type solutions [110] 2 d3r+ (4—n) lV2xl2d3r— (2—in) lVxl2d3r = 0 (5.45a) n)
~2(2_
whence
S~=
f
f lVxl
f
~s2S*,
~
(5.45b)
where n is the dimensionality of space (i.e., the number of those spatial dimensions in which the soliton solution under consideration in the usual three-dimensional space is bounded). Therefore in what follows we use n as the dimensionality of the soliton in this sense. We thus have from (5.45b) for the plane (n = 1) soliton* (i) S~=
—
~J.L2S*;
(5.46a)
for the “cylindrical” one (ii) S~= 0;
(5.46b)
and for the three-dimensional soliton (iii) S~
(5.47)
2~
This means that only the plane solitons have negative energy while the energy of two-dimensional and three-dimensional ones is equal to or greater than zero.** Using eqs. (5.44) one can eliminate the parameter ~t from (5.46a) and (5.47) and obtain the relationships between the integrals S and S 2 for the stationary solutions:
(i)
3/3C~,
S~=
_(S*)
(ii)
S~= 0,
(iii)
S~
C~/S*,
~ = S~/C
1
~sis arbitrary,
(5.48a) (5.48b)
C3/S*.
(5.48c)
=
The particular case of a stationary three-dimensional soliton is the approximate spherically symmetric solution (5.31c) obtained in the preceding section. For this solution it follows from (5.32), (5.33a), (5.37c) that =
—A =~((S*)2/16_2R02)=
=
~
(5.49)
or 1aR~= I (see eq. (5.35)). 2/2 = —S2/16 (see (2.31)). This formula coincides with (2.34) with v = 0, ii one takes into account the equality ~ 0, =A Besides strictly stationary solitons (solitons at rest), eqs. (5.39) and (5.23) allow solutions in the form of moving solitons ~(r) exp( — iAt — ikr) (see, e.g., subsection 2.4 for the plane case). For those A is greater than for the solitons at rest. * **
~‘
=
V.0. Makhankou, Dynamics of classical solitons
82
Inserting this expression into (5.41c) we get the relation (5.36), S2 = 48/ S* which we have found above from the condition of extremum of the spheriton energy, dS2/dR0 = 0. This extremum, as 2S follows from fig. 5.3 and because of the sign of the second derivative d 2fdR02, is a maximum. As we saw above when S2 < S~the packet collapses and for S2> S~it spreads. 3. The evolution of various spherically symmetric initial packets was investigated in refs. [108— 110]. In the papers [109,1101 it was found that under the condition S2 < 0 (in fact S2 < S~)a two-scale wavepacket after becoming self-consistent, which gives rise to the soliton emitting ion-acoustic waves and taking it on the quasi-planar soliton shape (5.1k), starts to move to the centre, i.e., it collapses, However, the rate of the packet collapse in the dynamical system cannot exceed v’s’ = I/VS. as follows from the difference of the integrals S2 in the quasi-static and dynamical models. In fact, in the latter model instead of (5.33b) we have
s2=~(2+R02.~_~j(l_sv2)). From the condition S2 V~Eesh=
—
<
S~we get a limiting value for the packet collapse rate: R~a~ 1,
~
(5.50)
where a = S2/S and K is a numerical factor of order of unity. The evolution for various initial data is represented in fig. 5.4 (see [54—55]). Analogous results for the packet collapse have been obtained for a cylindrically symmetric geometry [44]. Finally note, that eq. (5.43) in axially and spherically symmetric cases is reduced to 2 —~
n—l (p+Vrr(p~~2~(p+l(pl 420. 2
4./p
2
~ ~
\
t~6O.
‘~ t~59.
~E’~::~ ~frts;~
:~ ~ *‘
~
~‘
:~/~tTstJL ?29
~)~)‘
t’~g5 ~
r
~
t-1jo.
t—’t3.
‘.1
~541’
i
2f
(a)
—~m36
-
(b)
Fig. 5.4. SS-collapse of two different initial f-wave packets.
V.G. Makhankov, Dynamics of classical solitons
83
This equation has an enumerable set of solutions which are bounded at the origin and vanish at infinity. Since for these solutions 42(r = 0) = 0 the maximum is situated at r R0 l/~.This fact leads to that, for the same dimension of the solitons, CS and SS solutions have an integral S which is greater than S for the solutions with a smaller degree of symmetry for which the maximum of ç may be at the centre. Just the latter solutions are the most probable ones (at any rate these solutions are more favourable from the energy point of view). 5.5. Dipole CLW
The simplest solution has been investigated in a[110].For this solution theplane density of the oscillating 2s/i has a distribution in the form of dipole which lies in the perpendicular to the charge p = V dipole moment. In the centre of such a dipole the field ~pis a maximum. Before a description of the results [1101obtained by numerical calculations, let us consider some properties of the solutions of eqs. (5.23) and (5.39) following from self-similar substitutions [110,103]. 1. For the complete system of eqs. (5.39) one can find the self-similar substitutions in two limiting cases; in the quasi-static one when it is reduced to eq. (5.23) and in a “supersonic” limit when in the right-hand side of eq. (5.39a) one can neglect the quantity ‘F.* In the first case, i.e., for (5.23), the self-similar substitution is [13] s/i
=
exp{—i~s2ln(t 0
—
t)}X(~),
~ = r/\/t0—t,
(5.51)
with the following equation for x(~) 2( ,..t2x + ~ + V2~)+ div(lVxl2Vx) = 0. (5.52) V In his early paper [13] Zakharov obtained a spherically symmetric solution of (5.52) in the region of —
1/it where x satisfies the equation i~V~ = 2,A2X,
~‘
so that xl~l
Xo’
This solution, as was noted in [131,leads to a non-physical divergence of the integral S; moreover, it follows from (5.51) that S 2(t) = S2(0)/V~, which leads to S2 = 0 as dS2/dt = 0. The first difficulty may mean that the self-similar approximation is valid only in a restricted region of space but also can indicate the absence of a region of applicability. The second one is an essential restriction on possible initial states.t 2(t)s/i (which corresponds to the In the “supersonic” limit the substitution [1101is/i1 —p. adiabatic approximation) andwe wemake get from (5.39) —*
2
2
2
V (—p. (t)~+V~)—div(’FV~)=0, =
(5.53a) (5.53b)
En the system of eqs. (2.24) which is more translucent from a physical point of view the “supersonic” limit corresponds to the neglect of the V211 in the wave equation. t The paper [105]also notes that this substitution is erroneous and gives the formula ~ = r(t 2t5 for n = 3. 0 — tY
* term
V2lVxl2.
84
V.0. Makhankov, Dynamics of classical solitons
This system permits among other things a substitution of the form [1101* 2,.
p
2
~p.0
~
fl(~)
—
—
x
—
—
—
~(~)
,h
#~l—2fn,
—
i~ii~ I)
~, I~i~
-2/n
—
\4Tfl ,
—
r(t~—t)
,
(5.54)
H
where n is the dimensionality of the solution as before. Details of the study of the solutions of (5.54) can be found in the original papers [103,1101. For what follows we need the following properties of the solutions of (5.54): (i) in all cases (5.54) leads again to S2 2’F — (t 2 (t V 0 t) 0 r~ , —
=
0;
—
(5.55)
—
2 lVx(0, t)l2 = f(t) = (t
(iii)
(42(0,
t)l
(S.56a)
0—~~’ 413.
(5.56b)
(iii) l~(O,t)l = 420(t0 t) From (5.54) it follows that as t t 2’Ff’F 0 for a plane soliton V 2’F/’F 1, ~ collapse 2’F/’F —*0.for Thea two-dimensional condition S V 1, const., and finally in the three-dimensional case V 2 = 0 which is necessary for the self-similar solution leads to the following restriction of possible initial states in the three-dimensional case** —
—~
—*
—*
0~
(5.57)
The quantity S~for the spherically symmetric collapse was found by us above, formula (5.36). Finally, formulae (5.56) show that, if valid, the self-similar substitution (5.54) leads collapse. to a linear 314 during the three-dimensional time-dependence of the quantities 42(0, t)(~ and (‘F(0, t)( 2. These conclusions have been verified by means of a computer investigation in a series of papers [103,110]. The initial states were chosen in three-dimensional space (in an r, z coordinate system) as follows: —
~p 2s/i=
.~
.
rrz
0Vwsin—~-
w>0,
(5.58a)
p=V w= I—~(r2+z2).
For the system (5.39) we also gave the values ‘F(r, z, 0)
=
—
vs/il2,
‘F 1(r, z, 0)
=
0.
(5.58b)
The distribution (5.58) describes at t = 0 a dipole-type charge directed along the z-axis. The results of the computations are shown in figs. 5.5—5.7. One can see from them that for some
time the collapse behaviour is in fact satisfactorily described by the self-similar substitution (5.54). Given 5, the collapsing soliton has a disk shape, independent practically of the initial data. The deviation from self-similarity is observed at about t **
in all the cases studied; this may be
Note that the substitution (5.54) in the two-dimensional case (n = 2) is valid for the complete system (5.39).
The domain of the initial values (5.57) instead of the point S, = 0 exists, allowed which can lead to a reduction of S, from ~ to zero. **
t0
in fact,
only for the exact system (5.39) where the emission of c1 waves is
V.0. Makhankov, Dynamics of classical t
Fig.
5.5a.
=0.0
~
-0.12
t
Collapse of a disk shaped initial packet of f-waves (see (5.58a)) at Po =
too
85
solitons
t=0.17
13
in the S3 model.
Ipi
t~O.14
100
.
N I
0.1
2 as function of the radius
Fig. 5.5b. ~j coordinate z.
r
~
0.2
and of the longitudinal Fig. 5.5c. Self-similarity curve
t0.0
as function oft.
t=0.30 t=O!iO
IhL
_
t’~O.5O
_
Fig. 5.6a. Collapse of a disk shaped initial packet of f-waves for p 0= 13 in the dynamical model. 2 lP~
_____________________
100
~
0 —iso
——
t
I’P15
‘O.4~
(00
002c
0 —100
0.1
“ ,/ Fig. 5.6b. jg2 and V as functions of r and z.
0.2
Fig. 5.6c. Self-similarity curves g~ and ~ correspond to cI~and the filled dots to it”4.
The open dots
86
V.0, Makhankov, Dynamics of classical solitons
: 0.025
Fig. 5.7. Self-similarity
curves for
~
—
t
2it is neglected in the sound equation (supersound collapse) for p~=
the collapse when V
13.
associated either with computational algorithm defects (this explanation is preferable to the authors) or with the properties of the solution itself. This question is at the moment still unclear.
5.6. Spheriton collapse in the system of coupled Schrodinger and Boussinesq equations We shall now discuss the effect of the non-linear and dispersion terms in the low-frequency equation on the collapse of a spherical wavepacket [60].In this case we have jçt+V2rr 42_~42
/
i9
(5.59)
=‘F~,
2
2\
~Vrr~VrrVrr)’FVrr(’F
2
2
2
2
)Vrr(lçl)
= 4me/3m, is a small parameter). The exact solution of this system is not so far available so we can give only a qualitative picture, which may throw light on the phenomenon under consideration. If the amplitude of the initial packet is sufficiently small, 142m1 ~ 11, then the picture of the formation of a self-consistent spheriton and its initial movement to the centre coincides with that described above in subsection 5.3. The spheriton velocity reaches its limiting value v” = l/’~/~, the spheriton amplitude then increases with decreasing radius according to (5.32), viz.,
and
A/A1
2, =
(5.60)
(R0~/R0)
R 0—R01-\/~t.
After a time t0 R01/V~the packet in the (5.39) model should be collapsed and its amplitude should have become singular. In the model (5.59) considered here there arises a connection between the velocity and amplitude, of the soliton, if the amplitude is sufficiently large. This effect has for the plane case been discussed at length in subsection 2.5 (see formulae (2.40) and (2.41)). The collapse of the spheriton (5.60) is depicted in fig. 5.8 on the A, v-plane as a motion of a point in parametric space along the line II. When this point approaches the curve I the solution must be
V.G. Makhankov, Dynamics of classical solitons
87
17
Fig. 5.8. Spheriton collapse.
reconstructed turning from a one-hump into a two-humped one like the (2.40) one. Hereupon the representing point either goes along the curve I or the spheriton becoming unstable and breaks up. Moving along the curve I we have first
A =2/R~—~A2<0, due to the collapse condition A2> 4/R~.Then the second term in A grows faster than the first one so that
and
(Al ~ A2 ~ (R 4, 0Y ~ A2 ~ (R 8’3, 0)
for the system (5.39),
for the system (5.59).
The collapse results in A( increasing and hence the contraction velocity v decreasing (it is clear the mass of the soliton does not change due to the decrease in R 0).
Since 2OS~=
S
R
( 27)V4
2 3’2 = R 0A
coust.,
we have 2
R2A312— —
A312
0V~i
or
A/AL = (R
413
and
A/A, = (R
0~/R0)
instead of (5.32).
3,
(5.61)
01/R0)”
Finally, when A approaches the value Acr~~ 1
~‘,
(5.62)
the spheriton should cease to contract or becoming unstable it breaks into pieces. The packet width is then about i~r~ 7de. Note that the numerical coefficient in formulas (5.62) and ~r can alter by several times, if one considers instead of the Bq equation its modified variant (see [60] for details).
88
V.0. Makhankov, Dynamics of classical .solitons
5.7. One more example of an SS-packet collapse The equation for the scalar Higgs field in SS-geometry has the form 2
X
3
1VrrXX+X
=0,
(5.63)
and the energy is E=J~YCdr.
~
(5.64)
1)2}.
Hence we have for the quasi-planar bubble (see (3.2)) x(r, t)
=
tanh[~5~(r_R0— Vt)],
(5.65)
and E is proportional to R~. On the other hand the relativistic invariance of eq. (5.63) gives E = Mlr_o x y so that 2. (5.66) E o~R~/V1— V An initial packet in the form (5.65) with v = 0, which tends to decrease its potential energy U =~Jr2dr(x2_1)2,
will collapse. The energy conservation law, dE/dt = 0, then leads to the following equation of bubble motion for R 0~’1 [1121 (5.67) with the solution R0=R~~cn~—~-—,5j \L~0
(5.68)
/
2 = ~).Formula (5.68) describes well the bubble (here cn(x, ~)(extension) is the elliptical cosine k is very thin, R contraction stage as long with as themodulus bubble wall 0 1. When R0 = 1, (5.68) is no longer valid. We shall discuss pulsating solutions of equations of the (5.63) type with a degenerate vacuum in section 7. ~‘
6.
Stationary solitons
6.1. The stability of more-dimensional solitons We discuss some general properties of more-dimensional stationary solutions of some non-linear field equations which follow from their integral characteristics. 1. We start with the simplest case of a real (uncharged) scalar field described by the non-linear
V.0. Makhankov, Dynamics of classical solitons
89
wave equation (6.1) (here III
=
2/3t2
8
—
V2, F’( 42) = dF/d42) and by the Lagrangian
F((p). The Hamiltonian of the stationary field 42(x) is = ~[(42r)~ —
(42x)2]
—
E=J[~(42~)2+F(42)]dx_=K+V. A single integration of (6.1) with 8/at potential part of the energy V: =
(6.2)
(6.3) =
0 leads to the relation between the kinetic part K and the (6.4)
F(42).
The constant of integration is chosen so that for ~ = ~± at X—*±c,2the function F(42) vanishes. Otherwise, the value of V, together with the energy E, is infinite, i.e., it does not satisfy the soliton definition. Here the relation K=V
(6.5)
is the virial theorem. The result stated below was apparently first obtained by Derrick [1131and by Hobart [115],it has a somewhat different form in the paper by Goldstone and Jackiw [114].The result is that soliton-like stationary solutions of (6.1), if they exist, are stable only in the plane geometry. Consider the scale transformation 42~= q(ax). Then the solution of (6.4) will be 42a for a = 1, but this solution gives an extremum of the Hamiltonian (6.3). By transforming the variables in (6.3) we get E[42a]=aK+’ V,
(6.6)
or da
=K-V=0,
i.e., we have the theorem (6.5) again. From eq. (6.6) it also follows, that the value a = 1 gives a minimum of E[42], because K >0 and =2K>0.
da
Similarly, a consideration of the system of interacting scalar fields in an n-dimensional space with the Hamiltonian E=
~f
~ (V42j~d~x+
f
F(421) d~x,
(6.7)
gives after the scale transformation E[coa]a
2—n
K+a —n V,
so that we have instead of (6.5)
(6.8)
90
V.0. Makhankov, Dynamics of classical solitons
~
(6.9)
By calculating a2E2[42a]/da2 for a d2E —~
=
I and taking into account eq. (6.9) we have
=—2(n—2)K,
da
(6.10)
which gives rise to a maximum of E[
42~] for n > 2 and to a point of inflection for n = 2. It means that the solutions under consideration are unstable with respect to a~oscillations*. At once we make a reservation that the statement proved above is valid only for scalar real fields with the charge, Q=if(42*42t_42~42)d3x,
(6.11)
vanishing. The presence of the additional conserved quantity Q may, as we shall see Later, qualitatively change the situation (see also [10]and [117]). 2. As a simple model of a charged field we consider the so-called co theory. Its Lagrangian is 2 ~ m2l4212 ±g2(ço (~, (6.12) = 42t(~ V42( and the Euler equation of motion is —
2—
2
2
L11142±mç±2gj42(42=O. (6.13) We restrict ourselves to solutions depending on time in the formt (we call them below quasi-
stationary solutions) =
~3=
i/i(x) e’~.
(6.14)
Then we have instead of (6.l2)—(6.l3) QpJs/2d3x~p.s1~fr1 = ~{p.2s/i2 —
(6.15)
(Vs/i)2 ~ m2s/i2 ±~g2s/s4}+ const.,
(6.16)
V2s/i + (p.2 ~ m2)s/i ±g2s/i3 = 0.
(6.17)
The energy density corresponding to (6.16) is = ~{(Vs/i)2
+
(p.2 ±m2)s/’2 ~ ~g2s/’4}+C.
(6.18)
The value of the constant C in (6.18) depends on the asymptotic behaviour as x --* ~ of the solution s/i(x). Here we consider only so-called non-topological solitons for which 42(x) as x ~ goes to the constant vacuum value The necessary condition for the existence of topological solitons is the degeneracy of the vacuum. For such solitons the boundary conditions at infinity are different from the boundary conditions for the
~.
*
by
—~
Note that this proof is easily generalized to the case
of vector
fields for which the operator V2 is replaced in the spherically symmetric case
D=V2—21r2. These solutions have
a minimum energy for
Q 0 (see
section
3).
V.0. Makhankov, Dynamics of classical solitons
91
physical vacuum state. In the one-dimensional case examples of topological solitons are the kinks of the equationfor the Higgs field and of the sine-Gordonequation. In three-dimensional space such an example is the magnetic monopole solution of ‘t Hooft [87]and Polyakov [85].More recent results on topological more-dimensional solitons are summarized in [118]. For what follows we rewrite (6.18) in the form 2)(s/i2 s/i~)r ~g2(s/’4 — = ~{(Vs/i)~+ ±m —
(~~2
One of the possible vacuum values of (6.17) is s/i,,~= 0, i.e., we get the theory based on the Klein—Gordon equation and havingsoliton solutions for m2> p2 (e.g. see [115]).The second value of sfi is obtained if one drops the first term V2s/i in (6.17), so that s/i~= (p2+ m2)fg2 which leads to the solutions (including non-topological ones), for which, however, the vacuum value depends on the charge. The physical meaning of this is not quite clear. Therefore we consider the equation V2s/i + (p2
m2)s/1 + g2s/? = 0,
—
(6.19)
which may be obtained by the variation of the Hamiltonian
f
E = ~ {(V~)2+ (p2 + m2)s/? — ~g2çfr4}d3x,
(6.20)
keeping Q fixed. To do this one has to express p as a functional of p
=
s/i:
(6.21)
Q/S[s/j],
and vary E, keeping in mind (6.21). Let us write E in the form E
=
2 + p.2)S — S
S 1[s/i]
2[s/i],
(m
+
2f
~
d3x. Using an a-scale transformation we get
(6.22)
where S1[s/i] coincides with K while S2 = ~g E[s/in] = S 2)S/a3 S 3, 1/a + (p.~+ m 2/a and —
2
26
2
26
p.~—Qa/S
p.a,
or 2
3
2
—3
—3
E[sfi,,]—S 1/a+p.aS+tna whence
S—a S2,
2— ni2)S+ 3S
dE/dalai or
=
—S1
+
3(p.
2
0,
2—m2)S=~S (p.
1—S2.
. (6.23) 3x we get a second relation between the functionals S, S
Multiplying (6.19) by s/i and integrating over d 2 m2)S = S (p. 1 2S2,
1, S2:
—
—
or
(6.24)
2—p.2)S=~5 S2=~S1,
(m
1.
(6.25)
92
V.0. Makhankov, Dynamics of classical solitons 2
2
.
2
~.
Finding 8 E = (d E/da ~ and using (6.25) we see that 6 E > 0 for p > ~m or p > Thus the solitons are stable against a-oscillations (the scale transformation). Now we consider the second variation of the Hamiltonian (6.20) for arbitrary perturbations. Let = s/’~+ 5i/i, then ~
=
J
2
[Js/’~
&/iI~’5s/Id3x +
p.~
os/i
2
1
2
d3x].
(6.26)
We get (6.26) using (6.21), from which it follows that 2
5(p.2S)
=
/
S[s/i~] ~—2
J
s/’~6s/id3x
J_(Os/i)2
[J
d3x
s/,~s/id3x]
S[~ 5]
+2
- ~
In (6.26)* the operator .F~is a test Hamiltonian and is given by formula 2+m2—p.2—3g2s/i~. (6.27) I~I—V It is easily seen that in subsection 4.2 we had a similar operator during the investigation of the planar solition stability. As before we are confronted with an eigenfunction (EF) and an eigenvalue (EV) problem for the operator =
A.y,.
(6.28)
Because of the translational symmetry equation(6.19) has the solution s/’ 5(x + E),** as wellas the solution s/i5(x). For small translations e the function = V (k = 1,2, 3) is also a solution of eq. (6.28) with A = 0. (One easily verifies this by differentiation of eq. (6:19).) This means that the translational mode is a three-fold degenerate state of the operator H with energy zero. Hence, min [119,1161, a spherically andsymmetric there is at geometry the translational mode forms a p-state with! = 1, s/i~=(0s/i5/8r)Y1 least one s-state (1 = 0) with a lower energy, A <0. tFollowing [10]we show that if 1) there is not more than one negative eigenvalue of the operator H, and 2) the value d In Q/dp. is negative, then the soliton solution of eq. (6.19) is stable (at least with respect to perturbations which do not change the symmetry of the solution “longitudinal” stability; see subsection 4.2). Differentiating (6.19) with respect to p and introducing the notation s/i,. = ös/i 5/3p we get s//k
ks//S
—
Ifs/i,.
=
2p.s/i5.
(6.29)
Let us expand the functions s/i,~and s//5 in the orthonormal system of the eigenfunctions of the operator iTt (which we assume to be real) s/i = ~ s/t,.1y1,
(6.30a)
~ s/i~,y1,
(6.30b)
=
and
J
3x
=
61k.
yy5 d
* Note, that as the second term of (6.26) is positive, ,,3/Q>0, the presence of a charge leads to a stabilizing effect.
‘°‘This means
that a translation of the soliton centre from the origin does not change its shape. In general, there would be two s states with zero and unit radial quantum number, respectively.
V.0. Makhankov. Dynamics of classical solitons
93
Inserting (6.30a) into (6.29) and using (6.30c) we have = 2
p.s/i51.
(6.31)
Hence s/’~= 0 for A, = 0. Moreover, we note that 3x = ~ s/i,~A s/i,.iTt~/~,. d 1,
f
but from (6.29) we get 3x = p. s/~,jTts/i,.d
f
f
~—
s/.i 3x 5~d
=
p. ~
(6.32)
~.
Therefore s/I~
~‘
1A,=
or
~/A1
p.~-.Q, =~
(6.32a) (6.32b)
The prime in (6.32b) means that during the summation we omit the terms with A, arbitrary perturbation Os/i as a power series in the eigenfunctions y~, Os/i =
=
0. Expanding an
~
we have from (6.26) 2E =
~
~‘
O~A
3 1+
I5
~‘
s/s~,s/’~O
p.
151 ~~ 51T1183,
(6.33)
where the test matrix T~is defined by 3s/i T1, = A130 +~ p. 51s/i51.
(6.34) 2E it is necessary to find the eigenvalues of the matrix I~,i.e., to
For of the sign of 5 solvethethedetermination equation ~ T 083—flO,.
(6.35)
Inserting (6.34) in (6.35) we get 3s/’
(11— A1)~ p. 51
~‘
s/i~~6~ = 8~.
Multiplying both parts of this equality by ~ and summing over i, we have 3 ~ —1=0. (6.36) g(tI) p. Let us see whether eq. (6.36) has negative roots fl < 0. Suppose the operator If has only one negative ~‘
94
V.G. Makhankov, Dynamics of classical solitons
Ill
ft1 ~\ ) ,\
I
______
Fig. 6.1. The function g(fl) in the case of one negative eigenvalue A in the KG3 model.
eigenvalue A1
=
-
Fig. 6.2. The function g(fI) with A~and A2 both negative.
—A then for Cl<0 —1.
g(fl)=_.4_p.3~I1115~fr~
If the function g(fl) crosses the abscissa axis for fl <0, it means that eq. (6.36) has a negative root. The approximateform of the function g(fl) is shown in fig. 6.1. It is seen that a negative root(1 <0 exists only. if the function g(uI) is negative at the point ~1= 0 (curve II in fig. 6.1). From (6.32b) it is easily seen that (0~=—--~(zI~./A.~— 1= (63~ g / p. ~ ‘.~rSi L’ d ‘ ‘~‘
~
and hence the negative root together with the instability occurs for -~~>0. Qdp.
(6.38)
The existence of two negative eigenvalues of the operator iTt leads inevitably to a negative eigenvalue of the test matrix T11* and hence to soliton instability (see fig. 6.2). The result of this theorem can be formulated also in a negative meaning: it is sufficient to have two negative eigenvalues of the test operator H in an arbitrary geometry (n = 1, 2, 3) for the instability of a stationary charged soliton; in the three-dimensional SS case the sufficient condition for the instability is inequality (6.38) and there exists always a negative EV of the operator H (s-state). The stability theorem for a potential of a rather general form is given in the Appendix. We check the known soliton solutions from this point of view. In the ~ theory (plane geometry) the equations permit the transformation s/15(x,
p.) = p.
whence
2~const. p. The translational mode of the Higgs kink soliton with zero energy, as we saw above, is the lowest one (nodeless), and, hence, all A. ~ 0, i.e., the soliton is stable. For the bell soliton the same mode is not the lowest one (one node), therefore there is one lower EV, A <0 as can be seen in figs. 6.3a and 6.3b. The
Q = p.
*
—
This root is situated between
A~and
A 2.
V.0. Makhankov, Dynamics of classical solitons
95
4~
_____
_______
-~
(a)
~,
(b)
Fig. 6.3. Field function and (a) its derivative (translational mode) for the LGH kink-soliton: (b) its derivative for the KG3 bell-soliton.
second condition leads to I dQ p.2 constdp. vm —p. —
/
2
V 2
2
22
2=
p.
m
m
2
p..
Hence, in the complete accordance with the study of section 5, the bell soliton is longitudinally stable for p.2>~m2. For the SS-solitons we have
Q=
~
2~const.,
Vm-p. and dQ/d~>0, which is in agreement with the result by Anderson and Derrick [116] for the absence of stable time-independent SS-solitons for the ~ (Klein—Gordon) theory.* Given a non-linearity of the form 422 in the equation ofmotion it is easy to check that all the solutions starting with n ~ 1, if such exist, are unstable. The equations with the non-linearities of a more general form demand an additional investigation of the properties of its stationary solutions, in particular, finding the behaviour of Q(p.). In what follows we give some examples. 3. We generalize the theorem for the case of a Schrödinger-like equation 2
2
2
(6.39) Eq. (6.39) is obtained by the variation of the action with a Lagrangian ~
I.
*
*
421)—(V42( —-2gjço(. 2
12
Changing again to the real field function s/i = stationary solutions: —p.~s/i+ V2sfi + g2s/i3
*
(6.40) 2t}, we have the following equation for
4
~
exp{ — ip.
= 0,
It is interesting to note that the same solutions for ~s> m12 turn out to be stable with respect to the scaling mode.
(6.41)
96
V.0. Makhankov, Dynamics of classical 301 itonS
and a Lagrangian L=
—
J
d3x. ((Vs/i)2 + p.2sft2 ~g2i/I4} —
(6.42)
Eq. (6.39) has the first integrals* S E
=
J1~I~ d3x,
=J {(
(6.43a)
2I~(4} d3x.
V
42 ~— ~g
(6.43b)
Eq. (6.41) can be derived by the following variational principles: (1)
OLE s/i]/Os/i = 0,
(6.44a)
given that the variations Os/i are arbitrary; SE[s/i]/&/i
(2)
~s/,,
=
(6.44b)
given that the additional condition OS[s/i] = 0 holds (this condition may be considered as a constraint, and p.2 is a Lagrangian multiplier). Here, to first order, the variation Os/i is orthogonal to the solution
f
ifr
3x =0; 58s/,d (3) The variational principle can be formulated by introducing some function F(S) of the functional (6.43), H = E[i/i] + F(S), into the Hamiltonian (6.43b). Equating the first variation oH/Os/i to the variation OLfOs/i, we get eq. (6.41) and we determine the function F(S) dF/dS =
(6.45)
2
In this relation p. is a function of 5, and the variations Os/i, are arbitrary as in the case (1). For the study of the stability of possible stationary (soliton) solutions of eq. (6.41) it is necessary to find the second variation of the Hamiltonian, 52E in the case (2) and 82H in the case (3): 02E =
f
OI/iIfOs/I d3x,
(6.46a)
given that the additional condition holds
J
{2i/i~Os/i+ (Osfr)2} d3x = 0,
and ITt =
—V2 + p.2
—
3g2s/i~.
(6.47)
Hence it follows that if there is even one eigenfunction Yt of the problem Ify 1 = A1y1 with negative A. <0 satisfying the condition (6.46b), the solution s/i5 is unstable. The presence of two negative EVs A1 and A2 means that the soliton is unstable as the condition (6.46b) is satisfied by the corresponding selection of the constants a and b in the linear combination Os/i = as/i1 + bsfr2. *
The soliton momentum integral in the rest frame is not interesting.
V.0. Makhankov, Dynamics of classical solitons
97
Let us consider instead of eq. (6.39) an equation of a more general form to include so called “saturable” non-linearities (e.g., see [120]) i42,
+
2 V 42 + QF(~42r)= 0.
(6.48)
2)~M as IQt2~~x. Eq. (6.48) realizes the extremum of
A non-linearity the Lagrangian is called a saturable one if F(~q,~ ~~i(ço,*~ç — 42*42,)
=
—
~V
where
J
2U(~
—
42(
2),
(6.49)
42I
l~l2
U(l
2)= 42~l
—
F(x) d3x.
(6.50)
Eq. (4.80) has the integral S (eq. (6.43a)) and the integral 2)}d3x.
E=J{1V
(6.51)
4212+ U(j42~
Changing to the real function s/s = ~ exp{—ip.2t} leads to V2~ p.2~+ sfrF(s/i2) =0. —
(6.52)
This equation is obtained by the variational principle (6.44b). Calculating the second variation of E by analogy with (6.46) we get (6.46a), where
If =
—V2 + p.2 — F(s/i~)— 2s/i~dF(s/i3/d(s/i~).
(6.53)
As in the Klein—Gordon equation case considered above, the translational symmetry leads to the presence of a three-fold degenerate p-state with energy zero (the lowest non-symmetrical state). Hence there is at least one symmetrical s-state with A <0. Applying the method of Lagrangian multipliers we write the problem of the minimization of the functional 02E in the form
Ifo
42 = fl042 + as/is,
(6.54a)
and
J
3x = 0. Oq,s/i5 d We expand 042 and ~ in the complete orthonormal system of EF of the operator (6.53), .FTty, = 842
=
~ CjY~,
s/i 5 = ~ s/i5~y~,
and we substitute these expansions into (6.54); then we have 842
=
a~AQ
Yt.
From (6.54b) we have
a ~‘ As—fl = ag1ffl) = 0.
(6.55)
98
V.G. Makhankov, Dynamics of classical solitons
-_______ ____________ Fig. 6.4. The function g(f)) when there is only one A
1 <0 in the non-linear Schrodinger model.
For a = 0 the function 842 is the same as one of the y~and Cl the same as one of the A1. As we saw above, a negative EV A, exists only for the spherically symmetrical s-state. If such a state is the only one, it is the ground state and if it has no zeros, it cannot be orthogonal to the nodeless solution s/i~~ so that a 0 (in any case for the study of the nodeless soliton stability). Hence g(Cl) = 0.
(6.56)
Differentiating eq. (6.52) with respect to p. and using the above mentioned method (see eqs. (6.30)—(6.32)) it is easy to obtain g1~,1—~A,
4p.dp.~
6
The function g1(fl) when there is only one A1 <0 is represented in fig. 6.4, where we see that the negative root of eq. (6.56) occurs for g(0)<0, or dS/dp.>0.
(6.58)
Hence we may formulate the theorem on the stability of solitons of eq. (6.52) in a negative aspect: the presence of one negative EV of the test Hamiltonian H (and this exists always) and the satisfying of the condition (6.58) are sufficient for the instability of the soliton 42~o~ We note that taking into account the second-order term in the perturbation (042 = y and ~~ I) in the orthogonality condition (6.54b) leads to
iTty = uly +-~as/i5+~y, and hence we have instead of (6.56) g1(Cl) = 0—
The roots of this equation are the points where the function g(fl) intersects some line g~which lies under the abscissa axis. Therefore the soliton instability condition for infinitesimal is dS/dp.~0.
(6.59)
This analysis allows a simple and clear geometrical interpretation. The extreme ~ is a saddle point in
99
V.0. Makhankov, Dynamics of classical solitons
a functional space, and the orthogonality condition determines a surface on which the variations 842 are allowed, given that these variations cross the saddle along some curve. Depending on the sign of dS/dp., the saddle point in this curve is either a maximum or a minimum. Hence the value of dS/dp. is connected with the angle between the variation surface and the plane containing the steepest descent line. Therefore, although the saddle point is unstable, the presence of the additional constraint (6.54) can lead to a stabilization of the solution. Finally we note that the stability theorem for the Klein—Gordon equation (6.19) can be easily generalized for equations of the form V2s/i—(1—p.2)~+s/IF(s/i2)0. 6.2. Some properties of spherically symmetrical solitons Eqs. (6.59) and (6.52) will be identical if one introduces the notation 2_li~p., 2, K ~ p. In that case one has
for for (6.59), (6.52).
—
V2sfr
—
1(21/, +
sfrF(s/I2)
Let us show that as [761*
K —*0
=
0.
(6.60)
the SS solutions of (6.60) are the same as the solutions A~(r)of the equation
1 d / 2dA~\ —~--~~r -~--j_A~+A~_0
(6.61)
(n gives the number of nodes of the function A(r)). Introducing the new variables 42
=
s/i/ic
and
(6.62)
p = Kr,
instead of s/i and r, we have 1 22 V Q—(p+42---2F(I( ~ )0. If one looks for the solutions in a class of bounded functions ç ~ N, we have for KN 4 1 an expansiont 2
x
2, if we redefine 42 by
i.e., we have eq. (6.61) with the accuracy of a zeroth approximation in K 42
2dF(x)
2
=A. 2) be upper bounded (a “saturable” non-linearity) as Let the function F(ifr *
uX
x~’O
~
then one can write
We studied in detail an analogous equation in section 2 (see (2.59)).
fit is natural tosuppose that for small in the definition of ,e.
~‘
the expansion of F(~’2)starts with a term ~
i.e.. F(0) = 0. Otherwise the term F(O)
0 could be included
100
V.0. Makhankov. Dynamics
of classical so! itons
U.
—
,1~
Fig. 6.5. Approximate form of a poteniial in which a fictitious particle moves within the framework of an equation with a saturable non-linearity,
Fig. 6.6. Approximate plot of S as function of , equation with a saturable non-linearity.
= (I
2)112 for an
—
w
its asymptotic behaviour as k=
Fi(~2)=M_(C~)2k,
1,2,...,
(6.63a)
if there is a power series representation, or as F
2) M — C~exp{—C 2} 2(s/i 2s/i for an exponential non-linearity (C
(6.63b) 1 and C2 are some constants which depend on the form of the
function F). Inserting the expansion (6.63a) into eq. (6.60) one can easily see that the extremum points of the potential u
F(x) dx
= _~1(2~2 +
determining the trajectory of the solution in the phase plane* are ~=
*(K2+F(~2))(K2+M_~)0
Hence, besides the point 1/’ = 0, we have also 1
C
~1m~(M_’K2)
112k
.
(6.64)
Therefore 2
ln
(6.65)
C
(6.66)
1 2m
— ~2.
Eqs. (6.64) and (6.66) are valid as *
K2~~*
M.
An approximate form of the potential a is represented in fig. 6.5.
V.0. Mak)iankov, Dvna,nics of classical solitons
101
For the simplest models of saturable non-linearities F1 = [5//2/(1+ ~~2)]q (see, e.g., [120]) and 2}) (see, e.g., [64,68]) we get from (6.64) and (6.65) exact formula F2 = 1/C(1 — exp{— l/q Cs/i ~/iim =
±(1
(6.67a)
2fa)1/2~
1
1
(-~ In ~ — CK2)
5/i2~~ = ±
1/2
.
(6.67b)
Using these results we can study the behaviour of the integral S[s/i] as a function of K. For small follows from the substitution (6.62) that s
=
f
~2
d3x
=
-i--f
K
it
2d3p =
increases as K—*0. As K2—* M —0 the behaviour of the integral S will be determined by the extremum points they determine the scale of the solution as a function of the parameter K. Hence
5/Im
as
S ~ s/’~-~.
It follows from this formula that for equations with a saturable non-linearity there are solutions for which S increases as K2 —~M. The approximate form of the dependence of S on ic is represented in fig. 6.6. As S must be a continuous function of the parameter K, it has a minimum at some ~cr~ It is natural that the derivative dS/dic at Kcr changes its sign. Hence for the class of equations considered here with bounded F(s/i2) there is a domain K> Kcr, where dS/dK > 0. Finally we show that soliton solutions can exist for which the test Hamiltonian H(s/~’ 5)has only one solution with a negative eigenvalue. For this, following the paper [120],we note that as ic—*0 we arrive at the operator of a cubic 3y. = —V2y 2y, equation 3ic21/r~y If 1 + ic 1= A1y1. —
By a numerical investigation [120] it was shown that among the spherically symmetric solutions of this equation there is only one solution with A1 <0. All the remaining EV of SS-solutions are greater than zero. The energy levels of the operator (6.53) are shifted when K increases. Let us show that no one of them passes into the negative region. It is sufficient to show for this that H has no bounded SS solutions with zero energy for 0< ic2< M. Then, for increasing K none of the EV, A1(K), corresponding to a symmetric function, vanishes or, a fortiori, becomes negative. As we saw in subsection 2.7 any trajectory in the phase plane starting from the axis s/It = 0, with increasing r ends at 5/s = 0 or at the points 5/’m (it can also be seen in fig. 2.24). Only for definite “initial” values of the function 5/’(r)Iro do we obtain trajectories finishing at zero as r—*co. The slightest variation of the initial value s/i(0) leads to a trajectory which will finish not at zero, but at a point ~/~m (the well). The solution s/i. with the asymptotic behaviour s/si —*0 as r is unstable with respect to a variation -of initial data (Liapunov instability; see the paper by Zhidkov and Shirikov [76]). IF s~.is a solution of (6.60), which finishes at the point ~/‘m and has boundary conditions 5/i~(0)= 5/i~(0)+ e, ds/i5/dr = 0 and sfr5(r) is a solution of the same equation but with zero asymptotic behaviour, we have —*
e-*O lim5)~=8~1) E t95f’5(O)~°°
102
V.0. Makhankov. Dynamics of classical solitons
Hence, the function x~ as/i~(r)f3s/s5(0)is unbounded as r—* differentiation of (6.60)) an equation =
and satisfies (as one can easily see by
0.
(6.68)
This means that the operator If has no symmetric EF with zero energy, as we wanted to prove. Hence, for any values of sc in the interval (0, Vp.) the test operator H (see eq. (6.53)) has only one 2) spherically symmetric solution with negative energy. means that eq. (6.60) with function which is bounded as s/i—*~has stable solutions for Itsome values of sc> K~r andthe at any rate F(s/s when K2~* M. Note that in the case of a relativistic equation the charge Q is conserved, but not S, so that the sign of the derivative dQ/dp. determines the region of stable solutions. Here it is necessary to bear in mind also that K —*0 corresponds now to p. 1. The second asymptotic behaviour exists only for the case K2_* M or m2(l — w2)—* M so that it is necessary to have M < m2. The above treatment holds only for this case. The asymptote for Q for small w is the straight line co = (1 M1m2)”2, but not the ordinate axis. The decreasing branch of Q corresponds to a stable solution. —*
—
Examples of stable SS-solitons 1. One example of equations having stationary stable SS-soliton solutions was given by Anderson
6.3.
[1171 2~+ g242J42~2— a4214214 = 0. 42 ~- ~,, — m The corresponding Lagrangian and Hamiltonian are
V2
2=142,1 —1V 2
2
2
2
142! ~‘=I42~l +1V421 +m 1421 421
2
—m
2
2
4
12
12
6
—~aI42l,
+-2g 42! 2
1~
(6.69)
4
I~
6
—~g1421 +~aI42I Changing to a real function s/s = ~ e’~’making the usual substitution of variables, co 2 gs/i(1—co) 2)’12 a = &m2(1 w2)/g4, p = mr(1 — co we are led to a one-parameter equation:
(6.70)
.
=
p.1m,
=
—1/2
—
U’~P
-~~ bU’4’
3
5
—~+—---=~—q~ +a~. dp pdp The method of investigating this equation is analogous to the method of subsection 2.7. Rewrite (6.71) in the (anharmonic oscillator) form: —‘(a.
____
\2+i( 4)2~i(h4
d
—
‘
~2’.p/
2J~~
2
aq)6
—
3
—
+
U.
The function u is shown in figs. 6.7a and 6.7b for two characteristic values of the parameter a defined by the extremum points, du/d~,and the zeroes of the function u: 2 _________ F ii~Vi—4a\” 12=~ and Pl.2_~\ 2 ) 2a 3a —
From the boundary conditions ‘I~= 0 at p = 0 and t1(p) —*0 as p —~~ and from fig. 6.7a it follows that for a given a in the interval (0, ~) there are one or several solutions of eq. (6.71) (one solution is
103
V.0. Makhankov. Dynamics of classical solitons U. U-
/~
~1~77~2
(b ) 6) non-linearity model for two values of a: (a) a
(a) Fig. 6.7. Potential in the ~ — ag
(b) a>
<~,
nodeless). For a > ~ there are no values of 4(O) when the solution has the asymptotic behaviour as p—*ca~ (this is connected with the sign of d/dp<0). Anderson [117] gives nodeless solutions of eq. (6.71) and studies numerically their stability as a function of co for different values of the parameter a. In fig. 6.8 [117] we give the soliton instability growth rate as a function of co. This shows that the instability region is reduced with an increase of a. For a ~ the soliton is stable in the interval (0.18< co < 1) (nevertheless the region of small co <0.18 is unstable). The presence of a stable solution for this model, in contrast to the 424 model, can be clearly illustrated by means of the potential part V 5 of the soliton energy density for both modest (see figs. 6.9a and 6.9b). It is seen that in the 424 model the only extremum of V5 (besides a trivial one) is the maximum, hence the soliton “rolls down” either to s/i—*0 (“dissipative mode” of instability in the terminology of [116]) or to 5/,-÷cc(“singular mode”). In the model under consideration the absolute maximum of the function V5 arises at a fixed value of s/i = F2. The soliton has two possibilities; either to roll down from the hump to s/i —*0 (region I) (dissipative mode as above) or to roll down to the well s/i—*F2 (the singular mode disappears!). However, for Q w = 0 this does not yet mean soliton stability. In fact, it can conserve the value s/s = F2 when it increases its dimensions. This gives a negative contribution to its energy. The soliton can compensate this negative contribution by means of increasing the gradient ~/‘r(see fig. 6.10). A kind of a shock wave arises and moves to large r with increasing steepness of its front. The of the charge Q = bounded: const. can depress 3x =presence const. = of Q, the the integral soliton dimension is upper this instability. Because of co f ~2 d R<1~’3 ~ O~~rn~417.wF2(1_co2)) Tm (~)
:
Fig. 6.8. Growth rate of the Q-soliton instability in the
(~4 ag6) model as a function of —
w for various a.
* As in subsection 2.7, nodeless and oscillatory solutions with non-zero asymptotic behaviour are possible.
104
V.G. Makhankov, Dynamics of classical solitons
(a)
(b) 4 model; (b) in the (g’ — ag6) model.
Fig. 6.9. Soliton potential energy density: (a) in the g
For the appearance of the “dissipative mode” the soliton energy perturbation must be of the order of its initial energy E 0 which leads to a soliton climbing to the potential hump at the point F1. The instability modes have been both observed in the numerical experiments of Anderson [117] for the appropriate selection of parameters. Anderson describes a behavior of the soliton in a stable region of the parameters for sufficiently small perturbations of the soliton energy (for details see ref. [117]).The important fact is that the presence of a charge can stabilize the soliton (see also section 3). 2. The second example is the soliton of Friedberg, Lee and Sirlin (FLS) [10].*They find a solution to a system of two (charged and neutral) interacting scalar fields (see eqs. (1.14)). The first equation is an analogue of the Klein—Gordon equation (1.5) and the second one is an analogue of the Higgs field equation. Extracting as usually the time dependence of the complex field, 42(r, t) =
s/i(r) e~’,
3=
(6.72)
we get from (1.14) the system of equations 22 V2 5/i—ax sfr+co 2 sfr=0, 2
22
1
(6.73a)
2
the soliton charge,
Q=
wJ
3x
~2
d
wS[5/’],
Fig. 6.10. Dissipative instability mode in the (~4_~ a~6)model. The soliton ‘sits” in the potential well (see fig. 6.9b). *
An analogous but more complex (spinor field) SLAC bag soliton has been considered by Bardeen et al. fl22}.
(6.74)
V.0. Makhankov. Dynamics of classical solitons
105
and its energy E=
f
~
d3x,
(6.75a)
where =
~{(V~)2+ (Vs/i)2 + ~(w2
+
a2x2)5/’2 + ~(x2
—
(6.75b)
1)2),
and a = rn/p. is the ratio of the charged ~pmeson mass to the neutral x meson mass. Eqs. (6.75) can be derived by a variational principle in the following three formulations: (i)
=
or
= 0,
=
0,
(6.76)
where L
=f2 d3x,
2
=
—~{(V~)2 + (Vs/c)2 +
The variations Os/s and O~are arbitrary and w (ii)
~
=
=
0,
or OEJQ
~(a2~2 —
=
w2)çb2 + ~(x2 l)2}.
(6.77)
—
const.;
(6.78)
= 0.
The variations Os/i and O~are also arbitrary but now co must be considered as a functional of s/i by using (6.74) =
(6.79)
Q/S[s/i].
The charge Q is fixed. From (6.75) it follows that (6.80) Finally, introducing a “reduced” energy ~w=~ü
and
E=f~rd3x~
(6.81)
we get as in the case (6.44b) -.
(6.82)
which means that OE/0~=0,
(6.83a)
OE/8s/i = w2çt’.
(6.83b)
and
Here as earlier w2 plays the role of a Lagrangian multiplier. From (6.83b) we have
~ E(s) = ~w2,
(6.84)
106
V.0. Makhankov, Dynamics of classical solitons
whence (6.85)
E=E—~wQ=—L+~wQ.
The variational principles (6.76), (6.78) and (6.82) are useful not only for soliton-type solutions but also for plane wave solutions (“free mesons”) if one considers the latter in a large but finite volume and sets periodic boundary conditions (see section 3 and [119]). Using the variational principle (6.76) we get the virial theorem by the scale a-transformation
f
~
(6.86)
where 2 + (Vsf,)2}, = ~{(V~k’) 1 2 22 2 1 2 = ~{(w — x )s/’ — ~(x —
a
(6.87a) 2
1)
}
From (6.86) and (6.87b) it follows that for co
(6.87b) = 0
the system (6.73) has no soliton solutions since
3. By using the method described in subsection 6.2 we can study the behaviour of soliton solutions as 1(2 = (1 w2/a2)—*0. Putting p = aicr and s/i = (K/a)42 we have from (6.73b) with an accuracy of —
(x2— t)x0.
The solution x = 0 is rejected since in that case s/i does not vanish at infinity as p solution x = I ± cy and have for it y = ±1(2422 or
—*
co•
We take the
22
X’~42~
(6.88)
Now the first equation of (6.73a) is also reduced to (see (6.19)) 2
3
V~~ 42—42+42=0. The behaviour of theUsing charge Q near zerothat K isthe determined by the 2 (see (6.63)). (6.25) we find soliton mass is relation Q ~ co/K, i.e., it diverges as a E = Qm(1 ~ O(ic~))=rQm + C/Q+~~~. (6.89) —~
We emphasize that as ic —*0 the soliton mass is greater than Qm. This value, as we shall see later, is the mass of “free mesons” in a box with total charge Q. In fact, let us place the system into a sufficiently large volume V. We suppose that the fields have periodic boundary conditions. Then from (6.73) it follows that one possible solution is a constant: ~=±~‘
~=_~(i_~).
Introducing the notation w = a cos E and using the normalization condition Q *
same.
(6.90) =
co5/2 V we get*
It is interesting to note that there are analogous solutions for “free mesons” also in the model (6.69) as V—*c~and formula (6.91) is exactly the
V.0. Makhankov, Dynamics of classical solitons
xcose,
~b(Q/coV)
5~fl~
1/2 ~
v2a E = Qm(sec )(1 — ~ sin2 ).
(sin2 ) cos = 2Qa/ V, For fixed Q we have in the limit V
a,
=
107
(6.91)
—* ~
E = Qrn.
(6.92)
A similar result was obtained by us also in the plane geometry for another models. In this (and only in this) sense the limit ic —*0 for sufficiently large Q can be considered as approaching the “free” field. In all the models considered above it reduces to the 424 theory. We estimate now the soliton mass for small to. For this purpose we use trial functions of the form x
—
—
1~—exp[—(r—
R)//1,
f(5/’~sin
r ~ R, r>—R.
wr)/r,
r~
9 (6. 3) 94 (6.
)
The form of these functions follows from the expansion (6.88) and formulae (6.90). The quantities I and R are parameters with the dimensions of length. Their values are taken from the condition of the minimum of the soliton energy. The condition* R = nIT/co
(6.95)
must be also satisfied. Here n is the mode number (n = I corresponds to a nodeless solution). From the normalization condition we have s// 0 (1/ir)V~j~. Inserting (6.93) and (6.94) into eq. (6.75) for large R and Q we get 3 + 0(R2). (6.96) Emin ~
+
R
Minimizing this expression we have Emin
~ ~1T2”4Q3”4,
(6.97)
which occurs for R
(2Q)”4.
(6.98)
The soliton mass is less than the mass of the “free” mesons in the box, if Emin < Qm,
or 14
Q>Q
4
(6.99)
5=m(s(irla)).
Hence for Q> Q5 the soliton is stable, at any rate with respect to a decay into free mesons. Finally 2). noteMoreover, that formulae are valid forand to =(6.99) (irIR)were ~ 1, when one under may drop (6.96) the since(6.96)—(6.99) the expressions (6.97) obtained the in assumption termsQ 0(R that ~ I, they are a very good approximation as a -40. In the paper [10] it was found that in the * The function ~ is then the exact solution of (6.73a), and
x has the correct asymptotic behaviour as
r-÷s~.
108
V.0. Makhankov, Dynamics of classical solitons
Fig. 6.11. Energy of soliton in the FLS model as function of the charge.
opposite limit, a
—*~,
Fig. 6.12. Soliton charge Q(w)
one has for Q5 the following formula
3,
Q5—= 111.8a i.e., Q 5 —*0 as a
as function of w.
(6.100)
we
~. More than that, for a > 5 have Q. < 1 and the quantum soliton for which Q must be an integer is stable for any Q 0.” The results obtained enable us now to answer the question about the stability of the soliton considered. As we saw above if there is one negative EV, A, <0, of the test Hamiltonian If the stability region of the solution coincides with the region dQ/dco <0. The existence of A, follows from the limiting case K—*0 when the problem is reduced to eq. (6.61) for which the test Hamiltonian (6.27) has one spherically symmetric state with negative energy [116] A, = —I5.6,c. The same reasons as in subsection 6.2 permit us to infer that when K increases, up to unity, the second negative eigenvalue of H does not appear (the same conclusion can be obtained by studying the conditional minimum of the functional E[5/i, xD. The behaviour of the function Q(co) as to —*0 and to known from (6.63), (6.98): as co—*0, Q(co) cx Q(w) ~ (a2—co2)”2, as co—*a. —*
—*
a is
4,
~
From the continuity of the function Q(w) it follows that for the whole of its descending branch the soliton is stable. The approximate behaviour of the functions E(Q), Q(w) and E(S) 110] are shown in figs. 6,11—6.13. In these figures the soliton stability and instability regions are also shown. The expressions for the second variation of the soliton energy (52E)( 0 and the test Hamiltonian H will be considered in the appendix. The details of a proof of the stability of the soliton under consideration (which is somewhat different from our proof) can be found in [10]. In this paper it is shown that with increasing node number (the mode number) the solitons become more unstable. By the method of ref. [18] the quantization of the soliton is carried out and a quantum correction to its mass is found. This correction has with an accuracy of 0(g°)the standard form (6.101) Here
~N
are the normal modes of the operator H2,
‘N
are the occupation numbers, the last term is the
vacuum energy. In this term the summation is extended over all the frequencies ~ of the solution in Here we may remain in the framework of the weak coupling approximation since the quantity a in (6.100) is determined by a relation *
between three constants: the masses of the charged and neutral mesons and the coupling constant.
109
V.0. Makhankov, Dynamics of classical solitons
Fig. 6.13. Plot of reduced energy E as function of S.
the form of “free mesons” (for the plane case see also [85]).The quantum mechanical corrections lead also to a situation when in the region Q~< Q < Q5, where the soliton energy is greater than Qm, the soliton is metastable because of acan barrier penetration since the latter isseeproportional 2) the soliton lifetime be quite long if g2effect. ~ I (gHowever is the coupling constant, subsection to exp(—1/g 3.6). Finally we note that the model permits an evident generalization. Let 2
=
—
8 ~*~9
—
~(D~x)2 —
a2X2!42!2
—
U(~),
(6.102)
where U(~)is an arbitrary polynomial of degree not higher than four in x~so that the theory remains renormalizable. Then if U has an absolute minimum at the point x = Xvac U(~)~ U(Xvac) 0,
0, putting
we have for the charged 42-meson mass m
=
and for the soliton energy density = ~{(Vx)2
+
(Vs/i)2 + (co2 + a2~2)+ 2’W(x)},
(6.103)
where W(x)
=
Hence Emin
~ IT~+ ~ir
U(0)R3 + O(R2g2),
and
g2=x~. Then we can consider two cases (i) U(0) 0, (ii) U(0) = 0,
Emin ~
Em,n~~ir(6Q/g)2’3,
where we have put U(~)= ~g2x2(xXvac)2 —
-
(6.104) (6.105)
110
V.0. Makhankov, Dynamics of classical solitons
and x
= (I +
e’~y’
as a trial function. For both cases the behaviour of the function Q(w) is analogous to fig. 6.12. The conditions of the stability theorem remain also useful, so that the left branch of the function Q(co) corresponds to a stable (in the classical sense) soliton. Finally we note that authors of [10] describe a numerical calculation for finding solutions of eqs. (6.73) and give the functions E(Q) and Q(w) obtained in that way.* For the case of a = I they have Q~= 3.03, and the point C corresponds to to 0.96. For a = 5 the value Q. is 0.21. 4. In concluding this subsection we give one more example of a stable localized spherically symmetric solution of the Higgs field equation [85]. In the simplest case when a uncharged field is an isovector and its direction coincides with radius-vector in any point of space (“hedgehog”) Q
5 = 3.47,
Xa
we have the equation 5’q(r) + i~(r) ~3(r) = 0, i5 = V~—21r2. .1 This equation has a solution with the asymptotic behaviour
(6.106)
—
? 7(r)=C/r,
asr—*0, as r-+~.
(6.107)
This solution is a bubble inside which the Higgs field vanishes. Because of the abnormal boundary2 conditions infinity which energy preventand the hindering bubble from expanding the rterm playing the at role of an (~(r)—* effective1) kinetic it from collapseand forof small such—277(r)1r a bubble is stable. This fact has been confirmed by a direct numerical calculation of Rastorguev at Dubna. We emphasize that the stability of the bubble does not contradict the Derrick—Hobart theorem (6.10) since the bubble is not the soliton of our definition since because of the boundary conditions at infinity and the term —(2/r2)’q(r) the total energy of the bubble is linearly divergent. The same considerations help to understand qualitatively the appearance of the stability region in the example of the two-field charged soliton in the Friedberg—Lee—Sirlin model: the Higgs field ~(r) prevents the soliton from spreading (the “dissipative” instability mode), the presence of the charge Q hinders its collapse (the “singular” mode) because of dQ/dco <0 which for small to leads to dQ/dR > 0, i.e., if Q = Q 0 is given the soliton radius cannot be less than some value R0 (see, e.g.. (6.98)). Hence the Higgs field realizes a kind of a surface tension and the charged field leads to an effective spatial repulsion and the stable soliton solution arises as a result of the balance of these forces. Finally we note that the Langmuir wave collapse considered in = section example of the an 2= — to2 1 —(1—5 is an = 2y~~~O when unstable “hedgehog” in a “non-relativistic” approximation it equation of vector 424 theory is valid. However, in contrast to (6.106) here we have ~)2
I.s/i(r)— 5/i(r)+ s/s~(r) 0, *
The fourth-order Runge—Kutta method was used.
(6.108)
V.0. Makhankoi. Dynamics of classical solitons
111
with normal boundary conditions at infinity, s/i(r)—*0 as r—*~and just this fact leads to the “hedgehog” instability. As we saw above, for scalar fields the soliton stability appears for sufficiently large it, i.e., for small to (“ultra-relativistic” approximation). Just in that region the saturation of the non-linearity and the presence of the “charge” S can lead to the appearance of stable (including the case of non-symmetric perturbations) soliton solutions of the plasma turbulence equations. Thus, for instance, for a model with a non-linearity in the form (6.63b), (6.109) 155/i(r) p.25/’ + s/s(l — e~2)= 0, —
it is easy to find from the virial theorem that —
p.2 ~ ~
J r2 dr(1
—
e~2)~ 0,
that is s/~,as~ 1,
and R 0 ~ const.
Numerical investigations [64] show that the constant is 4V2de. Let us show now that the presence of the charge and the saturation of the non-linearity lead in any case to a stopping of the collapse (i.e. to a suppression of the “singular” instability mode).* 2) M C/s/i2 as s/i ~, and then For this purpose we write as before (see (6.63a)) F(sft E = J r2 dr{~42rI2+ ~ ~ M~ço~2 + tn 42~2~} —*
—
—*
If the quantity S 3 = E + MS which is conserved in time is bounded for t = 0, the packet collapse is stopped. In fact the presence of the collapse means that for any in the interval 0< < S and arbitrary R we can find such an instant of time r starting with which the inequalities
Jr2
drI42(r,
)(2>
are satisfied, i.e., in any arbitrarily small spatial volume practically all the plasmons, which were contained in the initial packet, collect as a result of the collapse. Since S3=j
12 1 2 2 r dr~J42rI+~42I +IlnI42H 2
I
2
is the sum of only positive terms we have the estimate
~f
2 drI
r
2 ~~
42!
-~ ~
dr ~
~—
S 3,
2) =
whence it follows that as R
—~
0 the integral S3 cannot be bounded. For the rion-linearities F(1421
~For eq. (6.109) this result has been obtained in 1641.
112
V.0 5fakI~.~
Dynamics of classical solitons
ko~I(1+ I42I2),~and(6.109) one can obtain exact expressions for S3: S3 =
f
2
dT{142r12 +~
IøI~+ ln(1 + 14212)},
r
and s 3= Jr2dr{l42rl2+~I42l2+(l_e~2)}.
However, from this discussion the longitudinal stability of the “hedgehog” with zero asymptotic behaviour at infinity does not yet follow and even less the “transverse” stability. Nevertheless by an analogy with scalar solitons [1161one can expect that in any case the ground state (a nodeless one if one does not consider the origin) may be stable with respect to non-symmetric perturbations. The unformed two-scaled soliton is unstable with respect to the non-symmetric mode, as follows, for example, from section 4 and numerical calculations [92].
7. Long-lived pulsating solitons We have considered so far non-stationary solutions of non-linear wave equations. There were bound solitons of the sine-Gordon equation—bions, and quasi-stationary bound solitons of the Klein—Gordon and Higgs equations. All of these one-space dimensional solutions we call bions. The last two equations are not completely integrable and the life-time of their bions may be very long, though finite. Therefore the bions of the KG3 and LGH equations are only approximate solutions. There naturally arises a question whether there exist in the real three-space dimensional world certain analogues of the bions. We consider below non-stationary spherically symmetric solutions in the frame-work of the KG3, the sine-Gordon and the GLH equations and show there really exist long-lived pulsating solitons both of large and small amplitudes, which are something like three-space dimensional bions. The chief results have been here obtained by a computer and only in the small-amplitude approximation was it possible to construct a perturbation series for the solutions in a sufficiently consistent way and thus to find analytical solutions in analogy with the one-space dimensional problems (see section 3). 7.1. Meson bubble life-time Let us first consider as an example the evolution of a quasi-plane initial packet x = tanh[(r—R0)/V2], R0>> 1 in the frame-work of the Higgs equation [111,112]. Such a bubble may be regarded as a gluon-type meson without quark—antiquark pairs (for the physical sense of this model see for example [122, 112]. As we saw earlier (see subsection 5.7) the bubble starts to collapse to the centre.* Near the centre a reflection takes place as a result of which the compression is replaced by an *
When at the initial time we have a packet of sufficiently general form
x(r,
R0), but R5’> 1,
with the necessary boundary conditions at the
origin and at infinity, at first the self-consistency of the packet occurs as in the case of the CLW spheriton and then the packet takes the form of a quasi-planar soliton.
V.0. Makhankor. Dynamics of classical solitons
113
130 L
160
~IL ~
10
-
~ t~24
20
~‘III
~T~1C
70
-0.21
,‘
t~22B
~
20
Fig. 7.1. Evolution of a “meson” bubble in the LGF1 model for
R0= 10.
expansion. Because of the law of conservation of energy the expansion can happen until the radius R1 ~ R0, then compression starts again. The picture resembles strongly that of a conventional oscillator. It is clear that the lifetime of such a quasi-planar bubble is completely determined by the radiation of energy to infinity. In the plane world considered above the analogous oscillations of the field correspond to the bions which have a rather long lifetime. Is there the same situation in the real world? Unfortunately one cannot estimate analytically the energy emitted and the bubble lifetime which is determined by this energy. For this purpose Bogolubsky and the author have performed.a series of computer experiments [126]. 2XtXr emitted through a sphere of radius WeR have calculated the total flux of energy Q(t) = ~T rm ~‘ 0 and found the function E(t) and the distribution ~(r, t) (see (5.64)): E(t)= J~rdr~
~
l)2}.
It turns out that for different values of the initial radius the bubble emits over half of its initial energy after only 1 to 3 pulsations. In this process there is both the emission in the form of small-amplitude waves near the vacuum as defined by the boundary conditions and also a fractionation of the field energy into separate spherical layers, some of which move to the boundary rb and leave the domain r> ~,.The computations have been performed for Rc = 5, 7, 8, 10 and 15. In fig. 7.1 we show a typical picture of the bubble evolution (R0 = 10). Fig. 7.2 gives the dependence E(t) for several values of R0. From these curves one can see that the bubble lifetime is not greater than a few (3 to 5) times its radius R0. However, in this figure one can note that as a result of evolution of two bubbles with sufficiently large energies (R0 = 8, 10), identical long-lived one-scale states appear for which the
114
V.0. Makhankov. Dynamics of classical solitons
E 100
lao
200
300
Fig. 7.2. Bubble energy as a function of t for various initial radii R
0.
thickness of the transition layer is of the order of their radius. Both the energy and the field function are concentrated near the centre; moreover, in fact the energy distribution does not change with time and the field function oscillates around the vacuum fixed by the boundary conditions. Naturally we call such solutions “pulsons” (do not confuse with pulsars!). Below we discuss in more detail the pulson properties. Such studies have been performed in order to investigate the bubble evolution in the sine-Gordon model Xrt~rX+s11lX0.
(7.1)
The initial packets were chosen in the form of a quasi-planar solution x
=
4 arctan[exp(r
—
R0)],
R0 ~ 1.
(7.2)
Eq. (7.1) is completely integrable, does not permit stochastization or emission when solitons interact, and has bions as exact solutions in the plane world. Sometimes the hopes are expressed that these unique properties of completely integrable equations must somehow arise in more real world of two and three space dimensions [801.The authors of the paper [126] set their mind on testing whether these properties might possibly occur in the evolution of the bubble (7.2). In particular, one might assume the existence of SS analogues of the bions and, as a result, of infinite oscillations of the bubble. The computation for R0 12 shows a powerful emission already after the first reflection of the bubble from the centre. The picture of the evolution coincides qualitatively with one depicted in fig. 7.1 up to the formation of long-lived pulsons. Thus in this computation we do not observe any unusual properties of the sine-Gordon equation in the physical world. In the next two sections we study the properties of the pulsons of the KG3, the sine-Gordon and the Higgs equation on the basis of papers by Bogolubsky and the author [127—128]. 7.2.
Pulsons of the KG3 equation
Let us limit ourselves to the real scalar field 3=o. xo—V~x+x—x
(7.3)
115
V.0. Makhankov. Dynamics of classical solitons
As we saw earlier this equation has non-trivial stationary solutions, the plane and the spherically symmetric solitons. However, both those are unstable. On the other hand, eq. (7.3) in the plane geometry has rather stable (in any case in the longitudinal direction) localized oscillating solutions, the plane pulsons [82]. Using the method of [82] for small amplitudes and a computer we study the SS pulsons of eq. (7.3). Let ~(r, t) be given by x(r, t)
=
a(r) cos tot + b(r) cos 3wt ~
.
b ~ a.
(7.4)
Inserting (7.4) into (7.3) we get the well-known eigenvalues problem arr+~ar_K2a+~a30,
a~(0)=0,
K2= 1—to2,
a(oo)=0,
or, if A2 = ~it2a2, we have (7.5)
Arr+~ArA+A~0.
The solutions A~(r)of (7.5) are numbered as before so that A~has (n
—
I) nodes, thus
A 1(0)
4.34< A2(0)
14.10< A3(0)
29.13<~~ . <~4(~) <...
In the case of small amplitudes xm we have
2t)xrn A~(a~
x~(r,t) = V~x0A~(a~0r) cos (Vi~~
0r) cos(\/1 as a solution of eq. (7.3). The function b(r) in the same limit is b(r)=-
~X~A~(XOT).
2t), —
a
=
1,
(7.6)
X0
(7.7)
The function (7.6) for the first three modes (n = 1,2,3) has been investigated by a computer for the amplitudes x m = 0.2, 0.4 and 0.7. At xm ~ 0.4 the expression (7.6) is the solution of eq. (7.3) with high accuracy (errors are less than 1%). We note that for sufficiently small amplitudes x~‘~1 the radiation by the pulson is very slight and its lifetime r—~~ as xm—*ø. If in eq. (7.6) xm = 0.7, then the pulson amplitude Xm(t) slowly decreases to Xm(t) = 0.63 (at t = 80), and the characteristic radius grows. The stability of the solution (7.6) is tested by the variation of the parameter a. Values of a < I correspond to a broader initial field bunch with the same amplitude and frequency and values of a> 1 correspond to a narrower one. In the first case, the initial bunch has a greater potential energy than the “equilibrium” one (a = 1), in the second case it has a smaller potential energy. It is natural that the initial evolution of both bunches is determined by this deficit of the potential energy. The positive deficit leads to a compression of the bunch, the negative one to an expansion of it. This fact is confirmed by computer calculations; however, the evolution does not stop at the equilibrium value of a I and goes further. Moreover the first bunch is unstable with respect to a singular mode and the second one with respect to a dissipative mode. These instability modes are both inherent to the charged Q-soliton of eq. (7.3) [116]and were described in subsection 6.3.* Compare the growth rate of -~
* Note that although the solution (7.6) is the real part of the solution obtained by Anderson and Derrick 1116] for the charged bunch, there is a qualitative difference between them. The second one is gauge invariant and conserves strongly the charge Q.
116
V.G. Makhankov. Dynamics of classical solitons
i..L(0,t~
0.2~
(.0-)
~
20
t
~
~7
~
Fig. 7.3. Time-dependence of the envelope of the oscillation amplitude of the field function in the origin (K03 model). (a) a < l (dissipative mode), (b) a> I (singular mode).
Fig. 7.4. Singular mode of the Q-soliton instability in AndersonDerrick’s theory. is the soliton energy density.
both the instability modes for the pulson (7.6) [127](see fig. 7.3) and the Anderson—Derrick Q-soliton (see figs. 11, 12 of ref. [116]).It is easy to do it for the2 =singular mode. From figs. 7.3b and 7.4 we find 4.34 Vi —0.64 2.6. Bearing in mind that the that tp 40, Xp(O, 0) = 0.4; tQ 2.7, XQ = 4.34VI w’ instability growth rate (1/t) cx x2 we have tQ/tp = x~/x~ 0.024. Numerical calculations give tQ/tp 0.067, i.e., these values are of the same order of magnitude. The difference of a factor 2 is either connected with the rough estimate, or it indicates the larger stability of the Q-soliton as compared to the pulson stability. Analogous estimates can be obtained for the dissipative mode. We find the pulson energy —
J
E = 4’z~ r2 dr =
4irJ ~‘dr
= 2irJ
(x~+x~+x2—~x4)r2 dr
(7.8)
for small x
1. We note that because of the scale transformation x(r, Xo) t’°(x~)I~(x~) + I~(x~) = x~(I~’° I~)+ 5(n) + Xo = I~”~(x~) + S m~
—
~
—
=
~0y(~0r) we have
—
where j(n) =
J (~)2r2
2~
dr,
J y~r2dr,
I~ =
S
J
2~ y~r2dr,
I~=2~Jx~r2dr, and
n
is
the ~~1S°’1
mode +
number.
J
i~”~2~ ~
In
the
limit
0)(cos2 tot +
to2
as
Xo~°
sin2 wt)r2 dr
we get
117
V.6. Makhankov, Dynamics of classical solitons
I
Fig. 7.5. The field function
2
x~and
the energy of a spherical layer ~‘,as functions of the radius for the first three modes, n
with an accuracy of O(~~). Hence because E1~~2~J ~
3
&2 =
= 1,2,3.
1 — x2~,we see that
0)r2 dr,
(7.9)
and that the energy density LW,, is independent of time. The functions ~,, and x~for n = 1,2, 3 are represented in fig. 7.5. It is interesting to note that for the same values of x m ~1:2:3:4:9: 1 the pulson masses with the number n are and 5 approximately in the ratios proportional to S~/A~,° 7.3. Pulsons of the sine-Gordon and Landau—Ginzburg—Higgs (LGH) equations 1. Let us consider the properties of one-scale field bunches which are formed as a result of the evolution of “mesonic” bubbles (see section 7.1) in the framework of eq. (7.1), Xtt—~rX+smnX0. Three characteristic stages of the evolution of the pulsons of this equation are observed in computer experiments for the initial data in the form of the quasi-planar bion 1K coswtl ~(t0)=4arctan~— ~, t.w cosh KTJ
/
w= vl—ic
2
(7.10)
,
for
K/to = 10 or to In the first stage (t 0 to 200) the formation of a one-scale bunch with a bell-shaped form of the oscillating function x(r, t) occurs. Then, more than half of the initial bunch energy, E = J 4irr2 dr(r)
=
21T
J (x~+ x~± 2(1
—
cos x))r2 dr,
radiates away (see fig. 7.6 and compare with the bubble (7.2)). Practically from the onset formation process we see a quasi-periodic behaviour of the function x(r, t) with a period T
(7.11)
of 1
after that the period
the 7.5;
slowly increases together with a decrease of the pulsation amplitude C(t) (C(t) is
the envelope of the function Ix(0, t)I). In the second stage (t = 200 to 700) a weakly radiating pulson with an amplitude C(t) modulated by a period T2 10 arises, the amplitude slowly decreases from C(t) 2ir to C(t) ir (see fig. 7.7).
118
V.6. Makhankov, Dynamics of classical so/lions
~00
of Fig.t. 7.6.
Pulson
1000
energy for the sine-Gordon equation as
—
,~nction -
0t)I inEnvelope SCln~) WO]nt-i) Fig. y(0, 7.7. the sine-Gordon p~ricd of the oscillation o~er model r’.”O,l,S-,3... pulson. amplitude of the field function
The field oscillations are practically symmetric with respect to the zero vacuum x~= small change in C(t) in the period T2.
0
because of the
One of the very important facts is that the evolution of such a pulson from the second stage really does not depend on the choice of the initial data ~(R, 0). The fact of the formation of a long-lived heavy pulson of eq. (7.1) from various field bunches indicates its stability for i~~ C(t) ~ 2~.On the other hand, the same results permit to state with a large degree of certainty that even within the framework of the sine-Gordon equation, which is completely integrable in the plane geometry, in the spherically symmetric case there are no exact solutions that are analogous to the plane multi-soliton ones (in particular to the bions). When the amplitude C(t) is equal to C2 ~ir (t = 630) the third stage of the pulson evolution starts. A relatively rapid (~t 80) decrease of C(t) down to C3 4 1 takes place accompanied by a “swelling” of the field bunch. Then again a state with a weak radiation arises. An analytical description small amplitude pulsonsection. C(t)4 The I is principal obtainedterm by the userequired of the 3and of the amethod of the preceding of the expansion sin x x ~x solution can be expressed by the formula —
—
~
t)= \/~~ 0A~(~0r)cos(\/l —X~t),
(7.12)
where A~is the set of solutions of eq. (7.5). 2. The dynamics of the pulsons eq. (5.63) 3 =0, xtt — V~~xx + x for the Higgs field turns out to be very similar to the above one. We note only characteristic differences. Firstly, the oscillations are non-symmetric with respect to the vacuum x~= —l selected for the calculations. The form of the quasi-periodic function x(O~t) (its period gradually decreases from T 5.3 to T V2 1r) even in the second regular stage (t 100 to 1350) differs from a sinusoidal one. At once after the formation of a weakly radiating heavy pulson of the Higgs equation, the modulation amplitude of the function C~0(t)(the upper envelope of the field function x(O~t); because of a non-symmetricity of x it is worth-while to follow its upper and lower envelopes C~0(t)and Cd(t)) is relatively large (see fig. 7.8). When C~0(t)approaches to zero the modulation amplitude tends to zero and the modulation period monotonically increases from TM = 8T at t 10 to TM 17T at 1200. In the third stage, as in the case of the sine-Gordon equation, the heavy pulson becomes unstable. A rather rapid decrease of its amplitude C~~(t) from zero at t 1360 to C~0(t) —0.8 —
119
V.0. Makhanko r. Dynamics of classical solitons
1DOr~
~
~
~O,1,2,3.
jx(O, t)~in
Fig. 7.8. Upper envelope of the oscillation amplitude of field function
the LGH model pulson.
(t
1650) takes place. At this time the oscillations become symmetrical with respect to the vacuum —1 selected for the calculations. Thus the lifetime of the heavy pulson of the Higgs scalar field is more than two times larger than the analogous sine-Gordon equation pulson lifetime. Finally, in the last stage one can describe a Higgs SS-pulson with a small amplitude of oscillations around the vacuum x~= —1 by writing the field function ~(r, t)= u(r, t)— 1, u 41 in the form of power series [85]: =
u(r, t) =
2’~’f
2g 1(r) +
~[e
2”~2g 2~~1(r) sin[(2n
+
l)wtj +
2~~2(r) cos[(2n
+
2)wtlJ,
(7.13)
and making the substitution r = \/~t/(I +
~2)I/2
~ = \[2r/(I
+
~2)l/2
As a result we have for the function A(~)= V~7~f1(~) again eq. (7.5). By making the appropriate substitutions we get the algebraic relations 32
g1=—-4ft,
g2—4f~
(7.14)
With an accuracy of O(E2) we have 2A~(~)(l + cos 2wt),
u(r, t) = ~= A~(~) sin tot —
to2 = 2/(1
+ ~2)
(7.15)
It is easy to check that the masses of the pulsons (7.15), M~ En with different values of n and with the same amplitudes Urn = (2/V3)A~(0),are in the limit Urn 4 1 in the same ratios as S~/A~(0) (~1:2:3:4:9:...). The distribution ~‘(r) having the sense of energy of a unit spherical layer of radius r and the energy density e(r) for small amplitude (x m 4 1, Urn 4 1) pulsons are both stationary; this directly follows from the solutions (7.6) and (7.15) (see also subsection 7.2).
120
V.0. Makhankov, Dynamics of classical solitons
V 0,927 t~370
ft
_____
C~J
~‘
‘7.
Fig. 7.9. Distribution of energy density in a heavy poIson of the LGH model at two times. The distribution is seen to be non-stationary and to resemble the energy distribution in a plane bion.
~
Fig. 7.10. Picture showing qualitatively the evolution of a pulson in the LGH model. The ball dimension is associated with its mass. U~ is the potential energy of the poIson as a function of the field amplitude (and hence mass). The vacuum q~= — I is fixed by a boundary condition.
In this context, heavy pulsons with large amplitudes differ qualitatively from light ones by the essential non-stationarity of ~(r, t) and (r, t) (fig. 7.9). The lifetime of the (7.16) and (7.15) solitons increases with decreasing oscillation amplitude. We also note that solutions in the form of light pulsons for all three types of equations correspond to the so-called near free-field approximation it2 = m2 — w2-÷0. As in the case of the FLS stationary Q-solitons [10] it is natural to expect in the region of large amplitudes the existence, besides the nodeless pulsons, of long-lived heavy pulsons with nodes of the field function both for the sine-Gordon and for the Higgs equations (proceeding to the limit of the functions A~for small oscillation amplitudes). In the meanwhile an analytical description of the heavy pulsons has not been found, and the solutions (7.16) and (7.15) are obviously unsatisfactory for large amplitudes (see figs. 7.7—7,9). Finally we note an interesting feature of the behaviour of the heavy pulsons. It is remarkable that in the framework of the essentially different eqs. (7.1) and (5.65) the massive pulsons are “stable” for amplitudes close to the upper vacuum x,~ (which is distant from the “main” vacuum Xo selected by the boundary conditions). These are IT ~ C~ 21T for the sine-Gordon equation and 0 ~ C~~(t) ~ I 0(t)E of the amplitudes, [0, ir] and [—1,0],respectively, for the Higgs equation. In the intermediate domains the heavy pulsons “lose” their stability and rapidly “fall down” to the region of small oscillation amplitudes near the main vacuum ,v~.This effect can be apparently explained by a prevailing influence (if one wants by the “attraction”) of one of the vacuums. It is schematically illustrated in fig. 7.10 (the dimension of the ball is related to the oscillation amplitude and thus to the pulson mass). It is seen that in the “stability” region the pulson, because of its radiation, must overcome a potential barrier and this considerably increases its lifetime. More work on an elucidation of stability problems for both heavy and light pulsons is carried out now. For the present it can certainly be asserted that the Higgs equation pulsons turn out to be stable with respect to non-symmetrical excitations (the results of these and some other two-dimensional computations performed at Dubna by Bogolubsky, Shvachka and the author will be published elsewhere). Conclusions and acknowledgements Possible development of theory 1. Plasma. Construction of theory of Langmuir turbulence on the basis of “collapsons” as a phase
V.G. Makhankov, Dynamics of classical solitons
121
transition in the plasmon gas near a phase equilibrium point (the liquid phase consists of the collapsed plasmon “droplets”). In this respect it is necessary to study: a) the stability of Langmuir “hedgehogs” for equations with a saturable non-linearity, b) elementary processes of their interaction, even if only for cylindrical geometry. For the first part some arguments raising hopes was given in the text. In the work on the second problem numerical experiments are the most important. It seems to us that efforts in this direction will not perish to no purpose (remember the achievements accomplished in this way in the physics of large dimension excitons [132] and, in particular, the observation of exciton droplets). 2. In particle theory the soliton use for physical applications just begins. It is apparently connected with the fact that, firstly, even in the study of “plane” solitons a new still enigmatic world of a great number of phenomena is revealed, which is interesting by itself and attracts investigators. Secondly, so far one does not know enough about solitons and their behaviour in the real four-dimensional world. The most interesting problem, namely, the soliton interaction still remains terra incognita. However, an interest in this problem arising recently gives grounds for optimism. The first that comes to mind is the soliton form-factors and their comparison with experiment for the extraction of the models with most perspective [19,133]. The second one is the unique properties of QS as coherent states consisting in a strong suppression of a decay channel connected with the total dissociation of a QS into constituents as compared with the transition soliton—~soliton + constituents. Remember the behaviour of psions which can be apparently explained by soliton models without charm [134,135]. The third one is attempts to explain the quark confinement by soliton bags [122,123] and other models [18]. Finally, why can black holes not be solitons [136]and why do laser beams in plasmas not show the wave-particle duality [137]. In fact the transparency of active materials increases by five orders of magnitude when solitons propagate through them [131] and the first soliton had been observed experimentally [2]. The author is extremely grateful to his colleagues for valuable discussions and criticism. Special thanks are due to Prof. I.S. Shapiro for helpful discussions concerning the application of soliton theory to particle physics and for actual initiating our work on three-dimensional pulsons and to Prof. D.V. Shirkov for discussions stimulating our work in the direction of the study of models which are transient between q,” and sine-Gordon theories. The author also wishes to thank Professors K. Nishikawa and H. Flaschka and Doctors B. Fried and R. Shukla for sending many papers which were very useful in writing this review. We would like to express our appreciation to Professors M.G. Meshcheryakov and E.P. Zhidkov for their encouragement, to Dr. I.L. Bogolubsky for reading the manuscript and some remarks and to Prof. V.K. Fedyanin for many valuable discussions and suggestions. Much help by G.A. Makhankova in preparing the manuscript and by A.F. Rubtsova and Yu. V. Katyshev in the translation is also appreciated. Finally, the author would like to express his sincere thanks to Prof. D. ter Haar, since due to correspondence with him the idea of writing this paper was born. We wish to thankfully acknowledge Prof. R.G. Salukvadze and the Sukhumi Technological Institute (where a large part of the review was written) for their hospitality.
122
V.0. Makhankov, Dynamics of classical so)lions
Appendix 1 Let us look for a soliton-type solution of the system i~,+
xc°
~ —
0,
i3~(x2)~~
— 132Xxxft
in the form =
çfi(x
x = x(x
exp[i(~vx —
—
Vt)
—
Vt).
Introducing A
= ci
v2/4,
—
~= x
—
and (3
vt
=
v2(3 2,
we get
2X + f3~~2 = ~2 (A.1.1) 7 Let further x = —A sech2(a~)(the solution of the uniform Bq equation), then the first equation (A.1.1) is +
+
(A + A sech2 a~,)t/i=
Introducing the variable
~=
0.
tanh ~, we find (A.1.2)
+
where Aa2 (n
=
= 0, 1,2,3,. =
$~(l— ~2)A2+2$a2(1 — 3~2)A=
—(1-- ~
(A.l.3)
s(s + 1), r— V~/a. Solutions çl’(~)bounded at ~= ±1 (~= ±~) are found for s — r = n . .), and eq. (A.l.3) is satisfied if n =4~~s = 2. In this case from (A.1.2) we get
B~(i
—
B = const.
~2)(s~~J)/2
Eq. (A.1.3) gives the condition imposed on the constants A and B: (f3~+ /0A2 = B2,
— 2A 7
+ /3
2+~/3A2=
0,
1A
or B2
(13~+
~A2,
A
2(13
+~
(A.l,4)
Finally, =
B(tanh a~)sech a~,
~
x
=
—A sech2 a~,
A=Af6.
(A.1.5)
(A.I.6)
Analogous formulae can be obtained for the case of WS-solitons (the plus sign of the term which is proportional to x in the first equation of (A.1.1)). The solution exists for B2
((3k
—
f3)A2,
A
=
2),
~ = A sech2 a~
(A.1.7)
V.G. ‘lakhankot. Dynamics
the other formulae are the same. Inserting (3
= /3
of classical
and ~
=
so/thins
123
into (A.l.4) and (A.l.6) we have the relation
(2.41): A
=
31 —— 20 y
A model with the IBq instead of the Bq equation leads to
s=’
A=_~(10y2_1)_1.
by (B2/ôt2)~2we get Replacing (8~/~x2)~2
A
—~-(l+9v2)~ty2.
=
Finally, for (a2/ot2)x2 and (i94/9t28X2)~we have A
=
31 — ~—
—~--~.
It follows from these formulae that Acr is very sensitive to the model when
~2=
1
Appendix 2 We generalize the theorem on the stability of a charged stationary soliton to the case of arbitrary potential U(2~(2). 2
2
1
2
~II —~V~ —~U(2l~(), ~r=112+ V~2+~U(2(~(2). “~‘
For i/i e”°’
we have Q =
wS[~i]
~
f
~/,2
d3x,
(A.2.1)
E = ~ J{to2~~2 + (V~)2+ U(~i2)}d3x.
(A.2.2)
The second variation 62E for fixed (A.2.i) is easily obtained (see the main text) 52E =
d3x +
31(II~’Ô~/I
~J-(f
1frSÔ~/Id3x)}~
(A.2.3)
where =
and
i/i.
—v2— ~2 + 8(*2)!+
2~ 8~tp~2l
is the soliton solution of the equation, +
—
~,fr~F(t,fr~) =0.
(A.2.4)
124
V.G. Makhankov. Dynamics of classical solitons
Differentiating (A.2.4) it is easy to check that ~/i~ = ~ is the eigenfunction of the operator H with A = 0 (the translational mode). It means that there is a p-state i/ri = (3~~/’~)Y,m(1 = 1) with zero energy and hence there is at least one spherically symmetric s-state with A <0. Differentiating (A.2.4) with respect to to and proceeding as in the main text, we find V ~ 2fA
~j
—
I
+
.~
d
Q
and =
A.8 11
~
Therefore t = 0; g(fl) w~~ t~41(fl— A1) hence as before we have the soliton instability condition dQ/dw>0, independent of the form of the potential U. For the determination of the stability region it is necessary to investigate the behaviour of the solutions ~fr. as functions of to in order to make sure in the uniqueness of the s-state with negative energy.
Appendix 3 The second variation of the FLS soliton energy can be easily obtained if one puts x the column functions
=
x~+ &~and
= ~/i~ + ö~i. Introducing
‘=(~)
~=(~)~
(A.3.l)
we get (52E)
3x +
~{J~
0
d
~k~j-~ [J ~ d~x]}~
(A.3.2)
where —
(10
°)v2 + 1
(a2c1i~ ~ 2aX
—
1)
2a2~s~frs)
5lJJ. and ~3 is the transpose of i~. Then as in the main text, = 3,,,i~,
=
A1y1,
q= ~
~
~‘ ~/A1 =
.F~’i~ =
~—
2w~,
(Q/to),
(A.3.3)
V.0. Makhankor, Dynamics of classical solions
125
whence T0
=
A.811 + ~J— ~,
(A.3.4)
and g(fl)
~J—~(ci ~‘
—
A1)’
—
I
=
0.
(A.3.5)
References [1] 5. Boussinesq, J. Math. Pure AppI., ser. 2, 17(1872)55. 121 J. Scott-Russell, Proc. Roy. Soc. Edinb. (1844) p. 319. [3] E. Fermi, JR. Pasta and SM. Ulam, in: Collected Works of E. Fermi, Vol. 2 (Univ. Chicago Press, 1965) p. 978. [41N.J. ~abusky and D. Kruskal, Phys. Rev. Lett. 15 (1965) 240. (5] AC. Scott, F.Y.F. Chu and D.W. McLaughlin, Proc. IEEE 61(1973) 1443. 16] N.N. Bogolubov and DV. Shirkov, Introduction to quantum field theory (Interscience, N.Y., 1959). 171 E. Noether, Nachr. Ges. Wiss. Gottinger Mathem. Phys. Classe (1918) 235. [8] DV. Shirkov, Preprint JINR (1977). [915. Kubo, M. Namiki and I. Ohba, Preprint Waseda Univ. Tokyo (1975). 1101 R. Friedberg, T.D. Lee and A. Sirlin, Phys. Rev. D13 (1976) 2739: Nod. Phys. 8115 (1976) 1:32. [II] D.J. Korteweg and G. de Vries, Phil. Mag. 39(1895)422. [12] V.E. Zakharov and A.B. Shabat, Soviet Phys. JETP 34(1972)62. [13] yE. Zakharov, Soviet Phys. JETP 35(1972) 908. [14] V.G. Makhankov, Phys. Lett. 50A (1974) 42. [15]K. Nishikawa, H. Hojo, K. Mima and I-I. Ikezi, Phys. Rev. Lett. 33(1974)148. [16]J. Gibbons and D. ter Haar, Preprint Univ. Oxford Ref. 2/76 (1976); J. Gibbons, S.G. Thornhill, Mi. Wardrop and D. ter Haar, J. Plasma Phys. 17(1977) 153. [17] V.K. Fedyanin and LV. Yakushevich, Preprint JINR P17-9628 Dubna (1976) (in Russian). 118] N. Christ and T.D. Lee, Phys. Rev. D12 (1975) 1606. [19] R.J. Finkelstein, Preprint UCLA/75/TEP/16 (1975). [20] AS. Davydov and NI. Kislukha, Preprint ITP Kiev (1972); Theor. Math. Phys. 16(1974) 702. [2l[ Yu. A. Beresin and V.!. Karpman, Soy. Phys. JETP 24(1967) 1049. [22] Kh. 0. Abdulloev, IL. Bogolubsky and V.G. Makhankov, Phys. Lett. 56A (1976) 427. 123] D.W. McLaughlin, I. Math. Phys. 16 (1975) 96; A.V. Mikhailov, JETP Lett. 23 (1976) 320; F. Calogero and A. Degasperis, Bumerons, Preprint n.9 Istituto di Fisica, Univ. di Roma (1976); F. Calogero, Lett. Nuovo Cimento 14 (1975) 443; Nuovo Cimento 3lB (1976) 229; A. Newell and L. Rederkopp, Preprint of Clarkson College of Technology, Potsdam, New York (1976). Reviews: M. Toda, Phys. Reports 18(1975)1; BA. Dubrovin, V.B. Matveev and S.P. Novikov, Uspekhi Mat. Nauk 31(1976) 55 [translation in Russian Math. Surveys]; A. Newell, Lecture at the Conference on Solitons, Univ. of Arizona. Jan (1976) (to appear in Rocky Mountain Journ. of Math.) and others presented at this Conference; yE. Zakharov, Inverse scattering method, in: Springer Verlag Lectures (1977) Volume Solitons” (to be published); F. Calogero, A. Degasperis, Preprint Univ. di Lecce. Istituto di Fisica 73100 (1975). See also references cited in these papers. (241 IL. Bogolubsky, Preprint JINR P9-8698 Dubna (1975); JETP Lett. 24 (1976) 160. [25]Kh.O. Abdulloev, IL. Bogolubsky and V.0. Makhankov, Phys. Lett. 48A (1974) 161. 126] A.E. Kudryavtsev, JETP Lett. 22 (1975) 82. [27) B.S. Getmanov, Preprint JINR P2-5996 Dubna (1976); JETP Lett. 24 (1976) 291. [28] IL. Bogolubsky and V.0. Makhankov, Preprint JINR E2-9695 Dubna (1976); JETP Lett. 24 (1976) 12. [29] RB. Kadomtsev and V.!. Karpman, Soy. Phys. Uspekhi 14 (1971) 40. [30]G. Zaslavskii, Soy. Phys. Uspekhi 16 (1974) 761. [31] A.A. Galeev and R.Z. Sagdeev, Voprosy Teorii Plasmy (1973) v. 7. Atomizdat, Moscow (translation published by Consultants Bureau). [32] V.1. Karpman, Non-linear waves in dispersive media (Pergamon, Oxford, 1975).
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