Unstable quasi-gaseous media

= [1—

exp(—lrl)]1

(23.14)

-

This solution describes periodic hills whose maxima become infinite as r—* —0 and the minima are still short of zero, in contrast to solution (23.8).

All these simple solutions have not been found until now. But what is most important, our “evolutionary” solutions are non-existent in the limit as t—~—cc, i.e., are spontaneous, whereas in ref. [45] the initial conditions of the form p = 1, v ~ 0 are chosen at t = 0, implying an initial impact forcing the tearing of the current sheet. We should also like to emphasize the fact that it is not the current, but the plasma “material” that is

being torn in the simplified one-dimensional model under consideration. Examination of a true tearing instability would require studying a two-dimensional problem which, however, is not reducible to the “quasi-Chaplygin” system of equations. In conclusion, note that two implicit parametric solutions are presented in ref. [45]dealing with the tearing instability. The first one appears as x=~

+ (K/2)

arctan[2~t/(1+ 2) +

Z=

1 + 2(~2 —

t

(42

~2

+ t2)2,

—

v/v

t2)],

0 = (ic~/z)(1 + ~2 +

n/n

(23.15)

0 = zI[Z

+

Kt(l +

~2

—

where K is a constant, and ~ is a parametric variable. It is assumed here that t 0, but we find it also possible to proceed from the asymptotic limit t—~ x~ The second solution fails to comply with the “evolutionary principle” and is described by the formulae 2—~2), p2t~ x=~+Esinq~, E=Kexp(t (23.16) v/r’ 1. 0 =2(t sin ~ + ~ cos ~)E, n/n0 =[1 + 2(tcos ~ sin ~)E] When t = 0 here, we have an initial impact with v(t = 0) = 2V 2)at n(t = 0) = n 0K~exp(—~ 0. —

~

—

194

BA. Trubnikov and S.K. Zhdanos’, Unstable quasi-gaseous media

24. Buneman plasma instability Following refs. [46,47], we shall discuss the so-called Buneman instability in a plasma, which arises when electrons (e) move at a superthermal velocity, v0> r’Te, with respect to the ions (i) which are initially at rest and are then described by the equations of motion On 0 —+—nr’=0, Ot Ox

Or’ Or’ e eO —+v—=—E=———~, 0t Ox m m1Ox

(24.1)

r’1, n = ‘~e= n1 (due to the quasi-neutrality), and ~‘ is the potential of the electric field resulting from the instability. The magnetic field of the current and the pressure will be ignored in view of r’0> r’Te~ For the electrons, whose velocity is v~,= r’0 + 3, where 3 is the oscillatory addition, we have the same equations

where v =

On

+v0

—

Ot

On 0 + nr’ =0, Ox Ox .

—

—

03

+ r’0

—

Ot

03 O3 + v Ox Ox

—

—

=

e 0 me Ox

—

—

(24.2)

~.

However, considering merely the slow motions determined by the ion inertia, we shall assume that On/Ot 4 v0 On/Ox,

O3/Ot 4 v0 03/Ox,

cv 4 kr’0

(24.3)

and then, omitting the small terms with time derivatives in (24.2), we shall get two laws of conservation:

mev~/2 ep

nu~= const. = n0r’0,

—

=

const. = rner’~/2,

(24.4)

whence the following potential can be found: 2 1) mer’~/2e. (v~Iv~ 1) mer’~/2e= (n~/n Its substitution into (24.1) and the introduction of the “effective density”, p

(24.5)

—

=

—

=

n/n 0, lead to the

“quasi-Chaplygin” system O

0

~-jp+~--pr’=O,

Or’

Or’

2_30

-~--~+v~--=c0~ ~—p, (24.6) 2.Thus, the system coincides with the equations for the “Chaplygin gas”

where c0 = r’0(m~/m1)” (m=—~).

The simplest solution the first among the “fundamental” ones is given by (see fig. 13) —

—

n/n 0

[47]

(—sinh r)/(cosh r

cos x)~

r’~/c0= (sin x)/sinh r, (24.7) 2 ~)/sinh2 r, = (m~r’~I2 el) (1 —2cosh rcos x + cos where r = t/t~<0, x = x/c 0t~.When TI 41, it changes to the exact self-similar solution reported in ref. =

—

BA. Trubnikov and S. K. Zhdanov, Unstable quasi-gaseous media

n/n0=IrII(i—cosx),

195

u~Ic0=—~r(~sinx, (24.8)

2 r2

(m~r’~/2 el) (1 cos x) (with a minor distinction of x —~x + ir); but this solution is no longer “evolutionary”, since it does not vanish when t—~ and arises only with a certain choice of the initial impact. For completeness, we shall adduce another three “evolutionary” explicit but inverse solutions: (23.9—10) a single well or hill — and (23.13) periodic in x: =

—

—~

—

—

(±x/c

2]112,

(24.9)

0t*)12=A±arctan(A/Iri), A=[Irp/(i—p)I—r ±xlc 0t = A + arctan(ApI(i

—

p)),

A = [exp(2r) (i/p —

—

1)2]h/2.

(24.10)

The last solution, in a more complicated form, is presented in ref. [47], but the three simple solutions (24.7) and (24.9) are missing there. The authors of ref. [47] show that numerical calculations with regard to temperature and kinetics at the final stage of evolution have always resulted in the forms close to -the self-similar formulae (24.8), which justifies the “quasi-Chaplygin” approximation. It might be interesting to note that, as suggested by solution (24.7), at the time t = 0, when the entire plasma gathers into bunches, the function of ion distribution turns out to be (—~< r’
—

[24]). eiq/2inv~

f\ / \ I’~w~ I 2\ :

a)

i’~ -

~i

-

-

-.

x/c

—1

0t~

b)

/

___ -~

\e(q/2mvO

2 4

____

~

c)~

~

~ -~

.-~

o ~

Fig. 13. The growing perturbations of the potential (solid line) and plasma density (broken line) during the development ofthe Buneman instability at: a) -yt = —1/2, b) yt = —1/3, 2]. c) -yt = —2. At t = 0 the current is instantaneously broken and the ion distribution function takes the form dN/dv = (N/1Tv~)/[1 + (v/u~)

196

B.A. Trubnikov and S. K. Zhdanov, Unstable quasi-gaseous media

Note, in conclusion, that two implicit particular solutions are given in ref. [47] devoted to the t—~—cc, thus complying with the evolutionary

Buneman instability. The first of them goes to zero as principle, and takes the form v=v0exp(yt)cos(kx—v(2U)~’2),

n=n

1’2,

0(1+ep/m1U) 2) (2U)”2]2 U

eip/m

—

1

(24.12)

—

exp(yt) sin(kx r’(2U)~’ where r’ 2 are constants. The second solution does not vanish as t—~—cc and has a more 0, k, form y = k(2U)” complicated r’ =

= ~[r’~

—

(e/Z)(U/2)112 sinh(x* IL) sin(t(2U)’’2/L)

x~= x

—

(EL!2)

arctan[v/([2(w + U)]”2 — (2U)”2)] (24.13)

Z = cosh2(x* /L) cos2(t(2U)’’2/L) + sinh2(x* IL) sin2(t(2U)’’2/L) [2(w + U)]”2

—

(2U)’’2

(n

= (2U)’’2

2 cosh(x* IL) cos(t(2U)’’2/L) 0/n — 1)

where e, L, V are constants, and w

=

is a parametric

(r/Z) (U/2)’’ variable.

25. Plasma instability in an oscillating external electric field This is sometimes called the “aperiodic parametric instability” [48]. It bears a close resemblance to the Buneman instability, constituting its variety. It will be recalled that in the “quasi-Chaplygin” equations (24.6) for the Buneman instability 0

0

Or’

~p+~—pv=0,

Or’

2_30

-~-+r’~—=c

0p ~ui

(25.1)

there is a parameter c~= r’~me/mj,where v0 is the velocity of inertial motion of electrons, set up by the initial impact. Let us assume now that there was no such impact, but the plasma e.g., contained in a capacitor has an electric field, E = E0 sin(flt), which is time-dependent, but uniform in x, and acts on the ions —

—

and electrons in accordance with the equations =

eE0

sin(flt) = ~mete

,

r’e

=

~mjvj/me

= (eEo/meQ)

cos([It)

.

(25.2)

If the frequency f~is sufficiently high, compared to the slow averaged plasma motions, the velocity should be substituted for

r’e

in c~ and averaged over fast oscillations, which yields a new parameter 2E~/2memjQ2. (25.3) c~—~ c~,= (me/mi) ‘(v~)= e This parameter is to be used in the same equations (25.1) describing now the “aperiodic parametric” r’0

instability which, consequently, also behaves as the “Chaplygin gas” and is basically not different from the “purely Buneman” instability discussed above.

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

197

It remains to be added only that the linear growth rate is now ~ = kcAP = keEo/fl(2memj)”2, and the previous considerations will hold true provided the condition ‘y~~ 4 f? is met, the latter implying the slow plasma motions over the period of the field oscillations E 0 sin(flt).

D. “Quasi-Chaplygin” soliton perturbations The solitary waves (solitons), first discovered by Russell, have come to stay as a key notion in the

modern theory of non-linear wave processes. In this section, we are going to examine them in terms of their stability against transverse perturbations. We shall assume the transverse modulation to be “long-wave” in nature, with the soliton perturbations turning out to be “quasi-Chaplygin” and giving rise to new varieties of QCh media. Five types of solitary waves for the most important non-linear wave models are discussed below: i Korteweg—de Vries (KdV) solitons [52,53], —

2— Cnoidal KdV waves [11,54], 3 Two-dimensional solitons of the Kadomtsev—Petviashvili (KP) equation [55,56], 4— NSE solitons [37], —

5

Solitons of the sine-Gordon (SG) equation [21].

—

In analysing specific examples we shall follow Whitham’s method [21]. Its main points being generally known, we see no need to dwell on them here. It should be noted though that we are

following the “Lagrange” formulation adopted by Whitham [21].The Hamiltonian formalism in the theory of one-dimensional modulations is extended in refs. [57,58], while the Lagrangian approach, similar to ours but devoid of the long-wave simplification, is used in ref. [53]for the KdV, NSE, Klein—Gordon and Higgs solitons. However, as shown in ref. [59],without the long-wave approximation the method of ref. [53] can give rise to inaccuracies. 26. “One-atomic gas” of KdV soliton perturbations We are going to show in this section that long-wave transverse perturbations of the Korteweg—de

Vries solitons are described by exactly the same equations as is the ideal and, besides, one-atomic

(!)

gas. 2u~ The KdV equation u~+ uu~ sc0l 5= 0, where s = ±1 is the dispersion symbol, c0 is the sound velocity, and I stands for the dispersion length, has a particular solution in the form of the KdV soliton 2O, 6=(x—ct)/zl, c=—su 2, (26.1) —

u,=—su0cosh

0/3,

~=l(12c0/u0)~

whose perturbations are known to be described by the Kadomtsev—Petviashvili (KP) equation [60] ~

,

j2p~

,

~

j

\1_/’

p \

~ut uu~ sc 0~~

—

~c0~ ~ ~U)~

,

which is derived under the assumption that the inequality k~54 k~is satisfied. Note that this equation is not only suitable for the analysis of the stability of the soliton (26.1) but is of great interest in itself, since in a two-dimensional case it allows complete integrability [611by the inverse scattering method (ISM), given arbitrary initial conditions. The linear theory of soliton stability based on linearization, such as u = u1 + ~u, is covered in refs. [60,62,63], where the “linear long-wave approximation” (k~5—*0)has been obtained, with a two-

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

198

dimensional medium studied in ref. [60],a three-dimensional medium studied in ref. [62],and a case of a three-dimensional anisotropic medium investigated in ref. [63]. For instance, the authors of ref. [60] present the law of dispersion of small perturbations, cv = ~sc~,u0k~, which shows the instability of soliton (26.1) in a medium with positive dispersion, s = + 1. Zakharov [64] has applied the ISM for obtaining2 the exact solution of -the linearized KP = —Iw~~I(1 lk~Izu2/l 6h12), for s = +1, equation and thereby finding the exact law of dispersion, w which covers, among other things, the region of maximum growth rate of small perturbations at sufficiently small soliton amplitudes (u —

—

0 4 c0).

The present paper, however, seeks to study large, i.e., non-linear perturbations. In view of this, note that particular solutions of (26.2) can be represented by “chains” of two-dimensional solitons found in refs. [65, 66, 67] whose stability in terms of the linear theory is discussed in ref. [68].Using the general formulae of ref. [64]as a basis, we can consider, in particular, the example of an unstable (for s = + i) soliton, from which such a “chain” splits off. Thus, non-linear soliton perturbations can take various forms, which should be considered individually if their behaviour in the region of maximum growth rate is a matter of special interest. This example shows the impossibility of a general description of non-linear perturbations with arbitrary k1 - Such an attempt, however, based on a variational principle with a Lagrangian, was made in refs. [53,69], but, as shown in ref. [59],this method [53,69] fails to produce a correct description of the maximum growth rate region. It is only far from this region, with k1 4 kr”, that the general non-linear analysis proves to be still possible, which will be demonstrated below. Specifically, using the Whitham method [53,69], we are —

—

going to show that in the “approximation of long non-linear waves” the KdV soliton perturbations

behave like the bunches of an ideal one-atomic gas with adiabatic exponent y = ~, whose effective pressure is positive in the stable case (s = i) and negative in the unstable case (s = + 1). In the unstable case, the above “gas” equations reduce simply to the Laplace equation which can be easily solved (two particularly simple spontaneous solutions are presented). Oddly enough, this useful and (!)

—

clear “gas analogy” has never been noted by other researchers, although the stability of KdV solitons

can be justly regarded now as a well-studied problem. One of the equations of interest to us can be easily obtained from simple and clear considerations, following the linear approach shown in ref. [70].For this to be done, it should be merely taken into account that the total soliton is preserved front bends and since the energy density per 2,the energy following equation when shoulditshold unit length is u~i u~ -=

--

u~’2+ div(u~’2tY

1)=

0,

(26.3)

where tJ1 stands for the transverse components of the group velocity, for which it is possible, in turn, to

obtain approximate non-linear equations similar to those of geometrical optics. To make the derivation more rigorous, we shall proceed from the observation that a particular solution of the KP equation is an inclined soliton 2 0,

=

—su0 cosh

where the constants u0, c1

=

0

=

(x

—

x°)/Lt,

x°= x°(t,y, z) = c 1t

a, /3, c, are related as 2 + /32)12 , LI = l(i2c 2 —su0/3 + c0(a 0Iu0)’’ ~,

—

ay

—

f3z,

(26.4)

BA. Trubnikov and S.K. Zhdanov, Unstable quasi-gaseous media

199

Note that this solution is exact and has arbitrary a and /3, but since the KP equation itself supposes k~
rewriting (26.2) as 2co’~’~ + (c

+ ~‘p’~~ sc0l

0/2) (~‘;~ + ço’~)= 0.

—

(26.5)

In such a form, this equation can be derived from the Lagrangian

~J ~

dx dy dz dt =

2~o~ + c

~

0,

+

=

~/3

+

sc0l

2/2,

(26.6)

0(V~)

into which the potential =

—su

0(t, y, z) 4(t, y, z)tanh 0,

0=(x—x°)I4,

(26.7)

corresponding to particular solution (26.4) can be substituted as a minimizing test function. Here, in view of the above inequalities, no derivatives other than those of the function x°(t,y, x) will be retained, which results in 4 0 [—x~’+ c 2/2+ (su 2 0)], (26.8) = u~ cosh 0 (V1x°) 0/3) (1 cosh — 2

Next, by analogy with Whitham’s method, evaluate the integral

L

=

J

dx

~

=

8l(c

2/2]u~2 su~’2/5},

2 {[—x?’ + c 0/3)V

—

(26.9)

0 (V1x°)

which should be treated as a new Lagrangian describing the perturbations x°and 4 in the soliton. Then, the Euler—Lagrange equations lead to the system —x~’+ c

2)~ = div(u~2c

2/2= su 0 (V1x°)

0I3,

(u~

0V1x°).

(26.10)

Introducing the convenient notation 312 = —c p = (u01u00) 0 V1x° were u00 is the unperturbed soliton amplitude, we obtain the “gas” equations p~+divpU~ =0,

LJIj+(LYIV)r’±=~P~V±Pef.

(26.11)

200

B. A. Trubnikov and S. K. Zhdanov, Unstable quasi-gaseous media

Here, the “velocity” components are interrelated through Or’~/Oz=

0r’

5/Oy,

signifying the flow poten-

tiality, and the “effective pressure” is 5POOP

y

,

=

p

,

00 = 2c0u00/15

Pef =

>0

(26.12)

-

As seen from the above, the KdV soliton perturbations behave like a one-atomic (in view of y = ideal gas, if the medium has negative dispersion, s = —1, so that Pef >0. In this case, we get in the linear approximation the following wave equation [70] for the perturbations ,,

2

p11,

c00 ~ p1

—

c00

= 0,

=

1/2

(2c0u00) /3

(26.13)

and hence there is no perturbation increase. Such solitons are stable, but the gas analogy shows that in view of the finiteness of the modulational amplitude phenomena akin to wave “tipping” are likely to occur, which however are beyond the approximations used in the derivation of the system (26.11). As already established before, solitons prove to be unstable in media with positive dispersion (s = + 1). This instability is responsible for the soliton self-focusing [70]which is quite simply described by eqs. (26.11). So, in the case of a cylindrically symmetric perturbation with s = +1, they take the

form (cf. section 11) 2

p,+(rpr’)~Ir=0,

r’t+vr’r=cOop

—1/3,

(26.14)

Pr

and have, in particular, a simple self-similar solution

3r/5r,

r’ =

p

=

[p02/3

—

2(r15rc00) 2 ] 3/2 ,

p0

=

6/5 Ir* Ir~

,

(26.15)

the density on the r = 0 axis, r = t — tcr <0, and r~is an additional parameter. Here, the soliton self-focuses into a point r = 0 at the time t = tcr~ In a two-dimensional case, eqs. (26.11) are written in the standard “quasi-Chaplygin” form

where p0(r)

is

—cc < t < t~, and

p

(pr’)~= 0,

+

r’ + r’r’~=

mc~0(plm)~,,

(26.16)

which complies with the azimuthal number of the QCh medium m =

~. They can be thus integrated by the general methods expounded in part I. As an illustration, we give an exact solution for the

“Coulomb” case of periodic modulation (see section 8): 4=

~

r’1c00

r’

=

=

3z

=

(cosh

~ + cos ~)sinh’

3(cosh ~ + cos ij)’

~,

sin ~, (26.17)

2Q

yt=—4

=

1,

Q,=~coth~—1,

ky=fl—(3LIQ,—~)sin?)sinh~~, where ~ = kc00, k = 2iT/A and A is the spatial modulational period, the sole parameter of the solution. Evolution of the reduced soliton thickness LI(t, y) is shown in fig. 14.

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

201

7

k~1.85

+~/2T

1.36

1.34

__~—_=~____~_L

0

—

Fig. 14. Evolution of the reduced thickness of a KdV soliton in the process of self-focusing. Numbers at the curves show the time in terms of (—yt) with —~
In conclusion we would like to emphasize that our gas equations (26.11) contain only the velocity v which is perpendicular to the direction of the soliton propagation. This is what makes them different from the “general geometrical beam” equations of ref. [71] which, for some reason, contain also the component u~along the normal to the soliton front. Note, besides, that in a particular two-dimensional case (t, x, y) the “gas” equations (26.11) can be derived from the equations of ref. [69], if 7 “superfluous” terms with the derivatives LI~),5 are rejected in the latter. Their retention, as shown in ref. [59], leads to inaccurate results in the region of maximum growth rate. 27. Cnoidal KdV waves The KdV solitons studied above are known to be a particular case of the more general, periodic solution of eq. (26.1). These waves can be described using the Jacobian elliptic cn-functions (hence the

term cnoidal). The general-type cnoidal wave is obtained under the assumption that the solution of the KdV equation (26.1) can be presented as

2(0,m)+B,

0=(x—ct)ILI,

u=—Acn

(27.1)

where A, B, c, LI and m (the latter standing for the elliptic function modulus, 0< m < Their substitution into the original equation yields but two additional relations

A

=

i2smc

are constants.

2,

0(1ILI)

B = c +4s(2m

1)

—

(27.2)

1)c

2, 0(i14)

so that any three parameters among those listed can be looked upon as arbitrary. The results of the linear stability analysis of the waves similar to (27.1) made for various wave models, such as isotropic, anisotropic and “velocity” non-linear models, are summarized in ref. [54].In —

—

general, they boil down to the assertion that the cnoidal KdV waves are usually unstable under the

same conditions as are the solitons. Here, we deliberately confine ourselves to a particular class of one-parameter cnoidal waves presented in ref. [11],for which the modulational equations prove to be similar to those of gas dynamics. We require that the wave described by (27.1) should be strictly periodic in x, with a period, A and, besides, that u = 0 in the period average. This results in two additional conditions LI = A/2K(m),

B = A (cn2),

(cn2)

=

(E(m)/K(m) —1 + m)/m

(27.3)

202

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

where K(m), E(m) are elliptic integrals. The modulus m of the elliptic functions can be conveniently employed as a parameter. In this case, in the limit as m—~0, the amplitude of the wave (27.1) is small, the latter approaching the harmonic form 2, i~ = 2ir(x Ct) IA, c sc 2 (27.4) u0 6smc0(’rrl/A) 0(2irl/A) For highly non-linear waves, m—~1, solution (27.1) changes to a periodic sequence of solitons similar to (26.2): u

u0 cos

—

-

i~ ,

—su

2 0, 0 cosh

u 0

=

12c0 (1/4)2,

c

—su013,

LI = AI2K(m)—~0.

(27.5)

Let the transverse perturbations of the en-wave be determined by the isotropic KP model (26.3), (26.6) as in the case of the soliton in section 26, our analysis being confined to the unstable medium with s = + 1. Formulate the averaged variational principle. The following en-wave potential is to be introduced into the exact Lagrangian (26.6) as a test function: =

—A(m) Z(0, m) LI(m)Im,

=

(x — x°(t,y, z)) /4(m),

(27.6) 0

m = m(t, y, z),

where Z(0, m) is the Jacobian zeta-function [15], and the parameter m is a slowly varying function of time and transverse coordinates, such that Im~5I ~ Since the initial wave is strictly periodic in x, the perturbations of x°and m in the en-wave (27.1—3) can be naturally described by the Lagrangian integrated with respect to the wavelength: -~

L=J~dx,

(27.7)

with A assumed to be fixed. This results in 2/2]+ c L = I[—x~’ + c0 (V1x°) 0 F(J),

(27.8)

with the following designations:

I = 1(m) =

f u2 dx;

F=

J (12u~2+ u3/3c 0)

dx

(27.9)

-

Setting up the Euler—Lagrange equations with the aid of the Lagrangian L, we get a “gas-like” system for transverse modulations of the cnoidal wave O1/Ot+

div(1ii~)=

0;

Ot3iOt +

(tiV)iJ= —c~F”(I)VI;

~=

—c,, Vx°-

(27.10)

The function F(1) included here (see fig. 15) is determined parametrically by the relations (27.9) which give explicitly

B.A. Trubnikov and S. K. Zhdanov, Unstable quasi-gaseous media

203

o

0P(I)/I~’2~

Fig. 15. Function F (I) for the cnoidal wave:

= (12c,l’k~)’,k0 =

2a-/A.

3(m)[3(cn4) (cn2)—2(cn2)3—(dn6)], F(m)=AA 1(m) = A A2(m)[(cn4) (en2)2].

(27.11)

—

The average powers of the Jacobian elliptic cosine can be conveniently determined by the recurrence rule (2n—i)mq~=2(n—1)(2m—i)q~..,+(1—m)(2n—3)q~

2

(27.12) 2~); q 2) = 1— m~+ E(m)/m K(m). (cn 0 = 1; q, = (en The general formulae (27.11) being cumbersome, it appears useful to adduce the results for the q~

limiting cases. In the limit (27.5) of large-amplitude waves (m F~—Iconst.I

—~

1) ,we find (27.13)

j5/3

which exactly coincides with the result obtained for the unstable single soliton (section 26). In the limit (27.4) of the near-harmonic wave (m—~0),we find [72] F”(I)= —i/12(c

2A<0; 01k0)

k 0 =21T/A

(27.i4)

so these waves are also unstable and in this case are similar to the “drops on the ceiling” (see chapter

C). Thus, perturbations of the cnoidal KdV wave offer an example of an unstable QCh medium of a more complex type, whose “azimuthal” number varies with the modulational amplitude from i

(harmonic wave) to 3/2 (soliton limit). For verification, it might be useful to derive the result (27.14) in a different way. It is quite feasible since the Stokes expansion [21]can be used for the slightly non-linear wave, so we seek the solution of the KP equation (26.3) through the expansion u

u0eos

i~+

au~cos2i~ +

i~=

k0(x— ct+ ~ (27.15)

c=c(~)=c°+/3u~+-”,

204

B. A. Trubnikov and S. K. Zhdanov, Unstable quasi-gaseous media

where the expansion coefficients c°,a, first coefficients, we get 0

c

22

/3,.

are determined by direct substitution into (26.3). For the

. .

2

c0l k0 + c0~/2, (27.16) 2k~, /3 = —1I24c 2k~. a = —1/12c0I 0l Considering that f3 ~0, the phase velocity of the wave (26.12) proves to be dependent on the =

amplitude, and in view of ~3<0 this relationship leads to self-contraction (see section 20). Then, as prescribed by section 20, we deduce the equations of transverse modulations which fully coincide with (27.iO),

(27.14).

Finally, we should like to point out that a different result was obtained in ref. [11] for a slightly non-linear cnoidal wave. It can be shown that in our notation it would correspond to the relationship F”~_j~2 rather than to the correct formula F” = —lconst.~(eq. (27.14)). Two-dimensional solitons of the Kadomtsev—Petviashvili equation [72] In the case of a medium unstable for solitons and cnoidal KdV waves with positive dispersion, the KP equation has a particular solution in the form of a two-dimensional or, as it is sometimes described, “rational” soliton. It was first found numerically in ref. [55] and then in analytical form in ref. [56]. It can be also obtained as a limiting case from soliton chains [66]. As applied to the KP equation in the 28.

form of (26.3), this wave appears as u = —12c 2 ~ ln[1 + ((x + ct)/4,)2 0l Ox LI~= 3c~1~I2c~ ,

+

(yILI 2], 2)

(28.1)

= 61/2421,

where c is the soliton velocity, and ~i2 stand for the scales along or across the propagation direction. As previously established, the rational soliton is stable for two-dimensional perturbations, but is destroyed by three-dimensional bends within the isotropic KP model [62].

In the non-linear stage, the bends of a two-dimensional soliton in the third dimension can be fairly simply described by the same method, as the one used for the one-dimensional KdV soliton (26.2). The Lagrangian KP form being presented above (see (26.6)), it remains for us only to evaluate the integral L=JJ~dxdy

(28.2)

for

the Lagrangian ~~93) with a properly chosen test potential function, ç~= u. The latter will be represented by the wave potential (28.1), though with a changed phase x + ct—+x + x°(z,t), and parameters 41.2 = LI, 2(z, t) represented by the slowly varying functions of z and t.

The result, as can be easily shown, will be L = 1[x~0,

+

c0(x50, )2/2]

where we write (q,~ u)

+

c0F(I)

,

(28.3)

BA. Trubnikov and S.K. Zhdanov, Unstable quasi-gaseous media

205

I=JJu2dxdy, -=

(28.4)

+=

F(I) = J J [u~/3c0+ l2u~2+ ~2/2] dx dy

=

—313/21T2(12c

4. 01)

It might be useful to note that the Lagrangian L for the I and x

0 modulations of the two-dimensional soliton is analogous in form to the one given in section 27 for the cnoidal wave (27.8) and, consequently, the equations of now modulations are the same as (27.10). It will be 2 1,one-dimensional with c convenient to define p = (c/ccv)~ 00 standing for the velocity of the non-modulated soliton, and introduce r’ = c0x~’,which will finally result in —

—

p~+(pr’)~0, (28.5)

+

r’v~= ~

eec =

2c0c00,

a “one-dimensional gas” with negative compressibility, which is another example of an unstable QCh medium with a new “azimuthal” number m =

The small value of the “azimuthal” number points to the possibility (see part I) of finding simple analytical solutions for the non-linear soliton bends, which will not contain elliptic functions. Thus, the spontaneous increase of periodic modulations is described by the following formulae (see fig. 16) p

=

(sinh ~)/(cosh ~ + cos ~),

r’ = C~~(51fl ~)/(cosh

~ + cos (28.6)

~ —ytp,

77

y(z—r’t)/c~~,

where —cc < t <0, and the parameter y >0 signifies the growth rate of the initial-stage instability. In this case, ~ = kzCef~ which coincides with the linear growth rate found in ref. [62]apart from the notation.

c’

1.5

0 2 with a periodic (with a period A) unstable bend of a two-dimensional KP soliton in an isotropic medium. The is equal pto= 1—4: yt = —~, —3, —2.5, —2. Fig. 16. Evolution of theparameter “effectiveytdensity” (c/c00)”

206

BA. Trubnikov and S.K. Zhdanov, Unstable quasi-gaseous media

The modulational minima and maxima, as can be seen from (28.6), follow the explicit but inverse relations yltl = p~

ln[(1

+

Pmin)’(’

—

Pmin)]’

(28.7)

-1

yltl = Pmax lfl[(Pmax

+

1)’~(Pmax—

1)]

and at the critical time tcr = —2/v we have Pmjn 0, Pmax ~ cc, which corresponds to the soliton tearing into separate “bunches” (in the shape of “necked sausages”) collapsing into points as t—~—0. It should be recalled that in this case the “effective” density determines the local velocity and amplitude of the soliton, c = c 2. 00padopted here for deriving the equations of long-wave modulations can be successfully The method —~

used for the analysis of even more complicated situations, when the perturbation evolution is not reducible to the isotropic KP model. By way of example with anisotropic dispersion we shall consider the stability of the rational soliton (28.1) the so-called oblique magnetosonic wave [55, 731. In the high-frequency region, this wave satisfies the following equation in the notation of ref. [73] + [qq~2

—

—

—s-- (r~ —1) ~ Wpe

CA

—

—~-~-

W,,~

(2a~~+

~~)]]

= —4 1~i,

(28.8)

‘1 ‘1

2 (aw/w.)2, z is the coordinate running along the magnetic field and ij is the where a is the slope, if one running in the direction of the wave propagation. The remaining notations are explained in ref. [73] and the parameters q, e and a are taken to be constant. After obvious changes in notation (e.g., 77—~X cArt/2, CA_~*c 0 etc.) we reduce (28.8) to a form approaching our KP representation of (26.3) 2i c [u + uu~ c01 0l~(2au’~~~ +~ = —c0(u’~~ + u’~,,)/2, (28.9) 2I2w~ 2 — i)c2/2w~~, assuming that 12>0 so that (28.1) will appear as a nonwhere 1~= c 1, 12 =(u singular solution of this equation. It can be shown that the Lagrange function ~AN for eq. (28.9) is —

—

—

~‘AN

=~ +

~

(28.10)

~se=

12 c0

~l2

i(~xz

2 ~

where ~Z’is the isotropic part (26.6), and ~ is the flow potential determined from the condition u = q~. Before selecting the test function, note that the solution of eq. (28.9) can be represented not only by the soliton (28.1), but also by a soliton with a constant inclination K. Its description requires merely the following replacements in (28.1): 2/2, X~ X + KX, C~ C + c0K (28.11) 12_* 12[1 + K(K + 2a)l~/l~] It is important that now the length I changes with the given inclination, in contrast to the isotropic case

BA. Trubnikov and S.K. Zhdanov, Unstable quasi-gaseous media

207

with l~= 0. This means that the length l becomes a new independent parameter in the anisotropic case with a bend. Therefore, we shall substitute the potential ~‘ of the initial soliton (28.1) into (28.10) as a

test function, adopting the following form =

—12c0l~.~ ln[1 + ([x + x0(t, z)]/41)2 + (y/42)2],

u = ~,

4~=

3l~c

(28.12)

2. 0Ic,

LI~= 3I~c~I2c

Subsequent integration of this result over the variables x and y with retention of no derivatives other than those of the “fast” phase x 0(t, z), yields

LAN = JJ ~

dx dy = I (x~,+ c0x~/2) + c0F + c0Gx~(4+

2a)

,

(28.i3)

with the notation 3(2

I = I~j-~q, F = I*p. 1/2 2 1~= 3i~2 (4c

0l)

,

—

3q) /3,

G = 21*p.3(l,/I)2/3, (28.14)

2

~ = (c/c0)

22 ,

q=

L/i

It can be easily shown that the Euler—Lagrange equations determined by the Lagrangian LAN (28.13) will appear as + VV~=

(pq)

c~pp~,

3]~ = 0,

+

(28.15)

[pqv + c~~pq(u)/ 2

22

q = q(v) = 1 + l,v(u + 2ac 0)/c01 2, u = c where p = (c/coo)U 0x,~, as written above. These are by far more complicated than the

“isotropic” equations (28.5) and are obviously not “quasi-Chaplygin”. Note, that the spectrum of small oscillations, v 0, p i 0, for the system (28.15) fully coincides with the spectrum found in ref. [73] by perturbation theory. instance, the characteristic directions, dzldt, in the region of initial 2 4For 1, are defined by the relation inclinations, me/mi 4 a (dz/dt u,)2 = c~(1+ K)2 [i0p2c~~/3a2c~ + K)2] —

—

— (1

(28.16)

K =

v/ac

2c~fIV(i+ K),

0,

V1 = V + 2Kp

which implies that the soliton is stable when 2 < 10c~~/3c~, given small bends [73],but also suggests the possibility of bifurcation, with the system (28.15) changing from the “hyperbolic” to the “elliptic” type, given large bends.

B. A. Trubnikov and S. K. Zhdanov, Unstable quasi-gaseous media

208

29. Perturbations of NSE solitons [12, 53, 72] The non-linear Schrödinger equation and its one-dimensional soliton have been already mentioned above (see chapter B, sections 18 and 19) in connection with numerous applications to the problems of self-focusing and self-contraction of wavepackets. There, reference has been also made to the stability of the NSF soliton in a one-dimensional case, as well as to the possibility of its breakage and ensuing collapse due to non-one-dimensional perturbations, discovered by Zakharov [37]. It will be demonstrated here that the soliton collapse proves to be “quasi-Chaplygin” in the simplified long-wave limit. The result looks particularly simple if the NSE is taken in the following dimensionless form 2tfr+~i~2=0,

i~+

(29.1)

~V

where V2 is the three-dimensional Laplacian. In this case the NSF soliton is described by the formula =

A

0(x) exp(i~), (29.2)

2t/2. A0(x)=acosh~ax,

tpa

Here, a is the wave amplitude which unambiguously determines the scale and phase of the solution. It can be easily shown that eq. (29.1) is matched by the following Lagrangian, with (*) symbolizing complex conjugation 5~?=~[i(i~~’

—

~p*~y~+ v~v~

—

k~4].

(29.3)

Now, following the adopted procedure, we have only to choose a suitable test function. The transverse modulation of the soliton affects primarily its phase, while its amplitude and size are only slightly varied. Therefore, let us take the following expression as a test function ~1i=

A 0(x, a(t, i~j)exp(i~(t,i~)),

~

~

~

(29.4)

and substitute it into the integral

L= J~dx=[~+(V±w)2/2}I+F(I),

(29.5)

with the notation 1= JA0(x)dx,

F= ~ f[A~_A~]dx.

(29.6)

The averaged Lagrangian L (29.5) is the same in “appearance” as L in (27.8) for the cnoidal wave, which is why the modulational equations for the NSE soliton will coincide in the form with the 3/3, system the (27.10). Inasmuch as the calculation made for the NSF soliton yields the values I = 2a and F = a final form of the equations [12, 53]

209

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

a~+ div

av~ = 0,

t3~ (29.7)

+ (ti~V)U~ = a Va

proves to be similar to eqs. (28.5) describing the self-focusing of the two-dimensional KP soliton. Thus, the collapse of the NSE soliton offers another, now two-dimensional, example of a QCh medium with “azimuthal” number m = ~. Therefore, for instance, the dynamics of a two-dimensional collapse for the periodically modulated NSE soliton is described by formulae (28.6—7) where it will be sufficient to take Cef = 1. Finally, a three-dimensional collapse is illustrated by the cylindrically symmetric self-similar solution

of eqs. (29.7); 2

p(r, t) r’ =

=

2 1/2

[p0(t) —2v

r/3r ,

p0 =

(29.8)

213

1r0/r1

where r=t—t,

r’O~/Or,

1<0.

(29.9)

In the generally established terms, the NSE (29.1) is called cubical due to the degree ofnon-linearity in i/i, this being a case of the most frequent natural occurrence. Of certain interest are also other NSE modifications; thus, the problem of the effect of dispersion corrections on perturbations of QCh media (see appendix B) should be handled with the use of the NSE of order 2N + 1: i~+ ~

(29.10)

the particular solution of which for N >0 is represented by the soliton =

AN(x)

exp(i~N),

2/Ncosh ax]1”,

WN =

a2t/2N2,

(29.11)

AN =[a((N+ 1)/2)”

whose stability to transverse modulations is our present interest. The main points of the derivation being comprehensively covered above, we shall briefly present only the principal results. The Lagrange function for modification (29.10) appears as ~ [i(~,

-

~

+

V~V

9~ N+ 1 -

On integration with the test function

i/i

=

~j2~1)].

AN(x, a(t, F1)) exp(i~(t,F1)) including the “fast”

and “smooth” a(t, F1), we get, as in (29.5) 2/2]+ F(I), L = J ~ dx~I [~ + (V1~)

(29.12) ~(t,

F1)

(29.13)

B. A. Trubnikov and S. K. Zhdanov, Unstable quasi-gaseous media

210

where, now 1=J A~dx,

F= ~ f[A~

—

2A~~/(N

+

1)] dx.

(29.14)

Evaluation of the integrals I and F is quite simple and results in F(I) = —I~(I/I~)(2+N~(2_~(2 — N)/(2 + N) (N +

1)

(29.15)

I~=(ii(N+ 1)/2)1/2 F(1IN)INF(1IN+

which corresponds, for

tJ

1 =V1q

~)

and p = 1II~,to the modulational equations of the following, also

quasi-Chaplygin form +

div prY1

= 0,

(29.16) 2’~2~/(N+1)). (rY1V)111 =V1(p It is worth mentioning that when N> 2 the soliton turns out to be stable to transverse perturbations, +

but not so, as shown by more thorough analysis, for one-dimensional perturbations which cause it to collapse.

30. Solitons described

by

the sine-Gordon (SG) equation

The SG equation can be found in the Josephson effect theory, in the problem of a laser beam in a two-phase medium, in some models of the theory of elementary particles, etc. It is traditionally written

as —~,=sinc~

(30.1)

where x and t are the dimensionless coordinate and time. Let us examine its three-dimensional modification V2~ —

p’,.

usin~,

(30.2)

where V2 is the three-dimensional Laplacian, and the factor a- = ±1 (in the conventional alternative

(30.1)) is added for generality. Consider the transverse stability of a one-dimensional soliton described by the SG equation in the particular case of the “2ir-kink” [20].This is a particular solution of eq. (30.2), appearing as = 4 aretan(exp(~)), (30.3)

~=(x+ct)Iii,

c2=1—u42.

In writing down (30.3) we have chosen a wave of a definite direction.

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

211

The SG equation allows a variational formulation d3r dt 0, = p2 (V~)2]/2 2a- sin2(~’/2),

~J ~

—

(30.4)

—

which will be helpful in describing the long-wave modulations of the kink soliton. We shall adopt

4 arctan(exp( i)),

= =

(30.5)

(x + x

0(t, F1))/zI(t, F1),

as a “test” function to generalize (30.3), the modulation being considered smooth, 41~~j ~ Evaluate the integral of £~?(30.4) with ~ = ~. Retaining the derivatives of the fast phase x0 alone, we get L= J~dx~[x~_(V1xo)2_u42_1]I,

where allowance is made for the relationship appearing as =

(2/LI) sin(ç1/2)

=

~

1= ~ J~dx, = (2 /4)

(30.6)

sin(~1/2).The form for the kink soliton

2/LI cosh ~,

(30.7)

we get the integral I = 4/LI in (30.6) and, finally, the averaged Lagrangian becomes 2 1 a-LI2]. —

L = (4/LI) [x~

—

(30.8)

—

(V1x0)

Setting up the associated Euler—Lagrange equations, we obtain the system (LI~x~,) = div(LI’V

1x0), 2+1—a-LI2,

(30.9)

x~’(V1x0) which describes the transverse modulations of the kink soliton. Writing p = x

00ILI and ü =

—

(V1x0)/x~,,

the system (30.9) is reduced to

p~+divptY0, (rYy1)

+

V1y1 = 0, 2

y~(1—v +crp

—2 —1/2 )

which bears a remote resemblance to the equations of relativistic gas dynamics.

(30.10)

212

BA. Trubnikov and S.K. Zhdanov, Unstable quasi-gaseous media

The “quasi-Chaplygin” case can be obtained on the assumption that v ~ 1, p simplified equations p+divprY=0,

1. Here, we find the

tJ~—V1x0, (30.11)

2/2,

+

~‘

(iJV)rY~a-V1p

which fit the dynamics of the two-dimensional “Chaplygin gas” with “azimuthal” number m = unstable when a- = —1. So, the simplest solution, according to (12.6), for the periodic modulation along a single axis, e.g., —

the y-axis, will be p = (cosh T

—

cos xi’

v=sinh’rsinX,

sinh~r~ r=yt<0,

x=~y.

(30.12)

It should be emphasized that the result obtained in this section being approximate, it should be treated merely as an illustration of the “quasi-Chaplygin” approach.

31. Conclusion Touching on the background of the present work and the contribution of its authors, we shall summarize this review in the following ten points: 1. D.J. Korteweg and G. de Vries obtained their equation for “non-overturned” shallow water in 1895. About the same time, a particular example of a hypothetical gas with the adiabatic exponent = —1 was analysed by S.A. Chaplygin who delivered a report “On Gas Jets” in 1896 and published it

in 1902. Time has proved both works to be rich in content and applications.

2. Several years ago, the authors of the present review, who had long been interested in the theory of current-carrying pinches, took notice of a publication by Book, Ott and Lampe [6], where constrictions on an incompressible skin pinch were studied numerically. The numerical method [6] failed to provide an answer to the question of whether constrictions could break completely. We, however, have succeeded in finding an analytical solution of this problem and getting a positive answer. 3. In doing so, we have found that, as noted in ref. [6], the problem of constrictions is described by the same equations as is the ideal gas with adiabatic exponent y—* 0, or to be more exact, with a pressure of the form p — ln(1/p). Then, we have studied analytically the problem of constrictions on a compressible pinch (which reduces to the previous one), followed by the problem of “overturned shallow water” (previously studied numerically in ref. [3] and analytically, though erroneously, in ref. [4]), and the problems of a cylinder of liquid with surface tension, of evolution of a one-dimensional Chaplygin gas (for Chaplygin himself studied a steady-state two-dimensional flow), of a “cold surface flow in the ocean” (equivalent to the “overturned shallow water”), and of a tangential discontinuity of the velocity. We have also discovered that perturbations of various solitons described, e.g., by the KdV, NSE, KP and SG equations in the long-wave approximation, which is often employed in a linear analysis of soliton stability, are also defined by similar non-linear “gas” equations, which result in a clear pattern of soliton behaviour. Besides, as it turned out, our simplest “evolutionary” solutions are

absent from almost all previous studies made by other authors and containing similar equations. 4. The authors of this review have noticed that all these media are equally described in the long-wave approximation by gas dynamics equations with two parameters

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

1i=—pdivrY,

213

(31.1)

ii.=c~mVpvm,

where p is the “reduced density”, and the dot signifies the total derivative with respect to time, e.g. p = op/~9t+ (tYV)p, etc. Specifically, when m = we have a “Chaplygin gas” and hence refer to system (31.1) as “quasi-Chaplygin” - There will be no instability unless c~>0, which can be called “the generalized Chaplygin—Lighthill criterion” (Lighthill studied a case with m = 1). —

5. Given symmetric conditions, eqs. (31.1) have, in particular, simple self-similar solutions for any number of measurements, N = 1, 2, 3 Of greatest interest, however, are either a one-dimensional ease with time-dependent evolution, or a steady-state two-dimensional flow which, with the supersonic velocity v0 ~ c0, reduces to the previous case allowing an exact analytical solution under any initial conditions (Cauchy problem). 6. For ordinary gases with c~<0, the solution can be found, for instance, by the well-known Riemann wave method, assuming that p = p(v), but bearing in mind that this method is inapplicable when c~>0, since the equations become elliptic. However, the “hodograph” method known in mathematics, as well as in gas dynamics, permits reducing any pair of non-linear equations ~a~1Of/Ox1=0,

i,j1,2

(31.2)

with coefficients of the form a.1( f1, f2) to linear equations for the inverse functions x, ,2( f,, f2). 7. We have discovered that in the case of c~>0, given any m, the hodographic method with a number of subsequent substitutions (see above) does yield linear equations, but reduces system 2!P(r, ~, not z) =only 0. This necessitates introducing cylindrical (31.1) to the Laplace V a spurious “angle” ~, which is devoid of any physical coordinates r = simple (p/p 1’2m, z —V uequation as well as 0) meaning, but elucidates the solutions through expansion in “electrostatic” multipoles composed of “charges” in the space r, ~, z.

8. Seeking to demonstrate the distinctions of the medium behaviour, many authors who dealt with equations similar to (31.1) studied usually one or two examples of solutions, made as simple as possible, using either numerical or analytical methods. As a rule, though, the perturbations described by those solutions were caused by an “initial impact” introduced by the researcher. In the numerical approach, in particular, this is the only possible formulation, which is responsible for substantial arbitrariness in

the solutions. The main shortcoming of such “random” solutions, however, consists in the fact that even if at the moment t = t

0, taken as the initial time, the amplitude of the perturbation introduced is chosen to be adequately small, but the profile is not properly selected, one can easily see, on finding the

analytical solution or making a count down in the numerical solution, that the amplitude was actually great at a certain instant preceding the initial time, tantamount to a heavy “impact on the medium”. 9. Meanwhile, unstable media, unlike stable ones, can have perturbations such that they vanish in the asymptotic limit —cc, leaving an unperturbed background, p(_cc) = p0 and V(—cc) = 0. This means that the “electrostatic” potential ~i(r, ~, z) should be engendered by no other “charges” than those located on the unit circle r = 1, z = 0 in the r, ~, z space. We call this requirement the ~

“evolutionary principle” of selecting spontaneous solutiq~ns,the simplest ones among them being the Coulomb and dipole potentials which give four “fundamental” solutions for the time t—

—r~Q~(a),

t—

rv Q~(a)cos(~+ ~‘) <0,

(31.3)

where a = coth ~, and ~, ~ are toroidal coordinates. These solutions show all the typical features in the behaviour of an unstable medium, which permits restricting the study to these four solutions and

214

BA. Trubnikov and S.K. Zhdanov, Unstable quasi-gaseous media

“unifies” the theory of all “quasi-Chaplygin” media, which were partly investigated before us. However, there are no publications known to us, where the authors would bring to view precisely these analytical solutions which are the simplest ones in the “spontaneously evolutionary” class, all the solutions available in the literature are more complicated. The only exception is the self-similar solutions though not spontaneous, nor vanishing in the limit —cc which necessitate an “initial impact” of a quite definite profile. —

~

—

In view of this gap, we have supplemented the theory, including all the previously known “quasi-Chaplygin” media. 10. The medium type can be conveniently characterized by the “azimuthal number” m, which most naturally fits in with the Laplace equation with solutions of the form IP(r, ~, z) = qi(r, z) cos(m~).Our studies have shown the media found “in nature” to have only six “quantized” values of this number, m = —2, —1, —1/2, 1/2, 1, 3/2, whose growth entails an increase in the complexity of the mathematical description of the medium due to the complication of the Legendre functions, Q,,,,2. Therefore, the “Chaplygin gas” proves to be the simplest and the cylinder with surface tension the most complicated examples of such media. However, all these media are basically similar in their behaviour which, besides, is fairly uniform. An example most easily visualized is the behaviour of a layer of “water on the ceiling”, devoid of surface tension. A phenomenon of general prevalence is the tearing of a medium into separate bunches “drops” or —

“streams”. It would be reasonable to hope that our “unification” of both the solutions and the patterns of spontaneous behaviour of “quasi-Chaplygin” media will turn out also useful in other more complicated cases, such as the formation of galaxies in the expanding Universe (?). The authors would like to express their appreciation to DN. Zubarev, Yu.A. Danilov, B.B. Kadomtsev and V.D. Shafranov for their valuable comments on the present work, and to L.M. Melnikova for careful translation into english.

Appendix A. Formulae and tables of functions In our parametric solutions for the time t( ~, 7)) and the coordinate x( i~) there are three special functions: Q,~,Q’0 and Am~containing the subscript v = —m ~which hereafter is assumed to be either integer or half-integer. Below we shall show the way various functions are calculated. The Legendre functions, Q~,are to be sought using the well-known recurrence relations ~,

—

1

2

Q~(a)~(a —1)

—1/2

(aQ,,—Q,,~),

Q,,(a) = a (2— 1/v) Q~~i —(1—

1/u)

Q

(Al)

2

—

i)h12

However, it would be of more to ause= (k new arguments, k = tanh(~/2) a —rewritten (a k’ = (1 k2)1’2 instead a =convenient coth ~. Since + k~)/2, formulae (A.1) will= be as

and

—

Q~,= v[(1 + k2) Q,, 2kQ,,_ 2) 1]/(1 k Q~= (k + 1/k) (1— 1/2v) ~ —(1— i/i.’) Q,~

(A2)

—

—

2-

Reducing the order of the function, we arrive at the elementary expressions

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

215

Q0=ln[(1+k)/(1—k)],

Q1

2)/2k] ln[(1

+

k)/(1

—

(A.3)

k)] —1

k and at the complete elliptic integrals

= [(1 +

for integer

is,

Q_

1’2 K(k),

1/2=2k =

(A.4)

2k1’2 [K(k)

—

E(k)]

for half-integer is. To make tabulation convenient, we shall take k = sin a°(a°~ a) and measure ° in degrees. The values of the 4 functions of (A.3) and (A.4) are given in table 4 and depicted in fig. 17. Consider, finally, the lambda-function, Am (7.9). In view of the recurrence relations (7.7) being satisfied for the integrals R~(7.6) and considering the relations (A.1) for Q,,, it can be easily shown that Am satisfies the simple recurrence equation relating but two functions Table 4 Values of Q,, and related quantities as functions of the angle a’: k=sina°;sinhe=2k/(1—k2);cosh~(1+k2)/(1—k2);coth~a(1+k2)/2k a’

10°

200

300

400

50°

~O

7Q0

0.34 0.12 0.77 1.26 1.63 8.54

0.50 0.25 1.33 1.66 1.25 4.00

0.64 0.41 2.19 2.41 1.10 2.42

0.77 0.59 3.71 3.84 1.04 1.70

0.87 0.75 6.93 7.00 1.01 1.33

0.94 0.88 16.1

~

0.17 0.03 0.36 1.06 2.97 33.2

0 0

0.35 0.04

0.71

1.10

1.53

0.16

0.37

0.68

2.02 1.09

0

1.32

1.89

2.38

2.86

0

0.11

0.33

0.62

0.98

00

k k2 sinh~ coshe a

k2

0 0 0 1

=

~Øo

900

1 1

1.00 1.13

0.98 0.97 65.3 65.3 1.00 1.03

2.63 1.66

3.47

4.87

~

2.48

3.87

=

3.39

4.01

4.86

6.26

~

1.44

2.03

2.86

4.26

~

16.1

= ~ 1 1

Q_,, 5

00 Fig. 17.

30°60°90°

Legendre functions Q0(a) f(a°);a°= arcsin k.

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous med,a

216

Am = Am_i

z r~’Q~_1/i~,

—

is

—m

=

(A.5)

.

—

Successive use of this formula results in the following expression containing a sum Am = AmN

—

(z/~) ~

rV+fl

~

(A.6)

Given integer m, we shall arrive at 1’2. (K E)R1/2)Ilrk If m is half-integer, however, we shall obtain the function A_ 1/2, which with regard to + irP_,. cot ~ri-’ will appear as

A0 = (Q~1/2R~,/2 — Q,/2R1/2)/27r = (kKR112

= (1/n-) lirn

= 2k~ J

~)

—

k’2

sin2

~r

Q~_,= (A.8)

—

(1

(A.7)

—

(vQ,_,R~~, isQ~R0)= R1/ir.

It will be convenient to introduce k’ = (1 ~

—

2)112 for the integrals R,, (7.6) and represent them as —

k

d~.

(A.9)

Hence we find the incomplete elliptic integrals —1/2

R_ 1/2 = 2k

E(k ,77/2), (A.10)

R1/2=2k

1/2

F(k,7712),

and the elementary function =

R,hi- = (2hr) arctan[k tan(77/2)].

(All)

The A0 function proves to be equal to the so-called “Heuman” lambda function A0(2/ir){K(k)E(k’,77/2)—[K(k)—E(k)]F(k’,ij/2)}

,

(A.12)

whose values are given, among other things, by Abramowitz and Stegun [15]. In the limiting cases, it is equal to sin(77/2) when k = 0 or to 77/IT when k = 1, and if the required accuracy is not too high a quite

simple approximation of the form 2(77/IT ~

=

sin(77/2)

+

k

—

sin( 77/2))

(A.13)

can be used for this Heuman lambda-function. Whenever a more precise calculation is required, use can be made of table 5 which gives the correction (A.13).

K

(accuracy percentage) for the approximated formula

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

217

Table 5 Correction K in the formula for the Heuman lambda-function: k

0.87

0.94

0.98

30°

400

500

60°

70°

80°

1.31 1.23 0.97 0.54

2.74 2.58

4.33

5.60

4.09

2.06

3.29

2.24 2.13 1.79

2.04

5.94 5.63 4.62 3.14

4.81 4.56 3.78

1.21

5.30 4.31 2.81

2.66

1.30

—0.05

—0.08

0.11

0.56

1.13

1.53

1.45

—0.28 —0.31

—0.68 —0.58

—0.78 —0.62

—0.56 —0.49

—0.13 —0.27

0.27 —0.06

0.45 0.07

076 0.29 0.07

0.50

10°

20°

90°

0.34 0.32 0.25 0.13

120°

150° 170°

600

1, k = sin a°

(1 + K/100)A~,°”

076

0.34

10° 30°

=

0.64

0.17

The formulae presented here are applied in part II for calculation of specific “quasi-Chaplygin” media, each type being characterized by a definite value of the “azimuthal” number rn.

Appendix B. “Quasi-Chaplygin” dispersion corrections and their relation to the non-linear Schrödinger equation The “quasi-Chaplygin” equations give a merely approximate description of the relevant media. If

small corrections are retained, they are usually included as the third derivative,

i.e. (B.1)

p+(pV)~=O, V~+ VV~=

p’~,

c~[p”p~+

together with a characteristic length 1, which is also small. It might be useful to correlate these equations with the non-linear Schrödinger equation (NSF) iji

—Ia /4~+J3I~/I”qi,

(B.2)

in which the potential energy itself depends on the psi-function. Its particular solution is a uniform background of the form ~ exp(iIlt) where Li = N = const. Here, it is convenient to introduce a new parameter the length I instead of al, taking IaI = NQ1l2, so that after elimination of al and /3 eq. (B.2) takes the form — /3

—

i~/~ =

—

—INQul2l/1’~~ f1I1I,/1froI’~cl’.

(B.3)

—

Its general solution will be sought in the form = ‘1’~p”2 exp(iq~)where p(t, x) and ç(t, x) will be real. If we write v = 21 NI) I l2co~,the following “near-Chaplygin” equations can be manifestly obtained for p and v ‘P

p~+(pv)~”O, ,

,

2

v,+Vv~c

—v

(B.4) 2

0(p p~+lX),

218

BA. T,-ubnikov and S.K. Zhdanov, Unstable quasi-gaseous media

where the condition of instability, Nfl >0, is assumed to be fulfilled and

is

= 1

—

N/2,

c0 =

Nfl!!, while

X becomes X(t, x) =

[P-h/2

~-

±

~

(~~I/2

a)].

(B.5)

If p = 1 + p1, where 1p11 41, then X=~p’~ and (B.4) coincides with eq. (B.l). Thus, the dispersion corrections for the “quasi-Chaplygin” equations could be allowed for in the solution, even though in approximation, if the “NS equation” (B.2) were solved. Unfortunately, it can be solved in the general form only for the case of N = 2, which was revealed in 1971 by Zakharov and Shabat in ref. [61]. However, when N is arbitrary, only “motionless” solutions with v = 0 can be shown (see below). The case of N = 2 leads to m = 1, which corresponds to the “water layer on the ceiling” with extra allowance made for the forces of surface tension a- which sets up an additional pressure in the layer 2)312 —a-H~(B.6) a- K(t, x) = —a- H~(1+ H~ Here, H(t, x) is the layer depth and K(t, x) is the curvature of its surface, so the total pressure in the “overturned” layer is given by =

p=

const.

p~gjH —

—

a-H~,

(B.7)

where Pm is the water mass density.

Thus, the layer is described by the equations (B.8)

+

vv~=

+

a-H~x/pm

which are reducible to (B.l), with p=H(t,x)/H

1’2 , 0,

l=(a-/gp~)1’2.

Comparing eq. (B.8) with the “NSF system” (B.4—5), given small ~ fl(p~H

(B.9)

c0—~gH0~

2/4a-)”2,

a((a-Ho/pm)1’2,

we find the parameters

/3=—fl/~<0

0g

of the corresponding NS equation (B.2) with N = 2, whose solution, however, is rather complicated and therefore will not be presented here. We shall restrict ourselves to the comparison of “motionless” solutions with v = 0 for eqs. (B.8) and (B.4) in order to bring out their similarity and distinctions. In the case of water (B.8), given v = 0 we have +

i2,i~ç~. = 0, (B.l0)

p=a+bsin~+ccosç,

ço=x/l,

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

219

i.e., a modulated “rippled” layer which is apparently unstable. When a < b, c, it breaks into separate drops. On the other hand, the “NSE system” (B .4) simulating water gives the following form for “motionless” solutions with V = 0 p + 12 X(x) =

p + 12 d

[~_1/2

d

(~_1/2

~

=

0,

(B.11)

where p >0, and should we now replace x by a new variable, y, assuming that dy = p1’2 dx the result

will be

p;+12

~

=0, (B.12)

pa+bsin’P+ccosçli,

‘P=y/I,

which coincides with (B .10). The only difference consists in the definition of the angles ~ = x/1 and = yll related as follows: ‘P

~ =J p”2 d’P = J(a

+

b sin

‘P

+

c cos

‘PY1’2

d’P.

(B.13)

In consequence, however, the function p(x) proves to be elliptic and is reduced to the so-called cnoidal

solutions of the NSE: p = a2 dn2(k, u),

dn2

= (1

—

k2 sin2

‘P)1’2,

(B.14) u = ax/21 = J(1

—

k2 sin2 a)~’2da,

which are unstable, like the drops on the ceiling. But in the limit k—*1, (B.14) yields the soliton (B.15)

p1’2 = a/cosh(axI2l), which is known to be stable to one-dimensional perturbations.

However, its similarity to the “water ridge on the ceiling” permits predicting by analogy that it will be unstable for two-dimensional perturbations and will break into separate oblong “drops”. This is exactly what happens in the theory of strong Langmuir turbulence in a plasma when a “layer” of waves excited therein is caused by the modulational instability to break into separate bunches growing to reach the stage of solitons which subsequently split into long “Langmuir cavities”, the latter undergoing a collapse at the final stage (see ref. [37]),induced by the aperiodic parametric instability discussed in section 25. In conclusion, it might be useful to note that when is = 1 N/2 is arbitrary, the motionless solutions of the general NSF system can be found from the equation —

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

220

p~p~+l2Xzr0, =

f(cip

+

1(!/2)(N+N2/2)1’2

c 2 2p

(B.16) —

p3_P)_i/2 dp.

With N = 2 or N = 4, the last integral is reduced to elliptic integrals, but complete integrability of the NSE is known to be possible only with N = 2. This is exactly the case for which refs. [74,75] give a number of exact solutions exemplifying the spontaneous evolution of a uniform background instability. In the notation of ref. [74], the NS equation can be written in the form iu+ u’~~I2+u(IuI2—1)=0,

(B.l7)

whose simplest periodic solution, in particular, is the function I

(K2/2)cosh~t+i~sinh8t1

u(x,t)=Ll_K

Keosh~t—~cosKx ]exp(_iz~), (B.18) where z~ç’= arccos(1 ,c2/2), ~ = K(l ,(2/4)1’2, and K is a parameter specifying the perturbation period. In the degenerate case of K —*0, solution (B. 18) passes into the “rational NSE soliton” of the form (see fig. 18) —

u

—

1—4(1 +2it)/(1 +4t2 +4x2).

(B.l9)

The perturbation evolution described by solutions (B.18) and (B.l9) is basically similar to the periodic and localized modes for quasi-Chaplygin media: the perturbations, infinitely small at the beginning (t—* —cc), break the uniform layer into separate bunches by the time t—* —0, these bunches resembling the “ridges” found in a layer of shallow water on the ceiling. Somewhat unexpected, however, is the fact that the exact solutions ((B.l8) and (B.19)) predict subsequent reversion to the “uniform background” when t—~+cc. This paradox becomes clear if we turn to the “hydrodynamic” NSE version (B.4). Thus, the “flow velocity” v for the localized mode

(B.19) is

Fig. 18. Exact solution of the NSE (B. 19) describing the instability of a background uJ quasi-Chaplygin approximation (broken line).

= 1,

uniform in the limit as 1—’

—~,

compared to the

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media v = c9

arg u/ox = xtl[t2 + (x2

+ t2

—

V=

2ITxl5(x2

—

~)

(B.20)

~)2/4]

which takes the following form in the limit

221

I t~—*0 (B.21)

sign t.

This means that an impulsive force arising at the moment

t=

0, causes instantaneous velocity reversal

and, consequently, restoration of the uniform condition. Appendix C. Energy principle in the theory of stability of solitons described by the generalized non-linear Schrödinger equation When discussing the soliton stability in various wave models in chapter D, we have dealt with merely transverse quasi-Chaplygin perturbations, leaving aside those modulations which do not break the soliton dimensionality. The picture would be incomplete, however, without mentioning the role of the latter. To those interested in this question we recommend refs. [12,76, 77], where the problem of soliton stability is covered at greater length. Small deformations of solitons are generally studied with the help of perturbation theory which traditionally requires solving complex high-order equations whose coefficients are explicitly dependent on the independent variable. If “rough” information on the initial wave instability is sufficient, it will be convenient to use a different method which will be called the “energy principle” by analogy with the well-known applications [78,79]. The essence of this method will be elucidated here by a particular example of the NSF with a power non-linearity (see section 29 and appendix B) i’P~+

~V2’P+‘PI’PI2~~~=0,

(C.1)

which is of interest, given arbitrary N, in the problem of the effect of dispersion corrections on the evolution of QChM perturbations. For instance, the case with N = 2 is realized in the problem of the transverse stability of the Langmuir soliton [12]which itself is a solution of (C. 1) with N = 1. The soliton for eq. (C.1) with arbitrary N>0 is described in section 29. Substitution of = A exp(iq~)with real A and ~ turns eq. (C. 1) into an equivalent “hydrodynamic” form: ‘P

p~+divprY=0,

(C.2) rY=Vfl,

Lianp~”-~-(~4)/2A

where p = A2 and tY=V~.In this case, the soliton of the original NSF (C.1) (see (29.11)): A 0 = (c/cosh ax)~”,

N>0,

(C.3)

2/(N + 1) = a2/2N2

11 = c

apparently complies with the static equilibrium

Li = const.,

V

=0 which is a particular form of this

BA. Trubnikov and S.K. Zhdanov, Unstable quasi-gaseous media

222

“hydrodynamics”. Therefore, the problem of its stability is equivalent to studying the conditions of static equilibrium stability (C.3) for the system (C.2). This enables us to resort to analogy and employ

the well-known methods of analysing the stability of hydrodynamic equilibria [78,79]. The system (C.2) can be shown to preserve the total energy H comprising the “kinetic”, K, and

“potential”, U, energy: K = fdv pv2/2,

H = K + U,

U=

J dV

[(VA)2/2

—

PN+l/(N + 1)].

(C.4)

Here, the original equilibrium (C3) will be stable, provided that the “kinetic” energy, subsequently varying in compliance with the expansion K=K 0

+

tK0 + tK0/2

(CS)

+~--

decreases upon introduction of any initial “test” velocity field, V0 = V(r, t = 0) [78].The expansion (C.5) apparently implies that K0 = 0 due to the equilibration of the initial state, so the stability requirement will be (C.6)

or, explicitly, S=

—

—f

=

dV

prY0

V(Li,),0

>0.

(C.7)

Eliminating the time derivative with the help of (C.2), the “energy principle” (C.6) can be written as S=

f

2

+

2[fl

—

(2N + 1)pN]~2}>0,

(C.8)

dV {(V~)

where =

—(A~) 0= (1 12A0) div p0rY0,

curl tY0 = 0

(C.9)

and p0(x) = A~is the profile of the initial equilibrium (C.3) with the parameter Li = const. Now, we shall confine ourselves to one-dimensional perturbations, assuming that V01 V(x), V10 = 0. It will be useful to rewrite the functional S directly in terms of the velocity V(x). Definition (C.9) gives ~ = A’(x) V(x) + A(x) v’(x)/2, which, on substitution into (C.8) and a number of obvious transformations, results in 2(x) + 4N (2— N) (1+ N)’ p~v’2(x)}/4, (C.10) S = dx p0[v” from which it follows that the value of power index N = 2 is critical for the stability of the NSE soliton (C.1). The case of N < 2. The functional S is non-negative and of greatest “danger” is the constant velocity field, v = const., for which S = 0. It corresponds to a mere displacement of the soliton as a whole, with

f

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

223

its shape undistorted and its constant “kinetic” energy imparted by the initial “impact”, which spells stability. The case of N = 2. The functional (C. 10) is also non-negative, but this time the field of velocities in the form V -~x is critical. Direct calculation proves that the “kinetic” energy acts as the integral ofmotion,

given such a field of “test” velocities. Nevertheless, the soliton is unstable! This is suggested by the fact that the system (C.2) has the following self-similar solution with N = 2 v(x, t)=x/(t+

T),

(C.11) p(x, t) = p0(x/(1 + t/r))I(1

+

tI’r)

which describes the self-similar gathering (T <0) or spreading (T >0) of the initial soliton, with the “kinetic” energy of the initial “impact” preserved. This example is yet another testimony to the fact that the rigorous inequality in (C.7) is essential to stability. The case of N> 2. Let us introduce a new “test” function, ‘P(x) = A(x) v ‘(x) instead of the velocity v(x) and write the functional S as

2(x)/2+ u(x) ‘P2(x)}/2,

sfdx {‘P’

(C.12)

u(x)= 11 —(N—1)(2N—1)Ar(x)/(N+

1)ns(1—(N—

1)(2N—

1)/cosh2 ax)a2/2N2.

In order to find the minimum functional S, it will be convenient to use the analogy of representing (C. 12) and the Hamiltonian in the variational formulation of the ordinary linear one-dimensional Schrödinger equation. Then it becomes obvious that the true minimum S is determined by the energy of the ground state E 0 in the potential u(x), while the wavefunction of this state complying with the equation +

[E0

—

u(x)] ‘P(x) = 0

(C.13)

describes the most “dangerous” perturbation. The Schrodinger equation (C.13) has a well-known solution [80]for the particular form of the potential u(x), which appears as ‘P(x)

—

(cosh ax)

E0=

—a

1/N—i ,

1

j dx

2 ‘P

= 1,

2(N—2)/2N.

(C.14)

The associated velocity field, limited with N> 2,

J dx(coshax)21~ yields S =

E

0/2 <0,

which makes the soliton unstable.

(C.15)

BA. Trubnikov and S. K. Zhdanov, Unstable quasi-gaseous media

224

Appendix D. On regulai-ization of the perturbation theory as applied to the soliton stability problem One of the most common methods of studying the stability of steady waves in non-linear plasma wave models is the perturbation theory (PT) employed in its simplified “long-wave” version. The first step in using it so was made by Kadomtsev and Petviashvili [60] who studied the two-dimensional stability of KdV solitons. An exact solution of the stability problem [64] is possible for the well-known “integrable” wave models [81], such as the Kadomtsev—Petviashvili (KP) equation. Not so for the “non-integrable” models found e.g., in the plasma wave theory, which is why the stability studies are generally restricted to the long-wave perturbation limit [54, 62, 73, 82, 83]. As applied to solitons, the existing version of the long-wave PT (“primitive” in terms of ref. [84],or PPT hereafter) has a certain distinction which is being discussed in the present section. It is common

knowledge that any perturbation theory seeks to eliminate secular terms. The soliton-oriented PPT is quite valid in this respect, in so far as it eliminates the dominant secular terms; however, it still needs regularization in order to advance into the short-wave region. Slight as it is, the irregularity in the PPT

will eventually cause divergences in higher-order terms. To make it clear, let us use the KP equation (u~+ uu~ u~11)~ = —u~,~/2

(D.1)

—

to study the KdV soliton instability as an example 21(77, u=

~ = x + ct,

c=

4K2.

(D.2)

u0(77) = —3c cosh~

This case is convenient since we know both the pertinent PPT result [84]and a relatively simple exact solution [64] for the linearized equation 2—

Lu

—k

-

—

u/2—iwOu/0rj,

(D.3) (021c?’q2)

(82/0772

—

u 0

—

c)

which describes perturbations of the form i7(77) exp(—iwt + iky) for soliton (D.2). The wavenumber k is considered to be small in the standard PPT procedure, and the solution of (D3) is sought through expansion in this parameter

(D.4) Under the restriction to the first three expansion terms, a limited solution is possible provided that 2/3, (D.5) = —2ck in which case it can be represented as —

~

1

8

-

8

k2 8

1 1/2

8 -~—(c~P 0)

3cIK)

= —(

(D.6)

tanh(K

77),

BA. Trubnikov and S.K. Zhdanov, Unstable quasi-gaseous media

225

where a is the perturbation amplitude, ‘P~is the flow potential (u0 = o’P018q), and the following derivatives with respect to the soliton c velocity are introduced for brevity d’P 018c

=

(‘P0 + 77u0)/2c,

8u0/Oc

=

(2u0

+ ~1 8u01877)/2c.

This, however, exhausts the possibilities of the PPT, since even the next term u3 is obviously unlimited. The way of eliminating the arising irregularity can be found by comparing the PPT result (D.6) with

the exact solution of eq. (D.3) obtained by Zakharov by the inverse scattering method [64].Before writing it down, note that the perturbation i7 in (D.3) is a two-scale function of 77. The dependence on the “slow” variable can always be taken to be exponential, which leads us to the representation =

x(n) exp( pi~).

(D.7)

Its substitution into (D.3) yields the following equation for x(’i) 2

(8/877 +

[(8/877+ p~)2 u

2~/2 iw(8Id’q + IL)x.

—

—

0

~t)

—

c] ,y

=

—k

(D.8)

Zakharov’s exact solution will be restored now under the requirement that x(’i) should be representable as a sum of derivatives of the potential ‘Po(77) x = a[82’P 2 + /3 ô’P~/8q+ y (‘Pa ‘P*)],

—

01877

(D.9)

where a, /3, y, ‘P * are constants. The x(ii) thus selected, is a solution if the following relations are satisfied $=2~,

~

),~2

=2~(c+2~2)+~t2’P*,

(D.10) k2 = 2~2[3(c+

For definiteness, let =

2) + ~‘P*].

j.~>0,

then solution (D.7—9) will be limited with adoption of (D.11)

‘Po(77_~*~c) = —3CIK,

which gives the parametric representation —iw 2

=

2~(c1’2 I.L)(cV2 —

2

—

(D.12)

1/22

k =6~(~—c

)

of the available ISP result [64]: w2 = —2k2(c 2(213)”2lkI)13. It can be easily shown that the PPT result (D.3) can be obtained from the exact solution (D.7, 9, 12) by a direct expansion in the parameter —

k.

This is exactly the cause of the irregularity, since such a procedure is valid only when I kr~4 1. Our representation (D.9—12) of the exact solution is useful due to its being actually a power series expansion of the parameter m, which is why it can be restored by the PT methods but, in contrast to the PPT, necessitates proper regularization. We shall seek a solution of eq. (D.8) as a power series in ~

226

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

X~Xo+LX~+PX~+

assuming that

w(~),k =

w =

(D.13)

k(s),

while

k2=~2/32+~3/33+---.

~

(D.14)

The coefficients in (D. 14) will be adopted so as to completely eliminate the irregularity of the expansion (D.13). Just as in the PPT, (D.13—14) yield the following in the zero approximation after substitution into (D.8) LX00,

X0—a8u0/O~.

(D.15)

The following equation is obtained in the next approximation

Lx, = —a ~

877

+2c)u0]

[u~+(61

(D.16)

with a solution (I x1 I
=

—a[81u0!c

+ (1 +

81/2c) ~18u0/877].

Elimination of the irregularity —n by adoption of 8,

(D.17)

=

—2c gives x,

=

2au0. Sticking to the same

procedure, it is possible to obtain the second order X2

=

a [‘Po(7j)—

‘Po(cc)],

(D.18) /32=6c,

82_’P0(cc),

2 and

as =well to show that x~3 0 inexact the next approximations, provided that 83 = —4, f3~= _12cU 6 inasfull compliance with the solution. Thus, the regularized perturbation theory permits us to restore the exact solution of the soliton stability problem without resorting to the ISP techniques. The associated re-normalization method /34

(D.7) applied to the KP equation proves to be relatively simple and appears to be universally applicable

to the generation of regular approximations for a wide range of wave models employed in the study of non-one-dimensional perturbations of solitons in plasma and plasma-like media.

Appendix E. Chaplygin’s method

To complete the picture, let us consider a steady-state flow of an unstable gas described in our notation by the equations divrYpet=0,

(rYV)rY=c~mVp~m.

In the case of a potential flow, the second equation yields the Bernoulli relation

(E.1)

B.A. Trubnikov and SK. Zhdanov, Unstable quasi-gaseous media V~=

—

2mc~(p~m 1)

Pet

—

= [1

—

(i4/2mc~)Jm(1

—

v2)~,

227 (E.2)

where V= rYI\I v~ 2mc~is the dimensionless velocity. Thus, the system (E.2) reduces to the equation —

div[V(1_V2)m]=0,

(E.3)

V~Vp,

which is of certain interest in itself owing to other possible applications. In the N-dimensional case, its particular solution is the point source V(1

v2y’ = const. ~

—

(E.4)

However, of greatest interest is the case of a two-dimensional flow, when the general solution can be obtained by Chaplygin’s method [1],assuming that (1— z)m OpIdy— —8’P/Ox

V~(1_V2)m=(1_z)m 8~I8x=t9’P/c9y,

where z

=

V2

=

(V~~)2. If the inverse functions x =

X(c,’P)

and

y=

(E.5)

Y(~,’P)are introduced, (E.5) will

give

(1

z)m Y~= X~,

—

(1

—

z)m X~ =

—

Y~,,

(E.6)

z = (X~,2+ Y~,2)1.

Next, Chaplygin introduces the velocity slope 7 = arctan(u~/V~) = arctan(Y~/X~) = arctan(—X~/Y~),

and considers the functions

and z(~,’P)whose derivatives are easily shown to be related as

~‘,‘P)

8~9(1—z)m8z 2z 8’P’

867 8’P

—

—

E8

(1+2m)z—lc7z 2z(1 Z)m+l ~

-

—

Finally, on introduction of the inverse functions ~ = ~(z, 7) and relations from the system (E.8): —

~

2z (1—z)m

, ~

(E.7)

‘P

=

cli(z, 0) we can get two linear

E9

(1+2m)z—1

,

cc

2z(1_z)m+l ‘Ps-

5

(

-

)

The condition of their compatibility is the equation 2(1

—

z) ~

+ z(1 + mz

z

—

z) ‘P

whose general solution is a sum ~ ‘P~(z,0)

=

‘P~

+ (1

—

1

+2m )

‘P~

= 0

(E.10)

of particular solutions of the form

F?’2 sin nO

where F = F(a, /3; -y; z) is the hypergeometric function with the parameters

(E.11)

228

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

a=(n—m+K)/2,

y=1+n,

f3=(n—m—K)/2,

Substituting (E.11) into the first equation of (E.9) and integrating it with respect to second function ço~(z,7) = z

(1 —

(E.12)

K~Vm2+(1+2m)n2. 7,

we obtain the

z)m (Fz’~2)cos nO.

(E.13)

These were the solutions given by Chaplygin in ref. [1]. The simplest case is the one with m = ~ corresponding to the “Chaplygin gas” proper, for which eq. (E.3) takes the form div(V/V1 V2) = 0 and can be solved in a simpler way since when m = eqs. (F.8) can be reduced to the harmonic forms —

—

8)

8a-

-;~--=~,

8)

—

i—Vf~

dz

if

Oo’

a-=~j

-~=—~—,

~

(E.14)

~

and, hence, the solution is any complex analytic function a- + i7 = f(~ + i’P). Let us deal now with the question of selecting “spontaneous” solutions, for which the perturbations would go to zero at the left-hand infinity x —cc, where the flow is considered to be unperturbed and has a velocity V 1 = v0, v~,= 0, which corresponds to —~

7=

0,

z = z0 = V~,

V0

=

v0I\/v~ 2mc~,

(E.15)

—

as well as to the unperturbed potentials ~cc0xV0,

(E.16)

‘P=’P0=y~~(1_V~)m.

This suggests that for spontaneous solutions the function ~(z, 7) should become infinite (—cc) at the point 7 = 0, z = z0. However, Chaplygin’s particular hypergeometric solutions (E.13) show no singularity at this point and, consequently, are not spontaneous. For them to be constructed we would have to consider an infinite sum of particular solutions, which is certainly difficult. It is only for the “Chaplygin gas” with m = ~that simple examples of such solutions can be found. Such, for instance, is a solution of the form —

a-—a-o+iO=ln(1+

a-0=lnA0,

~•‘P)’

where L is a certain specified length. In the limit cc is exactly what makes the perturbations small as x

Ao~V1+(coIvo)2_(coIvo)

—~

—cc

—*

—

we are sure to have a-

=

a-0 and 0

(E.17)

= 0,

which

On using the following notation for brevity’s sake ‘P~ccIL,

~P=’PIL,

N°°r(1+’P)2+!1’2

2+11’2, M’P

(E.18)

we find the following functions from relation (E.17) -‘P

0

=

0(cc, ‘P)= arctan ‘P + M

z

=

z(cc,

4A2MN

‘P)°°° (M +NA~)2

(E.19)

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

229

which permits to us eqs. (E.6—7) to find the coordinates ‘P+~lnM—~-inN,

~=

(E .20)

y

1—A~

L =

2A

~P —

A0 -~-

~1’

arctan

-~

—

1 ~-,~--

____

arctan 1

+ ‘P~

This offers direct evidence of the fact that perturbations are small in the limit as x

—*

—os.

References [1] S.A. Chaplygin, Selected Works [in Russian] (Nauka, Moscow, 1976) p. 94. [2]MA. Leontovich, Introduction to Thermodynamics and Stastical Mechanics [in Russian] (Nauka, Moscow, 1983) p. 239. [3]DL. Book, E. Ott and AL. Sulton, Phys. Fluids 17 (1974) 676. [4]E.E. Son, Pis’ma v ZhTF 4 (1978) 1023 [Soy. Tech. Phys. Lett. 4 (1978) 413]. [5] BA. Trubnikov and 5K. Zhdanov, Prepnnt MEFI N 001-84, Moscow, 1984. [6]DL. Book, E. Ott and H. Lampe, Phys. Fluids 19 (1976) 1982. [7]BA. Trubnikov and S.K. Zhdanov, Pis’ma v ZhETF 41(1985) 292 [JETP Lett. 41(1985) 358]. [8] S.A. Akhmanov, A.P. Sukhorukov and RB. Khokhlov, UFN 93 (1967) 19 [Soy. Phys. Uspekhi 10 (1968) 609]. [9] A.V. Gurevich and A.B. Shvartsburg, ZhETF 58 (1970) 2012 [Soy. Phys. JE1’P 31(1970) 1084]. [10] S.V. Bulanov and P.V. Sasorov, in: Nonlinear and Turbulent Processesin Physics, Vol. 1 (Gordon Breach Horward Acad. PubI., N.Y., 1984) p. 149. [11] D.Yu. Manin and VI. Petviashvili, Pis’ma v ZhETF 38 (1983) 427 [JETP Lett. 38 (1983) 517]. [12] L.M. Degtyarev, V.Ye. Zakharov and LI. Rudakov, ZhETF 68 (1975) 115 [Soy. Phys. JETP 41(1975) 57]. [13] Ya.B. Zeldovich and ID. Novikov, Structure and Evolution of the Universe (Nauka, Moscow, 1975). [14] L.D. Landau, and EM. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1959). [15] M. Abramowitz and IA. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). [16] V.Ye. Zakharov, YeA. Kuznetsov and S.L. Musher, Pis’ma v ZhETF 41(1985)125 [JEPT Lett. 41(1985)154]. [17] V. Nardi, W.H. Bostick, J. Feugeas and W. Prior, Phys. Rev. 22A (1980) 2211. [18] R. Lee and M. Lampe, Phys. Rev. Lett. 31(1973)1390. [19] B.L. Rozhdestvensky and N.N. Yanenko, Systems of Quasi-linear Equations and their Applications to Gas Dynamics [in Russian] (Nauka, Moscow, 1968) p. 172. [20] G.L. Lamb and D.W McLaughlin, in: Solitons, eds. R.K. Bullough and P.J. Caudrey (Springer-Verlag, Berlin, Heidelberg, New York, 1980). [21] GB. Whitham, Linear and Nonlinear Waves (J. Wiley, New York, 1974). [22] VI. Karpman, Non-Linear Waves in Dispersive Media (Pergamon, Oxford, 1975). [23] BA. Trubnikov, Plasma Physics and Controlled Thermonuclear Reactions 4 (1958) 87. [24] BA. Trubnikov, Plasma Physics and Controlled Thermonuclear Reactions 1 (1958) 289. [25] BA. Trubnikov, Pis’ma v ZhETF 42 (1985) 317 [JETP Lett. 42 (1985) 389]. [26] B.I. Samoilenko and Yu.B. Slezin, Proc. 10th Conf. on Vulkanism, Novosibirsk, 1979. [27] V.A. Veretennikov, SN. Polukhin, 0G. Semenov and Yu.V. Sidelnikov, Fizika plazmy 7 (1981) 1199 [Soy. J. Plasma Phys. 7 (1981) 656]. [28] V.A. Veretennikov, V.A. Gribkov, E.Ya. Kononov, 0G. Semenov and Yu.V. Sidelnikov, Fizika plazmy 7 (1981) 455 [Soy. J. Plasma Phys. 7 (1981) 249]. [29] V.V. Vikhrev and SI. Braginsky, in: Voprosy teorii plazmy (Gosatomizdat, Moscow, 1980) p. 10 [Rev. Plasma Phys. 10 (1986) 425]. [30] VS. Imshennik, S.M. Osovets and IV. Otroshchenko, ZhETF 64 (1973) 2057 [Soy. Phys. JET!’ 37 (1973) 1037]. [31] MA. Leontovich and SM. Osovets, Atomnaya energiya 3 (1956) 81. [32] G.A. Askaryan, ZIIETF 42 (1962) 1567 [Soy. Phys. JETP 15 (1962) 1088]. [33] E. Garmire, R. Chiao and C. Townes, Phys. Rev. Lett. 16 (1966) 347. [34] A.A. Vedenov and LI. Rudakov, DAN SSSR 159 (1964) 767 [Soy. Phys. Doklady 9 (1965) 1073]. [35] AS. Kingsep, Itogi nauki i tekhniki [inRussian] (Energoatomizdat, Moscow). [36] T.F. Volkov, Plasma Physics and Controlled Thermonuclear Reactions 3(1958)36; 4 (1958) 98. [37] V.Ye. Zakharov, in: Osnovy fiziki plazmy Vol. 2 (Energoatomizdat, Moscow, 1984) p. 79 [BasicPlasma Phys. 2 (1984) 81]. [38] M.J. Lighthill, J. Inst. Math. AppI. 1 (1965) 269; Proc. Roy. Soc. 299A (1967) 28. [39] B.B. Kadomtsev and VI. Karpman, UFN 103 (1971) 193 [Soy. Phys. Uspekhi 14 (1971) 40].

230

BA. Trubnikov and 5K. Zhdanov, Unstable quasi-gaseous media

[40] V.L. Polyachenko and AM. Fndman, Equilibrium and Stability of Gravitating Systems [in Russian] (Nauka, Moscow, 1976) p. 347 [English translation published by Springer]. [41] MA. Leontoyich, Izy. AN SSSR, ser.fiz. 81 (1944) 16. [42] S.V. Bazdenkov and 0.P. Pogutse, Pis’ma v ZhETF 36 (1982) 190 [JETP Lett. 36 (1982) 234]. [43] S.V. Bulanov and P.V. Sasorov, ZhETF 86 (1984) 479 [Soy. Phys. JETP 59 (1984) 279]. [44] A.A. Rukhadze, L.S. Bogdankevich, S.Ye. Rosinsky and V.G. Rukhlin, Physics of High-current Relativistic Electron Beams [in Russian] (Atomizdat, Moscow, 1980) p. 87. [45] SV. Bulanov and PY. Sasoroy, Fizika plazmy 4 (1978) 746 [Soy. J. Plasma Phys. 4 (1978) 418]. [46] N.G. Belova, A.A. Galeyev, R.Z. Sagdeyev and Yu.S. Sigov, Pis’ma v ZhETF 31(1980)551 [JETP Lett. 31(1980)518]. [47] A.A. Galeyev, R.Z. Sagdeyev, V.D. Shapiro and VI. Shevchenko, ZhETF 81(1981)572 [Soy. Phys. JETP 54 (1981) 306]. [48] V.P. Silin, ZhEFT 48 (1965) 1679 [Soy. Phys. JETP 21(1965)1127]. [49] J. Fukai and E.J. Clothiaux, Phys. Rev. Lett. 34 (1975) 863. [50] BA. Trubnikov, Fizika plazmy 12 (1986) 468 [Soy. J. Plasma Phys. 12 (1986)]. [51] A.A. Galeyev, in: Osnovy fiziki plazmy Vol. 2 (Energoatomizdat, Moscow, 1984) p. 331 [Basic Plasma Physics 2 (1984) 305]. [52]D.J. Korteweg and G. de Vries, Phil. Mag. 39 (1895) 422. [53] Vi. Makhankov, Phys. Reports 35 (1978) 59. [54] A.B. Mikhailovsky, S.V. Makurin and A. I. Smolyako~,ZhETF 89 (1985) 1603 [Soy. Phys. JETP 62 (1985) 428]. [55] VI. Petviashyili, in: Non-linear Waves [in Russian] (Nauka, Moscow, 1979) p. 5. [56] L.A. Bordag, AR. Its, A.V. Matveyev, S.V. Manakov and V.E. Zakharov, Phys. Lett. A63 (1979) 205. [57] BA. Dubrovin and 5.P. Novikov, DAN SSSR 270 (1983) 781. [58] S.P. Tsarev, DAN SSSR 282 (1985) 534. [59] E.W. Laedke and K.H. Spatschek, J. Math. Phys. 20, No. 9 (1979) 1838. [60] B.B. Kadomtsev and VI. Petviashyili, DAN SSSR 192 (1970) 753 [Soy. Phys. Doklady 15 (1970) 539]. [61] V.Ye. Zakharov and A.B. Shabat, Funkts. analiz i yego prilozheniya 8 (1974) 43 [English translation in Functional Analysis and its Applications]. [62] YeA. Kuznetsov and 5K. Tuntsyn, ZhETF 82 (1982) 1457 [Soy. Phys. JETP 55 (1982) 844]. [63] A.B. Mikhailovsky, S.V. Makurin and Al. Smolyakov, ZhETF 89 (1985) 1603 [Soy. Phys. JETP 62 (1985) 928]. [64] V.Ye. Zakharov, Pis’ma v ZhETF 22 (1975) 364 [JETP Lett. 22 (1975) 172]. [65] A.A. Zaitsev, DAN SSSR 272 (1983) 583 [Soy. Phys. Doklady 28 (1983) 720]. [66] 5K. Zhdanov and BA. Trubnikov, Pis’ma v. ZhETF 39 (1984) 110 [JETP Lett. 39 (1984)129]. [67] L.A. Abramyan and Yu.A. Stepanyants, Izv. VUZoy, ser. Radiofizika 28 (1985) 27 [Radiophys. Qu. Electron. 28 (1985) 20]. [68] 5K. Zhdanov, DAN USSR 284 (1985) 115. [69]Yu.V. Katyshev and V.G. Makhankoy, Phys. Lett. A57 (1976) 10. [70]B.B. Kadomtsev, Collective Phenomena in a Plasma [in Russian] (Nauka, Moscow, 1976) p. 130. [71]L.A. Ostrovsky and VI. Shrira, ZhETF 71(1976)1412 [Soy. Phys. JETP 44 (1976) 7381. [72]S.K. Zhdanov, Pis’ma v ZhETF 43 (1986) 414 [JETP Lett. 43(1986)]. [73] A.B. Mikhailovsky, GD. Aburdzhaniya, 0G. Onishchenko and A.I.Smolyakov, ZhETF 89 (1985) 482 [Soy. Plasma Phys. 62 (1985) 273]. [74]N.N. Akhmedieya, V.M. Yeleonsky and NYc. Kulagin, ZhETF 89 (1985) 1542 [Soy. Phys. JET!’ 62 (1985) 894]. [75]V.M. Yeleonsky, I.M. Krichever and NYc. Kulagin, DAN SSSR 287 (1986) 606. [76] VI. Petviashvili, in: Voprosy teorii plazmy Vol. 9 (Atomizdat, Moscow, 1979) p. 59 [Rev. Plasma Phys. 9 (1986) 59]. [77] VI. Petviashyili and V.V. Yankov, in: Voprosy teorii plazmy Vol. 14 (Energoatomizdat, Moscow, 1985) [English translation in Rev. Plasma Phys., Vol. 14). [78] BA. Trubnikov, Phys. Fluids 5(1962)184. [79] RB. Kadomtsev, in: Voprosy teorii plazmy Vol. 2 (Gosatomizdat, Moscow, 1963) p. 132 [Rev. Plasma Phys. 2(1966)153]. [80]L.D. Landau and EM. Lifshitz, Quantum Mechanics (Pergamon, Oxford 1965). [81] V.Ye. Zakharov, S.V. Manakov, S.P. Novikoy and L.A. Pitayevsky, The Theory of Solitons. The Inverse Transform Method [in Russian] (Nauka, Moscow, 1980). [82] YeS. Benilov, ZhETF 88 (1985) 120 [Soy. Phys. JETP 61(1985) 70]. [83] V.P. Pavlenko and VI. Petyiashvili, Fizika plazmy 8 (1982) 206 [Soy. J. Plasma Phys. 8 (1982) 117]. [84) VI. Petviashyili, in: Voprosy teorii plazmy Vol. 9 (Atomizdat, Moscow, 1979) p. 59 [Rev. Plasma Phys. 9 (1986) 59].