- Email: [email protected]
Compact and noncompact dispersive patterns
9 October 2000
Physics Letters A 275 Ž2000. 193–203 www.elsevier.nlrlocaterpla
Compact and noncompact dispersive patterns Philip Rosenau School of Mathematical Sciences, Tel AÕiÕ UniÕersity, Tel AÕiÕ 69978, Israel Received 17 November 1999; received in revised form 23 August 2000; accepted 23 August 2000 Communicated by C.R. Doering
Abstract We discuss the pivotal role played by the nonlinear dispersion in shaping novel, compact and noncompact patterns. It is shown that if the normal velocity of a planar curve is U s yŽ k n .s , n ) 1, where k is the curvature, then the solitary disturbances may propagate like compactons. We extend the KP and the Boussinesq equations to include nonlinear dispersion to the effect that the new equations support compact and semi-compact solitary structures in higher dimensions. We also discuss the relations between equations sharing the same scaling. We show how compacton supporting equations may be cast into a strong formulation wherein one avoids dealing with weak solutions. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The underlying theme of this work is the impact of nonlinear dispersion on the formation of patterns. We shall be concerned with four topics; A: The motion of curÕes in a plane. This subject has drawn a considerable interest in recent years, see Refs. w1–4x and references therein, and provides a natural setup for the introduction of compactons – solitary waves with a compact support – which are perhaps the simplest manifestation of nonlinear dispersion at work. If the velocity of the normal to the curve is assumed to be U s yŽ k n .s , n ) 1, the nonlinear dispersion enters naturally into the intrinsic description of the curve, and supports the propagation of both compact and noncompact patterns. The resulting partial differential equations are exactly of
E-mail address: [email protected] ŽP. Rosenau..
the type encountered in our previous studies of compactons w5–7x. B: Compact structures in higher dimensions. To this point in time, the wonders of integrability – which provide the essential tool to study the interaction of solitons – have eluded us in higher dimensions. The equations derived using the reductive perturbation technique, do not seem to be integrable, unlike the 1-D case. Rather then resign ourselves to the view that in higher dimensions true solitons cannot exist, we argue that in order to support localized patterns, the enhanced spread of waves in higher dimensions has to be counteracted by adequately stronger nonlinearities. The nonlinear dispersion presented in this work seems to provide one such mechanism. We base our approach on a certain affinity between pdeX s having their highest order operator degenerate at the front. Consider u t s Ž u n . x x and u t q Ž u n . x x x s 0, which have the highest order operators degenerate at u s 0. In spite of the obvious differences between these equations, in both cases
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 5 7 7 - 6
194
P. Rosenaur Physics Letters A 275 (2000) 193–203
the degeneracy at the front generates sharp fronts, and supports the formation and propagation of, robust, compact patterns. The fact that compact patterns emerge in the parabolic case in higher dimensions as well as in 1-D, leads to the conjecture that analogous dispersive equations in higher dimensions will also support localized patterns. To this end we endow a number of well known equations with a non-linear dispersion and construct, at least formally, compact and semic-mpact structures in higher dimensions. Even though, at this stage, these equations are formal mathematical extensions of known models, if the resulting patterns turn out to be robust, this will provide an impetus to seek physical realization of these equations. We note that in the case of the anharmonic lattice, the motion of the compacton represents a regime wherein the anharmonic forces completely dominate over the harmonic ones. This regime is thus far away from the conventional soliton domain, rather than being a perturbation and occurs either at large amplitudes Žand thus is not accessible to perturbative procedures. or, if ab initio the harmonic forces are negligible Že.g; if the chain is made of hard spheres.. In this case the linear speed of sound is very small and thus the transition to a fully nonlinear regime takes place at very low amplitudes. Of course, equations derived in the weakly nonlinear regime may sometimes develop large amplitude andror gradients, but the value of the information gathered in the high end is quite questionable. For instance, the ion acoustic solitons are adequately approximated by the Korteweg–de Vries ŽKdV. solitons. In the weakly nonlinear regime their behavior at large amplitudes has nothing to do with the real phenomenon, since the actual ion-acoustic solitons completely disappear at a critical amplitude, and a collision-free turbulent regime sets in! C: Equations defined Õia the inÕariance of scales. In many cases, though the underlying strongly nonlinear processes are quite complex, and thus very hard to model, the emerging patterns are remarkably simple oÕer a wide range of scales. In an effort to exploit this simplicity within the context of dispersive phenomena Žthough by no means limiting ourselves to such phenomena., we propose in Section 4 the following phenomenological route toward this goal: It has been observed by many, in a variety of
contexts, that the scaling relations of a process are its vital feature. Thus, if we have a handle on that property, whether from first principles or empirically, and then construct mathematical modelŽs. endowed with the correct scaling, we can expect that the resulting model will correctly immitate the actual process. It is important to observe that a given scaling does not define a unique model, but rather a sequence of model equations. The examples studied in Section 4 indicate that equations sharing the same scaling share similar patterns. This also works vice versa- in the aformentioned ion-acoustic problem, if one compares the interaction of its solitons using the BBM equation Ž u t q uu x s u x x t . and the KdV equation, then the patterns rendered by KdV, particularly away from very small amplitudes, are in much better agreement with those of the true interaction. The superiority of the KdV equation is thus not due to its integrability, after all the actual problem is not integrable, but due to the the fact that its width-height scaling relations are much closer the to the actual process! The classical argument presented in favor of the BBM that it has a better linearized dispersion relation, does not come into play as far as the interaction of solitons is concerned. D: Strong formulation of compacton carrying equations. By definition compact patterns are nonanalytical entities on the frontlineŽs.. In dispersive processes the presence of even a weak singularity poses a formidable numerical challenge, which was partially described in Refs. w5–7x and in recent literature devoted to the numerical aspects of the problem. In Section 5 we show that, at least for the considered class of nonlinear problems, one can circumvent those difficulties by casting the problem into an alternative formulation in which all singularities are weakened to the extent that all the present derivatives are smooth. Preliminary numerical experiments reveal that this representation has a dramatic impact on the numerics, and problems which originally were unaccessible are now easily dealt with.
2. Motion of curves in the plane Let a plane curve be represented by r s Ž j ,h .
;
P. Rosenaur Physics Letters A 275 (2000) 193–203
where Ž j ,h .Ž s,t .. Let t s E rrE s be the tangent vector to the curve where s is the arc length. Define g' t P t ;
Eh Ej ns y , Es Es ;
ž
and
;
'g ,
dt
r Ž s,t . s w Ž s,t . t Ž s,t . q U Ž s,t . n Ž s,t . ,
;
Ej Eh
U s yk ,
g s x and
z s y Ž x ,t . ,
Ž 2a .
zx x 1 q z x2
s0 .
Ž 3.
Let V s z x , then taking one derivative in Ž3. we have
q
n s 0.
Ž 9. E3
E
nq1 E s
Žk
nq2
.q
E s3
Ž kn. s0
n / 0,
Ž 10 .
1qV 2 x
H0 Vdx ,
/
s0 .
Ž 4.
x
r s Vy1 and t s t ,
then the Lagrange map Ž5. yields rh rt q s0 . 1qr2 h
ž
/
Ž 5.
E 2U
q k 2U q
Ek
Ž 6.
Ž k n . , n / 0,
Es
ks
2
ž /
s 0 for
k
n s 0.
Ž 11 .
E um
Eu
E 3un
q q s 0. Ž 12 . Et Ex E x3 For n s 1, Eq. Ž10. reduces to the m-KdV equation. For n / 1, the dispersion is nonlinear and three values of n are of special interest: n s y1r2 which is related to an integrable case, and the n s 2,3 cases which support compactons. For n s y1r2 ;
E3
E
Ž k 3r2 . q
Ž ky1r2 . s 0 . Ž 13 . Et Es E s3 This equation is a Lagrange image of an integrable problem. Indeed, in terms of y
s
us
H0 kds,
t s t.
Ž 14 .
nk 2
Ek q
E3
E
n q 1 Eu
q
Eu 3
k nq 1 s 0 ,
n / 0,
which, for n s y1r2, we recognize in terms of r s 1rk as an integrable extension of the Hary Dym equation w5–9x. We turn to the n s 2 case. Now 2 E
E3
Ž 7.
k2s0 . Et 3 Es E s3 This is the K Ž4,2. equation. A family of travelling solutions is given, via h s s y l t,
Ž 8.
k s 8 l3 Z 2 l1r3ah ,
E Es
Es
K Ž m,n . ;
Ek
s
H kUds . Es
E2 q
We shall revisit Eq. Ž11. in Section 4. Eq. Ž10. is a particular Ž m s n q 2. case of the K Ž m,n. equation introduced in w5–7x;
Et
Et E s2 Let us first assume that Usy
Ek2 q
Eq. Ž10. is mapped into Žcf., ref w8x.
The invariance of Eq. Ž4. under Ž5. enables us to generate from a new solution form a given one. Direct use of the Serret–Frenet representation leads to an intrinsic representation of the motion via the curvature w1–4x; s
and
Ek
Vx
Now let
Ek
,
n
Ž 2b .
where k is the curvature, then in Ž x, y . coordinates this choice of U leads to
hs
Et for
Et
Eh Ej
y y g 1r2 U s 0 . Es Et Es Et Firstly, note that, if
ž
Ek
Ek
which leads to
Vt q
k Es
;
Ž 1.
zt q
1 Ek
Thus
a unit normal vector. The dynamics of the curve is then specified via d
and Usy
/
195
q
k4q
a s 2 1r3r'6 ,
P. Rosenaur Physics Letters A 275 (2000) 193–203
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and Z is given implicitly via Z
hs"
X
H0 dz Ž 1 y z
X 6 y1 r2
.
,
0 F Z F 1.
The compact solution is obtained keeping half a period of the solution Žbetween two consecutive minima. w5–7x, and setting k to zero elsewhere. The compacton thus obtained has a weak discontinuity at the edge of the pulse. Nevertheless, it is easily seen to be a conventional solution of Ž10. w5–7x. Similarly, for n s 3, the compacton of the resulting K Ž5,3. equation is given via the half period of k s " Ž 2 l.
1r4
Z Ž lr72 .
1r4
h ,
with Z cut out of Z
hs"
X
H0 dz Ž 1 y z
X 4 y1 r2
.
,
0FZF1 .
The compact span of k means that, at a given time, the deformationŽs. of the curve can be completely localized with other parts of the curve being completely unaffected by its motion. For a more general class of equations we replace U with F given via Us
1 EF
.
k Es
Et
E s2
Fs
ž /
q
E Ž kF . Es
k
.
Ž 16 .
Let us apply the Lagrange map Ž14. to Ž18.. We have
Ek Et
E2
EF
2
sk 1q
Eu
2
Eu
with
Us
EF Eu
.
Ž 17 .
In terms of radius of curvature, r s 1rk, the resulting equation is even simpler;
Er Et
3r2
Zt q Ž 1 y Z 2 . w Z x x x y Z x x s 0. Eq. Ž19. is an integrable extension of the integrable case, due to Wadati et al. w8x. It follows from Ž18. that Ž19. is also is invariant under the Lagrange map Ž14.; GŽ r . s rr '1 q r 2 . Thus GŽ r . s ry2 ,ry1r2 , and rr '1 q r 2 represent three integrable cases. In terms of Z the solitary structure of Ž19. is obtained via y2 l 1 y '1 y Z 2 y Z 2 q Z s2 s E . Taking E s 0, for l ) y1r2 we have a solitary wave with a sharp peak at Z s 1 Žwhich corresponds to k ™ `.. This is yet another manifestation of nonanalyticity due to nonlinear dispersion. For y1 - l - y1r2 a soliton with a smooth shape is obtained. 3. Compact structures in higher dimensions
E2 s
or
Ž 15 .
Instead of Ž7. we now obtain a local form
Ek
Eq. Ž18. could be solved exactly and the results re-mapped into other coordinates. This issue will be addressed elsewhere. For now let: F s 1r '1 q k 2 which leads to Ek E3 E k q y Z s 0 where Z s 3 '1 q k 2 , Et Es Es Ž 19 .
E2 q 1q
Eu
2
EG Eu
s 0 here
G Ž r . s F Ž 1rr . .
Ž 18 . It thus appears that u is the natural independent variable to use – it leads to what appears to be the simplest description of motion. Note the linear operator acting on GŽ r .. If GŽ r . were linear in r, or given via a linear operator acting on r; G s LŽ Eu .w r x,
In this section we explore a number of formal mathematical extensions of soliton-supporting equations with the aim of producing compact dispersive structures in higher dimensions. In what follows we shall use another variant of the K Ž m,n. Eq. Ž12. u t q a Ž u m . x q u Ž u n . x x x s 0, m ) 1. Ž 20 . The assumed nonlinear dispersion emerges in nonlinear lattices and reflects the dispersive effect due to the discrete nature of the intersite potential: Ž y kq 1 y y k . nq 1 w5–7x. For m s 2, and n s 1, Eq. Ž20. was also recently shown to describe the dispersion of dilute suspensions w10x. The solitary structures of Eq. Ž20. and the K Ž m,n. are quite similar. For instance, for m s n q 1 we have 1
u c Ž x ,t . s
½
2l a
cos 2
< x y lt < F p and u c s 0 otherwise.
'a 2
Ž x y lt .
5
n
,
Ž 21 .
P. Rosenaur Physics Letters A 275 (2000) 193–203
A. Now let us consider
E u t q 4 Ž u nq 1 . x q u Ž u n . x x
Ex
xym
Ž u nq1 . x x
To unfold the motion in the original variables we write h in terms of x and t which for n s 1 gives in 3 D;
l q = H2 u s 0,
Ž 22 .
where = H ' E y2 q Ez2 Ž N y 2., N s 2Ž3. in 2Ž3.-dimensions. Eq. Ž22. generalizes the well known K–P equation in two ways; Ža. it adds a dissipative component and, Žb. the dispersive part is nonlinear. Let us recall that the K–P equation supports supports one-dimensional solitary waves in space but these waves may have arbitrary spatial inclinations. A variant of Ž22., based on the extension of the K Ž2,2. was presented in w5–7x. We now define new variables Ž t 0 s const.. U Ž s,t .
us
Ž t q t0 .
ar2
a s N y 1,
,
N s 2,3,
Ž 23 .
us
2Ž t q t0 .
2
,
r 2 s y 2 q z 2 Ž N y 2. ,
Ž 24 .
and v
ts
½
t 0 Ž t q t 0 . qt 1 , ln Ž t q t 0 . ,
v '1y n Ž N y1. r2 / 0. . v s 0.
Ž 25 .
Note that v G 0 only if n F 2rŽ N y 1.. Clearly, the dimension has a crucial impact on the rate of propagation and the attenuation of the wave. Upon substitution and one integration Žignoring the constant of integration. we have
EU Et
E q
Es
U Ž U n . s s q 4U nq1 y m Ž U nq1 . s 4 s 0 . Ž 26 .
In particular, when m s 0, the last equation supports propagation of compactons, cf., Ž21., found via the solution of U yl q 4U n q Ž U n . hh s 0 where h s s y lt . Ž 27 .
(
2 t q t0
N s 2,3,
4 Ž t q t0 .
y lln Ž 1 q trt 0 . ,
,
l
4Ž t q t0 .
y2 qz2
Ž 28 .
and u vanishes elsewhere. Note that at t s 0 the support of the solution is < x q r 2r4t 0 < F pr2. The forward front is then given via r f2 s 2 t 0 Žp y 2 x . and the trailing front via r b2 s 2 t 0 Žy2 x y p .. Each front describes an evolutionary paraboloid, that propagates with a logarithmic speed llnŽ1 q trt 0 ., and Žsince r 2 is compressed by the factor 4 t . also deforms in time. In addition, the motion is accompanied by an attenuation of the amplitude. In 2D the motion takes the form us
ssxq
cos 2 x q
p
where r2
197
cos 2 x q
y2 4Ž t q t0 .
(
y l t q t0 ,
p 2
,
Ž 29 .
and vanishes elsewhere. In both two- and three dimensions the nature of the disturbance is such that an observer at a fixed point in space will first experience an oncoming front of the wave. This will continue for a while but will come to its end at some point in time, after which he will no longer observe any wave. We note in passing that, up to this point in our discussion we could have replaced the dispersive part with Ž u nq 1 . x x x and, apart of numerical factors, the results would essentially be the same. However, the present form is more convenient when dissipation is included Žsee the Appendix.. In conclusion we note that the obtained patterns are only partially localized and thus cannot be considered a genuine generalization of the solitary waves. These are considered next. B. We now present a formal ‘1–1r2’ dimensional extension of Eq. Ž20.; u t q Ž u 2 . x q d w un u x x s 0 where n ' Ex2x q E y2y .
Ž 30 .
P. Rosenaur Physics Letters A 275 (2000) 193–203
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Here the wave travels in x-direction but has also a structure in y-direction. Integrating once in a travelling frame, h s x y l t, d s 1, we obtain n u q u s l.
Ž 31 .
(
If r s h 2 q y 2 , then a radially symmetric, compact, solution reads usl 1y
J0 Ž r . J0 Ž ri .
,
0 F r F ri ,
Ž 32 .
and vanishes elsewhere. Here Jj Ž r . is the j-th order Bessel function, and ri are the roots of J1. Note that, since the Bessel function decays in space and the spacing of its roots changes, unlike the 1D case, the subsequent zeroes of J1 generate different multihump compact structures Žthe presence of mixed sign parts in all but the first mode, will probably cause only the first mode to be stable, cf., Ref. w5–7x.. Eq. Ž30. has a remarkable structure; it may be cast as L Ž u . s d n u q u, Ž 33 .
u t q Ex uL Ž u . s 0 where
is a linear operator which enables us to find explicit solutions. Let L be any linear operator with constant coefficients and u s F Ž x . a solution of LŽ u. s l. Then all that is needed to solve the original problem is to ensure that u t q l u x s 0 is satisfied. Take uŽ x,t . s F Ž x y l t . and you are done. Also note that, both the dissipative extension of Eq. Ž20.; u t q a Ž u nq 1 . x q u Ž u n . x x
xsb
Ž u nq1 . x x ,
and the viscous flow in a thin layer, u t q Ž u 2 . x q Ž uu x . x q d Ž uu 3 x . x s 0,
Ž 34 .
m s 1,2,
xx
,
Ž 35 .
describe, for a s 0, the vibrations of a purely anharmonic lattice Žsee w5–7x., and as can be easily seen, support travelling structures with a compact support. For a s m s 1 we rewrite Eq. Ž35. as u t t s u x x q Ž uLu . x x ,
where
(
u s F1 Ž x q l t . q F2 Ž x y l t . ,
(
where l s 1 q a 1 q a 2 is also a solution. We thus obtain a special case of a clean interaction of two waves. The above analysis gives the impression that two compactons will cross each other without really interacting, but this is wrong; the additive property holds either for analytic functions or noninteracting compactons. In the later case the nonanalyticity at the edge of each compacton is compensated in the product uLw u x by the vanishing u. During the interaction between two compactons, u does not vanish in the intersecting domain and this enables the product uLw u x to induce interaction between compactons.
4. Equations defined via scales invariance Consider the following two partial differential equations ux ut q Ž u2 . x q s 0, Ž 36 . u xx
ž /
which was introduced in Section 1 ŽEq. Ž11.. and the Benjamin–Ono ŽBO. equation Žcf. w11x. ut q Ž u2 . x q H ( u )
x x s 0.
Ž 37 .
H ( u ) is the Hilbert Transform of u. Insofar as travelling structures are concerned, the two equations share the same scaling; u s l F lŽ x y l t . .
Ž 38 .
Question: how close is the dynamics of two different equations, if their travelling structures share the same scaling? To look more closely into this issue, we assume a given scaling
have similar ’hidden’ linear features. C. The Boussinesq equationŽs. u t t s au x x q Ž u mq 1 . x x q b u Ž u m . x x
If Fi Ž x . is a solution of Lw u x s a i , a i s const.Ž) y1., then F Ž x q l t ., where l s " 1 q a i , is not only a solution, but the special superposition of the two solutions
L w u x s Ž bE 2 q a . u.
u s l a F Ž l bh .
where h s x y l t.
Ž 39 .
and look for differential equations sharing this property. In this setting the question is too general. To make a progress we narrow our scope and start with model equations of the type u t q Ž u m . x q E r Ž u n . s 0,
Ž 40 .
P. Rosenaur Physics Letters A 275 (2000) 193–203
with the adjustable ’parameters’ m,n, and r at our disposal. When the integer r is even, to ensure the dispersive nature of the third term, E r , will be understood via the Hilbert Transform. For instance; E 2 will be understood as E 2 H ( u ). Assuming a given scaling law Ži.e., a given a and b ., one has
199
For y1 - E - 0 we have u "s
l< E < 1 . '1 y < E < cos Ž l'< E < h .
.
Thus, m is fixed, but otherwise we have a one parameter family of possible equations corresponding to a given scaling law. In the example considered, in addition to the two cases presented, all equations of the form
As E x0 the uq soliton transforms into a conventional soliton. Its shape coincides exactly with the shape of the BO soliton Ž43.!. Thus Eq. Ž36. supports two kinds of solitary waves; a solitary family that decays exponentially and a solitary wave which decays algebraically. It would be of some interest to determine whether the BO can also support solitons with an exponential tail. As an another example let us look at the two equations
u t q Ž u 2 . x q E r Ž u 3y r . s 0
u t q Ž u 3 . x q uu x x q 12 Ž u x .
m s 1 q 1ra and 1 q a q b s a n q b r .
Ž 41 .
Ž 42 .
share the same scaling. Let us now look more closely into the affinity between Eqs. Ž36. and Ž37.. The BO is integrable and its one soliton solution is given via us
2l 1 q l2h 2
,
where h s x y l t.
Ž 43 .
We can easily calculate the travelling structures of Eq. Ž36.; let u s lV Ž s ., s s lh , then after two integrations we obtain
and
E s const.
Unlike the familiar semilinear cases, where the integration constant plays the role of a ’total energy’ of the system, in this case we have a fundamentally different situation where the integration constant E affects the actual shape of the potential function. The solution has two branches; E)0 u "s
lE "'1 q E cosh Ž l'E h . y 1
.
The most important thing to observe about these solitary waves is that for each speed l we have in terms of E an entire one parameter family of solitary waÕes. Note also that the uq solitons are taller than their uy cousins and that both solitary waves travel to the right. However, since, Eq. Ž36. is invariant under u ™ yu and t ™ yt, the yuy Ž) 0. and the yuq Ž- 0. solitary waves travel to the left.
x
s0
u t q Ž u 3 . x s u5 x ,
Ž 44 .
which share the same scaling. The chosen quadratic form for the dispersion is a particular case of w uu x x q a Ž u x . 2 x x considered in the next section and, as can be easily seen, has a Hamiltonian structure. Both equations conserve mass Ž H x udx ., momentum Ž H x u 2 dx . and energy. The solitary solution of the first equation is found solving Ž u s 'l U and s 4 s 'l Ž x y l t ... 2
Vs2 q w yEV 2 y 2V 3 q V 4 x s 0 where
2
yU q 12 U 3 q Ž Us . s 0,
UG0 ,
Ž 45 .
which yields a piece of an elliptic function. To find a solution for the quintic equation, more effort is needed. After two integrations and using U as an independent variable with W ' Ž dUrds . 2 we have; Y
yU 2 q 12 U 4 y W Ž U . W Ž U . q 14 W X Ž U .
2
s E0 .
A general solution of this equation is unknown but a cubic polynomial, W s P3 ŽU ., may be used to construct a solution Žsee also w12x.. Explicitly, we found that if E0 s 5r54, then W'
dU
ž / ds
2
2
s
5 3 '30 Ž 3 U y U .
is a solution. Now observe that, modulo scalable numerical factors, this equation is identical with Eq. Ž45.. Yet, due to the degeneracy of the first equation at u s 0, the resulting solutions are not identical. This can be only seen from the original Eqs. Ž44.. The third-order equation has a degeneracy at u s 0
P. Rosenaur Physics Letters A 275 (2000) 193–203
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which isolates every half cycle of the periodic solution from its neighbors, turning it into a compacton that does not interact with its neighbors. In the quintic case this solution is a periodic wave which extends over the whole line. This is not overly surprising: we expect similar scaling to imply an affinity between solutions - but certainly not an identity. Indeed, quintic equations also tend to support pulsating travelling structures w13x, a` la breathers, which typically are not observed in third order dispersive equations. Thus, if one has to choose between cubic and quintic equations, enough difference is left to make this an easy decision. Note that combining the two equations in Ž44. which share the same scaling, leads to a more versatile model: u t q Ž u 3 . x q uu x x q 12 Ž u x .
d
2
q x
10
u 5 x s 0.
Ž 46 . Eq. Ž46. has the same scaling as Eq. Ž44. and for d s 1 it is also recognized as the quintic member of the KdV hierarchy. We note in passing that occasionally even a nonmonomial scaling may be represented by a simple equation. For instance, the dynamics of thin layer of water is described fairly well via the Naghdi–Green equations that support the solitary wave Žcf. w14x.;
z s acoshy2 Ž kh . , where
k2s
3a 4 Ž 1 q a.
2 ny 1
of the front u ; x H Ž x ., n ) 1, where H Ž x . is the unit Heaviside function. The leading counterbalancing terms yield 3y n n
u t ; Ž u . x x x ; x ny1 H Ž x . . Thus for n ) 3 the compacton solution cannot satisfy Eq. Ž12. in the classical sense. Indeed, all our previous numerical studies of the K Ž m,n. were limited to the n s 2,3 cases. Similar conclusions apply to Eq. Ž20. as well. Now consider this: if a given function F which vanishes at, say x s 0, and has a weak discontinuity there of a certain order, say re-ex F Ž x . ; x j, j ) 0, then F k where k ) 1, is better behaved than F. Rather than following the evolution of u, we shall utilize this observation to transform the K Ž m,n. equation and follow the evolution of u k . Note that unlike conventional weak solutions, here the regularization is obtained by multiplying the weak solution by its own kind! It is convenient to start with u t q Ž u m . x q q Ž v ,u . where
l2 s 1 q a.
Ž 47 .
Ž 48 .
Eq. Ž48. shares with Eq. Ž47. the same scaling and the same solitary waves! To the extent that examples can be used to infer a rule, the presented ones clearly indicate the fundamental role of scaling.
5. Strong formulation for the flow of compactons The compact solutions generated by the K Ž m,n. Eqs. Ž12. are nonanalytical functions. In the vicinity
Ž 49 .
Ž u s . t q A Ž u sqmy1 . x q s Ž u s u x x . x s 0, ss1q2v ,
Ž 50 .
and A s m srŽ m q 2 v .. For m s 2, Eq. Ž50. may be cast into Õt q Ž ÕL w u x . x s 0 where
Reading the scaling off Eq. Ž47. we consider u t t s u x x q 32 Ž u 2 . x x q 13 u t t x x .
2
q Ž v ,u . ' uu x x q v Ž u x . .
Let u s be a conserved quantity, then Eq. Ž49. may be cast into;
where and
x s 0,
Õ s us ,
and L is a linear operator. This is another twist on the issue of ’hidden’ linear features of nonlinear equations. Clearly, if v ) 0 then s ) 1 and the evolving objects in Ž50. have a higher degree of smoothness. For instance, for m s 2 and v s 1, Eq. Ž49. is recognized as the K Ž2,2. equation, where on the front u ; x 2 H Ž x . and thus the second derivative is discontinuous. Eq. Ž50. is then an alternative description of K Ž2,2.; 2 3
Ž u 3 . t q Ž u 4 . x q Ž u 4 . x x y 3 Ž Ex u 2 .
2 x
s 0.
Ž 51 .
P. Rosenaur Physics Letters A 275 (2000) 193–203
Here, Ž u 3 . t ; Ž u 4 . x x x ; x 5 H Ž x .. More generally: A. We define: S s u nq 1 , a conserved quantity of the K Ž m,n. equation, and R s u n , to obtain 2 nq1
St q
where
2 l
ls
Ž R l . x q Ž R2 . x x y 3Ž R x .
mqn
2 x
s 0,
.
n
Ž 52 .
On the front 2n
R;x
ny1
ny1
thus Ž R . x x x ; x
HŽ x. ,
HŽ x. .
Thus, in terms of R and S, we have a strong solution. After each time step, R has to be re-calculated from S via R s S , therefore this procedure becomes impractical for odd n’s if, due to numerical oscillations, S takes negative values in the vicinity of the front. To overcome this difficulty we use an alternative description; B. Let Z s u ny 1 and re-express the K Ž m,n.-equation as n nq 1
1 n
nq my2
ž
Z t q B0 Z
ny 1
/ q q Ž a ,Z . x
x
3
s B1 Ž Z 2r3 . x .
Ž 53 .
Here, B0 s and
m Ž n y 1. n Ž n q m y 2.
as
4yn 2 Ž n y 1.
,
B1 s
27 Ž n y 2 . 8 Ž n y 1.
2
,
Ž 56 .
Eq. Ž56. is the Lagrange image of the m-KdV Žsee Eqs. Ž10. and Ž14... Near the front u ; x 2r3 H Ž x . and we encounter the same difficulty as for the K Ž m,n s 4.. In terms of Z s u 3 we obtain 3
Zt q 2 Ž Z 2 . x q w ZZ x x x y Z x Z x x x s y1.5 Ž Z 2r3 . x . Note that the second term due to dispersion acts like a nonlinear diffusion. In the K Ž2,2. case the positive sign in front of this term acts like a backward diffusion that counteracts the tendency of the compacton to develop oscillations on its rear part. The need for this kind of stabilization is perhaps best manifested noting that if in Ž49. you take v s y1r2, the resulting equation has only a purely dispersive term uu x x x and does not seem to support the propagation of compactons w5–7x. In our case the negative sign in front of Z x Z x x , will hasten the decomposition of the compacton into a train of waves. However, if instead of Z we introduce Y s u 3r2 , then 3
Yt q 4Y 2 Yx q Y Ž YYx x . x s 23 Ž Yx . .
.
Now near the front any K Ž m,n. solution yields; Z ; x 2 H Ž x .. The right hand side of Ž53. is the ’penalty’ for using a variable which is not a conserved quantity. To appreciate the benefits of using Z consider the K Ž m s n,n. case. Now the compactons take the form Z s l Z0 cos 2 Ž v 0 s . ,
Numerical experiments reveal that Eq. Ž55.is very easy to integrate and the interaction seems very close to be elastic. In contrast, the singular behavior at the edge of its ’raw version’, the K Ž4,4., poses a considerable difficulty and the interaction is not nearly as clean w15x. Numerical stability of the compacton poses another limitation on what is an acceptable representation. Consider, for instance, u t q Ž u 4 . x q Ž u 3 u x x . x s 0.
3qn 2
201
Z0 s const.
for <) < F pr2, Ž 54 .
and vanish elsewhere. For n s 3,4; Z0 s 3r2,8r5, and v 0 s 1r3,3r2. Thus, for instance, K Ž4,4. reads; 3
Zt q 2 Ž Z 2 . x q 4 w ZZ x x x x s 3 Ž Z 2r3 . x .
Ž 55 .
Ž 57 .
In Ž57. the diffusive part, YYx Yx x , has a favorable sign for stability. Although in terms of Y the solution is not as smooth as in terms of Z, the last equation is better suited to describe a stable evolution of compactons. In fact, its numerical evolution is straight forward w15x. A final comment is in order: the fact that in one representation the problem is numerically stable while in another it is not, makes one wonder whether these are merely numerical difficulties, or do they represent a deeper issue related to the fact that we are dealing with mapping of weak solutions. Although in general the mapping does not seem to have an effect on the emerging structures, a subtle differences emerge when a change of parity is involved. For
P. Rosenaur Physics Letters A 275 (2000) 193–203
202
instance, in the K Ž3,3. equation all terms are odd and thus it is invariant under u ™ yu. Thus both compactons and anti-compactons Žyu. move in the same direction. However in Z-representation Žsee Eq. Ž53.. the solution is invariant under Z ™ yZ and t ™ yt, which means that the anti-compactons move in opposite direction. C. In the case of Eq. Ž20. and m s n q 1, we use its conserved quantity Z s u n to obtain; Zt q
nq1
where
2
Ž Z 2 . x q n q Ž v ,Z . x s 0 ,
v s Ž 1 y n . r2 n,
Ž 58 .
which is recognized as Eq. Ž49.. Since Z ; H Ž x . x 2 on the front, Z satisfies Ž58. in the conventional sense. D. Finally, we comment upon the difference between admissible and non-admissible singularities. Let u t q 2 u m u x q u x x x s 0.
Ž 59 .
Integrating twice Ž s s x y l t, and A and B are consts.., yB y Au y l u 2 q
4
Ž m q 1. Ž m q 2.
u mq2 q u 2s s 0.
Ž 60 . For m s 2 and m s 4 there is a special trigonometric solution, m y KdV ; where
us"
4k cos 2 Ž k s .
,
3 y 2cos 2 Ž k s .
l s 4k 2 ,
Ž 61 .
and m s 4;
us"
10 1r4 'k cos Ž k s .
(3 y 2cos Ž k s . 2
,
now l s k 2 .
To find Ž61. set B s 0,Y s Y0 q 1ru and let the constants A and Y0 be chosen such that in Y units the new potential retains only cubic and quadratic parts. Then, in terms of Z 2 s 1rY, Ž61. is obtained. The second case follows from the first, if one takes A s 0, B / 0 and u ´ u 2 .
We continue with the m-KdV. Let Õ s u 2 , then in terms of Õ Eq. Ž59. reads Õt q Ž Õ 2 . x q Õ x x x y
3Õ x2
ž / 4Õ
s 0.
Ž 62 .
x
For which solution Ž61. becomes Õs
16 k 2 cos 4 Ž k s . 3 y 2cos 2 Ž k s .
2
.
Ž 63 .
Now suppose one tries to generate a compacton from the periodic solution Ž61.. To this end, restrict the solution to the < s < F pr2 k interval and set it to zero elsewhere. Since u ; x 2 H Ž x . near the edge of the compactified solution, the third derivative will generate a d Ž x . function which cannot be counterbalanced anywhere in the equation. Next, consider Eq. Ž62. and the the compactification of Õ in Ž63.. Now; Õ ; x 4 H Ž x . at the front and the third derivative is continuous! Does this mean that the compactified form of Ž63. is an accepted solution of Eq. Ž62.? It would be very nice if that was the case – unfortunately it is not because, A: since the m-KdV is integrable one can in principle, use the compactified version of Ž61. as an initial datum to solve the inverse scattering problem. The non-analyticity has to generate some radiation. Remapping this solution in terms of Õ, will not eliminate the radiative part. B: Unlike the K Ž4,4. equation where u ; x 2r3 H Ž x . near the edges of its compacton, but the singularity due to the dispersion is exactly counterbalanced by the singularity of u t , in Eq. Ž62. the singularity due to dispersion is not balanced by any other term in the equation. The use of Õ in Ž62. only postpones the appearance of the unbalanced singularity by two orders but does not change the effect. Taking two more derivatives in Ž62. will uncover it. Thus, the ’compactified solution’ of Ž63. cannot persist and has to decompose.
Acknowledgements I thank R. Camassa and P. Olver for a number of enjoyable and illuminative discussions. This work
P. Rosenaur Physics Letters A 275 (2000) 193–203
was supported in part by a grant from the Israel Science Foundation and carried out in part during a stay at The Center of Nonlinear Studies at the LosAlamos National Laboratory and visit to The Courant Institute of Mathematical Sciences supported via Air-Force grant a F49620-98-1-0256.
203
front of this kink is a paraboloid, but unlike the purely dispersive case, it is not followed by a receeding front. In the overdamped case the kink has a monotone profile; U n'Vs
l 4
½
1 y exp Ž m nh .
cosh dn Ž h q h 0 . cosh Ž dnh 0 .
5
, q
Appendix A. A dispersive-dissipative interaction
h F 0,
We consider travelling wave solutions of Eq. Ž26. Žsee Ref. w16x for another approach to dispersive-dissipative interactions.. We have to solve
and vanishes elsewhere. Here dn s Dn r2 and h 0 is found via dnh 0 s ytanhy1 Ž m nrdn .. Here apart from the front, dispersion plays a secondary role.
Ž A.3 .
(
Vhh y m nVh q 4V s l , where V s U n
and
If Vn s expŽgh . then m2n y 16. The modes pending on whether underdamped case let
mn s
ž
nq1 n
/
m.
Ž A.1 .
(
2g "s m n " Dn where Dn s are over Žunder. damped dem ) Ž-. 4 nrŽ n q 1.. In the v n ' y Dn r2 and set
(
l Vs
4
1 y aexp Ž hm n . cos
vn Ž h q h0 .
4q.
The constants a and h 0 are found demanding that at, say, h s 0 both V and V X vanish. This determines a, while h 0 is found via v nh 0 s tany1 Ž m nrv n .. The resulting solution reads; U n'Vs
h F 0,
l 4
½
1 y exp Ž m nh .
cos v n Ž h q h 0 . cos Ž v nh 0 .
5
, q
Ž A.2 .
and vanishes elsewhere. This is a semi-compact oscillating kink. In the original x y y y z space the
References w1x P. Pelce´ ŽEd.., Dynamics of curved fronts, Academic Press, 1988. w2x R.E. Goldstein, D.M. Petrich, Phys. Rev. Lett. 67 Ž1991. 3203. w3x K. Nakayama, M. Wadati, J. Phys. Soc. Jpn. 62 Ž1992. 473. w4x K. Nakayama, H. Segur, M. Wadati, Phys. Rev. Lett. 69 Ž1992. 3425. w5x P. Rosenau, J.M. Hyman, Phys. Rev. Lett. 70 Ž1993. 564. w6x P. Rosenau, Phys. Rev. Lett. 73 Ž1994. 1737. w7x P. Rosenau, Phys. Lett. A 230 Ž1997. 305. w8x P. Rosenau, Phys. Lett. A 211 Ž1996. 265. w9x R. Camassa, D.D. Holm, Phys. Rev. Lett. 71 Ž1993. 1661. w10x J. Rubinstein, personal communication. w11x P.G. Drazin, R.S. Johnson, Solitons: an Introduction, Cambridge University Press, Cambridge 1990. w12x S. Kichenassamy, P.J. Olver, Siam J. Math. Anal. 23 Ž1992. 1141. w13x J.M. Hyman, P. Rosenau, Phys. D 123 Ž1998. 502. w14x W. Choi, R. Camassa, to appear in J. Fluid Mech. w15x J.M. Hyman, P. Rosenau, in progress. w16x P. Rosenau, Phys. D 123 Ž1998. 525.
Physics Letters A 275 Ž2000. 193–203 www.elsevier.nlrlocaterpla
Compact and noncompact dispersive patterns Philip Rosenau School of Mathematical Sciences, Tel AÕiÕ UniÕersity, Tel AÕiÕ 69978, Israel Received 17 November 1999; received in revised form 23 August 2000; accepted 23 August 2000 Communicated by C.R. Doering
Abstract We discuss the pivotal role played by the nonlinear dispersion in shaping novel, compact and noncompact patterns. It is shown that if the normal velocity of a planar curve is U s yŽ k n .s , n ) 1, where k is the curvature, then the solitary disturbances may propagate like compactons. We extend the KP and the Boussinesq equations to include nonlinear dispersion to the effect that the new equations support compact and semi-compact solitary structures in higher dimensions. We also discuss the relations between equations sharing the same scaling. We show how compacton supporting equations may be cast into a strong formulation wherein one avoids dealing with weak solutions. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The underlying theme of this work is the impact of nonlinear dispersion on the formation of patterns. We shall be concerned with four topics; A: The motion of curÕes in a plane. This subject has drawn a considerable interest in recent years, see Refs. w1–4x and references therein, and provides a natural setup for the introduction of compactons – solitary waves with a compact support – which are perhaps the simplest manifestation of nonlinear dispersion at work. If the velocity of the normal to the curve is assumed to be U s yŽ k n .s , n ) 1, the nonlinear dispersion enters naturally into the intrinsic description of the curve, and supports the propagation of both compact and noncompact patterns. The resulting partial differential equations are exactly of
E-mail address: [email protected] ŽP. Rosenau..
the type encountered in our previous studies of compactons w5–7x. B: Compact structures in higher dimensions. To this point in time, the wonders of integrability – which provide the essential tool to study the interaction of solitons – have eluded us in higher dimensions. The equations derived using the reductive perturbation technique, do not seem to be integrable, unlike the 1-D case. Rather then resign ourselves to the view that in higher dimensions true solitons cannot exist, we argue that in order to support localized patterns, the enhanced spread of waves in higher dimensions has to be counteracted by adequately stronger nonlinearities. The nonlinear dispersion presented in this work seems to provide one such mechanism. We base our approach on a certain affinity between pdeX s having their highest order operator degenerate at the front. Consider u t s Ž u n . x x and u t q Ž u n . x x x s 0, which have the highest order operators degenerate at u s 0. In spite of the obvious differences between these equations, in both cases
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 5 7 7 - 6
194
P. Rosenaur Physics Letters A 275 (2000) 193–203
the degeneracy at the front generates sharp fronts, and supports the formation and propagation of, robust, compact patterns. The fact that compact patterns emerge in the parabolic case in higher dimensions as well as in 1-D, leads to the conjecture that analogous dispersive equations in higher dimensions will also support localized patterns. To this end we endow a number of well known equations with a non-linear dispersion and construct, at least formally, compact and semic-mpact structures in higher dimensions. Even though, at this stage, these equations are formal mathematical extensions of known models, if the resulting patterns turn out to be robust, this will provide an impetus to seek physical realization of these equations. We note that in the case of the anharmonic lattice, the motion of the compacton represents a regime wherein the anharmonic forces completely dominate over the harmonic ones. This regime is thus far away from the conventional soliton domain, rather than being a perturbation and occurs either at large amplitudes Žand thus is not accessible to perturbative procedures. or, if ab initio the harmonic forces are negligible Že.g; if the chain is made of hard spheres.. In this case the linear speed of sound is very small and thus the transition to a fully nonlinear regime takes place at very low amplitudes. Of course, equations derived in the weakly nonlinear regime may sometimes develop large amplitude andror gradients, but the value of the information gathered in the high end is quite questionable. For instance, the ion acoustic solitons are adequately approximated by the Korteweg–de Vries ŽKdV. solitons. In the weakly nonlinear regime their behavior at large amplitudes has nothing to do with the real phenomenon, since the actual ion-acoustic solitons completely disappear at a critical amplitude, and a collision-free turbulent regime sets in! C: Equations defined Õia the inÕariance of scales. In many cases, though the underlying strongly nonlinear processes are quite complex, and thus very hard to model, the emerging patterns are remarkably simple oÕer a wide range of scales. In an effort to exploit this simplicity within the context of dispersive phenomena Žthough by no means limiting ourselves to such phenomena., we propose in Section 4 the following phenomenological route toward this goal: It has been observed by many, in a variety of
contexts, that the scaling relations of a process are its vital feature. Thus, if we have a handle on that property, whether from first principles or empirically, and then construct mathematical modelŽs. endowed with the correct scaling, we can expect that the resulting model will correctly immitate the actual process. It is important to observe that a given scaling does not define a unique model, but rather a sequence of model equations. The examples studied in Section 4 indicate that equations sharing the same scaling share similar patterns. This also works vice versa- in the aformentioned ion-acoustic problem, if one compares the interaction of its solitons using the BBM equation Ž u t q uu x s u x x t . and the KdV equation, then the patterns rendered by KdV, particularly away from very small amplitudes, are in much better agreement with those of the true interaction. The superiority of the KdV equation is thus not due to its integrability, after all the actual problem is not integrable, but due to the the fact that its width-height scaling relations are much closer the to the actual process! The classical argument presented in favor of the BBM that it has a better linearized dispersion relation, does not come into play as far as the interaction of solitons is concerned. D: Strong formulation of compacton carrying equations. By definition compact patterns are nonanalytical entities on the frontlineŽs.. In dispersive processes the presence of even a weak singularity poses a formidable numerical challenge, which was partially described in Refs. w5–7x and in recent literature devoted to the numerical aspects of the problem. In Section 5 we show that, at least for the considered class of nonlinear problems, one can circumvent those difficulties by casting the problem into an alternative formulation in which all singularities are weakened to the extent that all the present derivatives are smooth. Preliminary numerical experiments reveal that this representation has a dramatic impact on the numerics, and problems which originally were unaccessible are now easily dealt with.
2. Motion of curves in the plane Let a plane curve be represented by r s Ž j ,h .
;
P. Rosenaur Physics Letters A 275 (2000) 193–203
where Ž j ,h .Ž s,t .. Let t s E rrE s be the tangent vector to the curve where s is the arc length. Define g' t P t ;
Eh Ej ns y , Es Es ;
ž
and
;
'g ,
dt
r Ž s,t . s w Ž s,t . t Ž s,t . q U Ž s,t . n Ž s,t . ,
;
Ej Eh
U s yk ,
g s x and
z s y Ž x ,t . ,
Ž 2a .
zx x 1 q z x2
s0 .
Ž 3.
Let V s z x , then taking one derivative in Ž3. we have
q
n s 0.
Ž 9. E3
E
nq1 E s
Žk
nq2
.q
E s3
Ž kn. s0
n / 0,
Ž 10 .
1qV 2 x
H0 Vdx ,
/
s0 .
Ž 4.
x
r s Vy1 and t s t ,
then the Lagrange map Ž5. yields rh rt q s0 . 1qr2 h
ž
/
Ž 5.
E 2U
q k 2U q
Ek
Ž 6.
Ž k n . , n / 0,
Es
ks
2
ž /
s 0 for
k
n s 0.
Ž 11 .
E um
Eu
E 3un
q q s 0. Ž 12 . Et Ex E x3 For n s 1, Eq. Ž10. reduces to the m-KdV equation. For n / 1, the dispersion is nonlinear and three values of n are of special interest: n s y1r2 which is related to an integrable case, and the n s 2,3 cases which support compactons. For n s y1r2 ;
E3
E
Ž k 3r2 . q
Ž ky1r2 . s 0 . Ž 13 . Et Es E s3 This equation is a Lagrange image of an integrable problem. Indeed, in terms of y
s
us
H0 kds,
t s t.
Ž 14 .
nk 2
Ek q
E3
E
n q 1 Eu
q
Eu 3
k nq 1 s 0 ,
n / 0,
which, for n s y1r2, we recognize in terms of r s 1rk as an integrable extension of the Hary Dym equation w5–9x. We turn to the n s 2 case. Now 2 E
E3
Ž 7.
k2s0 . Et 3 Es E s3 This is the K Ž4,2. equation. A family of travelling solutions is given, via h s s y l t,
Ž 8.
k s 8 l3 Z 2 l1r3ah ,
E Es
Es
K Ž m,n . ;
Ek
s
H kUds . Es
E2 q
We shall revisit Eq. Ž11. in Section 4. Eq. Ž10. is a particular Ž m s n q 2. case of the K Ž m,n. equation introduced in w5–7x;
Et
Et E s2 Let us first assume that Usy
Ek2 q
Eq. Ž10. is mapped into Žcf., ref w8x.
The invariance of Eq. Ž4. under Ž5. enables us to generate from a new solution form a given one. Direct use of the Serret–Frenet representation leads to an intrinsic representation of the motion via the curvature w1–4x; s
and
Ek
Vx
Now let
Ek
,
n
Ž 2b .
where k is the curvature, then in Ž x, y . coordinates this choice of U leads to
hs
Et for
Et
Eh Ej
y y g 1r2 U s 0 . Es Et Es Et Firstly, note that, if
ž
Ek
Ek
which leads to
Vt q
k Es
;
Ž 1.
zt q
1 Ek
Thus
a unit normal vector. The dynamics of the curve is then specified via d
and Usy
/
195
q
k4q
a s 2 1r3r'6 ,
P. Rosenaur Physics Letters A 275 (2000) 193–203
196
and Z is given implicitly via Z
hs"
X
H0 dz Ž 1 y z
X 6 y1 r2
.
,
0 F Z F 1.
The compact solution is obtained keeping half a period of the solution Žbetween two consecutive minima. w5–7x, and setting k to zero elsewhere. The compacton thus obtained has a weak discontinuity at the edge of the pulse. Nevertheless, it is easily seen to be a conventional solution of Ž10. w5–7x. Similarly, for n s 3, the compacton of the resulting K Ž5,3. equation is given via the half period of k s " Ž 2 l.
1r4
Z Ž lr72 .
1r4
h ,
with Z cut out of Z
hs"
X
H0 dz Ž 1 y z
X 4 y1 r2
.
,
0FZF1 .
The compact span of k means that, at a given time, the deformationŽs. of the curve can be completely localized with other parts of the curve being completely unaffected by its motion. For a more general class of equations we replace U with F given via Us
1 EF
.
k Es
Et
E s2
Fs
ž /
q
E Ž kF . Es
k
.
Ž 16 .
Let us apply the Lagrange map Ž14. to Ž18.. We have
Ek Et
E2
EF
2
sk 1q
Eu
2
Eu
with
Us
EF Eu
.
Ž 17 .
In terms of radius of curvature, r s 1rk, the resulting equation is even simpler;
Er Et
3r2
Zt q Ž 1 y Z 2 . w Z x x x y Z x x s 0. Eq. Ž19. is an integrable extension of the integrable case, due to Wadati et al. w8x. It follows from Ž18. that Ž19. is also is invariant under the Lagrange map Ž14.; GŽ r . s rr '1 q r 2 . Thus GŽ r . s ry2 ,ry1r2 , and rr '1 q r 2 represent three integrable cases. In terms of Z the solitary structure of Ž19. is obtained via y2 l 1 y '1 y Z 2 y Z 2 q Z s2 s E . Taking E s 0, for l ) y1r2 we have a solitary wave with a sharp peak at Z s 1 Žwhich corresponds to k ™ `.. This is yet another manifestation of nonanalyticity due to nonlinear dispersion. For y1 - l - y1r2 a soliton with a smooth shape is obtained. 3. Compact structures in higher dimensions
E2 s
or
Ž 15 .
Instead of Ž7. we now obtain a local form
Ek
Eq. Ž18. could be solved exactly and the results re-mapped into other coordinates. This issue will be addressed elsewhere. For now let: F s 1r '1 q k 2 which leads to Ek E3 E k q y Z s 0 where Z s 3 '1 q k 2 , Et Es Es Ž 19 .
E2 q 1q
Eu
2
EG Eu
s 0 here
G Ž r . s F Ž 1rr . .
Ž 18 . It thus appears that u is the natural independent variable to use – it leads to what appears to be the simplest description of motion. Note the linear operator acting on GŽ r .. If GŽ r . were linear in r, or given via a linear operator acting on r; G s LŽ Eu .w r x,
In this section we explore a number of formal mathematical extensions of soliton-supporting equations with the aim of producing compact dispersive structures in higher dimensions. In what follows we shall use another variant of the K Ž m,n. Eq. Ž12. u t q a Ž u m . x q u Ž u n . x x x s 0, m ) 1. Ž 20 . The assumed nonlinear dispersion emerges in nonlinear lattices and reflects the dispersive effect due to the discrete nature of the intersite potential: Ž y kq 1 y y k . nq 1 w5–7x. For m s 2, and n s 1, Eq. Ž20. was also recently shown to describe the dispersion of dilute suspensions w10x. The solitary structures of Eq. Ž20. and the K Ž m,n. are quite similar. For instance, for m s n q 1 we have 1
u c Ž x ,t . s
½
2l a
cos 2
< x y lt < F p and u c s 0 otherwise.
'a 2
Ž x y lt .
5
n
,
Ž 21 .
P. Rosenaur Physics Letters A 275 (2000) 193–203
A. Now let us consider
E u t q 4 Ž u nq 1 . x q u Ž u n . x x
Ex
xym
Ž u nq1 . x x
To unfold the motion in the original variables we write h in terms of x and t which for n s 1 gives in 3 D;
l q = H2 u s 0,
Ž 22 .
where = H ' E y2 q Ez2 Ž N y 2., N s 2Ž3. in 2Ž3.-dimensions. Eq. Ž22. generalizes the well known K–P equation in two ways; Ža. it adds a dissipative component and, Žb. the dispersive part is nonlinear. Let us recall that the K–P equation supports supports one-dimensional solitary waves in space but these waves may have arbitrary spatial inclinations. A variant of Ž22., based on the extension of the K Ž2,2. was presented in w5–7x. We now define new variables Ž t 0 s const.. U Ž s,t .
us
Ž t q t0 .
ar2
a s N y 1,
,
N s 2,3,
Ž 23 .
us
2Ž t q t0 .
2
,
r 2 s y 2 q z 2 Ž N y 2. ,
Ž 24 .
and v
ts
½
t 0 Ž t q t 0 . qt 1 , ln Ž t q t 0 . ,
v '1y n Ž N y1. r2 / 0. . v s 0.
Ž 25 .
Note that v G 0 only if n F 2rŽ N y 1.. Clearly, the dimension has a crucial impact on the rate of propagation and the attenuation of the wave. Upon substitution and one integration Žignoring the constant of integration. we have
EU Et
E q
Es
U Ž U n . s s q 4U nq1 y m Ž U nq1 . s 4 s 0 . Ž 26 .
In particular, when m s 0, the last equation supports propagation of compactons, cf., Ž21., found via the solution of U yl q 4U n q Ž U n . hh s 0 where h s s y lt . Ž 27 .
(
2 t q t0
N s 2,3,
4 Ž t q t0 .
y lln Ž 1 q trt 0 . ,
,
l
4Ž t q t0 .
y2 qz2
Ž 28 .
and u vanishes elsewhere. Note that at t s 0 the support of the solution is < x q r 2r4t 0 < F pr2. The forward front is then given via r f2 s 2 t 0 Žp y 2 x . and the trailing front via r b2 s 2 t 0 Žy2 x y p .. Each front describes an evolutionary paraboloid, that propagates with a logarithmic speed llnŽ1 q trt 0 ., and Žsince r 2 is compressed by the factor 4 t . also deforms in time. In addition, the motion is accompanied by an attenuation of the amplitude. In 2D the motion takes the form us
ssxq
cos 2 x q
p
where r2
197
cos 2 x q
y2 4Ž t q t0 .
(
y l t q t0 ,
p 2
,
Ž 29 .
and vanishes elsewhere. In both two- and three dimensions the nature of the disturbance is such that an observer at a fixed point in space will first experience an oncoming front of the wave. This will continue for a while but will come to its end at some point in time, after which he will no longer observe any wave. We note in passing that, up to this point in our discussion we could have replaced the dispersive part with Ž u nq 1 . x x x and, apart of numerical factors, the results would essentially be the same. However, the present form is more convenient when dissipation is included Žsee the Appendix.. In conclusion we note that the obtained patterns are only partially localized and thus cannot be considered a genuine generalization of the solitary waves. These are considered next. B. We now present a formal ‘1–1r2’ dimensional extension of Eq. Ž20.; u t q Ž u 2 . x q d w un u x x s 0 where n ' Ex2x q E y2y .
Ž 30 .
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Here the wave travels in x-direction but has also a structure in y-direction. Integrating once in a travelling frame, h s x y l t, d s 1, we obtain n u q u s l.
Ž 31 .
(
If r s h 2 q y 2 , then a radially symmetric, compact, solution reads usl 1y
J0 Ž r . J0 Ž ri .
,
0 F r F ri ,
Ž 32 .
and vanishes elsewhere. Here Jj Ž r . is the j-th order Bessel function, and ri are the roots of J1. Note that, since the Bessel function decays in space and the spacing of its roots changes, unlike the 1D case, the subsequent zeroes of J1 generate different multihump compact structures Žthe presence of mixed sign parts in all but the first mode, will probably cause only the first mode to be stable, cf., Ref. w5–7x.. Eq. Ž30. has a remarkable structure; it may be cast as L Ž u . s d n u q u, Ž 33 .
u t q Ex uL Ž u . s 0 where
is a linear operator which enables us to find explicit solutions. Let L be any linear operator with constant coefficients and u s F Ž x . a solution of LŽ u. s l. Then all that is needed to solve the original problem is to ensure that u t q l u x s 0 is satisfied. Take uŽ x,t . s F Ž x y l t . and you are done. Also note that, both the dissipative extension of Eq. Ž20.; u t q a Ž u nq 1 . x q u Ž u n . x x
xsb
Ž u nq1 . x x ,
and the viscous flow in a thin layer, u t q Ž u 2 . x q Ž uu x . x q d Ž uu 3 x . x s 0,
Ž 34 .
m s 1,2,
xx
,
Ž 35 .
describe, for a s 0, the vibrations of a purely anharmonic lattice Žsee w5–7x., and as can be easily seen, support travelling structures with a compact support. For a s m s 1 we rewrite Eq. Ž35. as u t t s u x x q Ž uLu . x x ,
where
(
u s F1 Ž x q l t . q F2 Ž x y l t . ,
(
where l s 1 q a 1 q a 2 is also a solution. We thus obtain a special case of a clean interaction of two waves. The above analysis gives the impression that two compactons will cross each other without really interacting, but this is wrong; the additive property holds either for analytic functions or noninteracting compactons. In the later case the nonanalyticity at the edge of each compacton is compensated in the product uLw u x by the vanishing u. During the interaction between two compactons, u does not vanish in the intersecting domain and this enables the product uLw u x to induce interaction between compactons.
4. Equations defined via scales invariance Consider the following two partial differential equations ux ut q Ž u2 . x q s 0, Ž 36 . u xx
ž /
which was introduced in Section 1 ŽEq. Ž11.. and the Benjamin–Ono ŽBO. equation Žcf. w11x. ut q Ž u2 . x q H ( u )
x x s 0.
Ž 37 .
H ( u ) is the Hilbert Transform of u. Insofar as travelling structures are concerned, the two equations share the same scaling; u s l F lŽ x y l t . .
Ž 38 .
Question: how close is the dynamics of two different equations, if their travelling structures share the same scaling? To look more closely into this issue, we assume a given scaling
have similar ’hidden’ linear features. C. The Boussinesq equationŽs. u t t s au x x q Ž u mq 1 . x x q b u Ž u m . x x
If Fi Ž x . is a solution of Lw u x s a i , a i s const.Ž) y1., then F Ž x q l t ., where l s " 1 q a i , is not only a solution, but the special superposition of the two solutions
L w u x s Ž bE 2 q a . u.
u s l a F Ž l bh .
where h s x y l t.
Ž 39 .
and look for differential equations sharing this property. In this setting the question is too general. To make a progress we narrow our scope and start with model equations of the type u t q Ž u m . x q E r Ž u n . s 0,
Ž 40 .
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with the adjustable ’parameters’ m,n, and r at our disposal. When the integer r is even, to ensure the dispersive nature of the third term, E r , will be understood via the Hilbert Transform. For instance; E 2 will be understood as E 2 H ( u ). Assuming a given scaling law Ži.e., a given a and b ., one has
199
For y1 - E - 0 we have u "s
l< E < 1 . '1 y < E < cos Ž l'< E < h .
.
Thus, m is fixed, but otherwise we have a one parameter family of possible equations corresponding to a given scaling law. In the example considered, in addition to the two cases presented, all equations of the form
As E x0 the uq soliton transforms into a conventional soliton. Its shape coincides exactly with the shape of the BO soliton Ž43.!. Thus Eq. Ž36. supports two kinds of solitary waves; a solitary family that decays exponentially and a solitary wave which decays algebraically. It would be of some interest to determine whether the BO can also support solitons with an exponential tail. As an another example let us look at the two equations
u t q Ž u 2 . x q E r Ž u 3y r . s 0
u t q Ž u 3 . x q uu x x q 12 Ž u x .
m s 1 q 1ra and 1 q a q b s a n q b r .
Ž 41 .
Ž 42 .
share the same scaling. Let us now look more closely into the affinity between Eqs. Ž36. and Ž37.. The BO is integrable and its one soliton solution is given via us
2l 1 q l2h 2
,
where h s x y l t.
Ž 43 .
We can easily calculate the travelling structures of Eq. Ž36.; let u s lV Ž s ., s s lh , then after two integrations we obtain
and
E s const.
Unlike the familiar semilinear cases, where the integration constant plays the role of a ’total energy’ of the system, in this case we have a fundamentally different situation where the integration constant E affects the actual shape of the potential function. The solution has two branches; E)0 u "s
lE "'1 q E cosh Ž l'E h . y 1
.
The most important thing to observe about these solitary waves is that for each speed l we have in terms of E an entire one parameter family of solitary waÕes. Note also that the uq solitons are taller than their uy cousins and that both solitary waves travel to the right. However, since, Eq. Ž36. is invariant under u ™ yu and t ™ yt, the yuy Ž) 0. and the yuq Ž- 0. solitary waves travel to the left.
x
s0
u t q Ž u 3 . x s u5 x ,
Ž 44 .
which share the same scaling. The chosen quadratic form for the dispersion is a particular case of w uu x x q a Ž u x . 2 x x considered in the next section and, as can be easily seen, has a Hamiltonian structure. Both equations conserve mass Ž H x udx ., momentum Ž H x u 2 dx . and energy. The solitary solution of the first equation is found solving Ž u s 'l U and s 4 s 'l Ž x y l t ... 2
Vs2 q w yEV 2 y 2V 3 q V 4 x s 0 where
2
yU q 12 U 3 q Ž Us . s 0,
UG0 ,
Ž 45 .
which yields a piece of an elliptic function. To find a solution for the quintic equation, more effort is needed. After two integrations and using U as an independent variable with W ' Ž dUrds . 2 we have; Y
yU 2 q 12 U 4 y W Ž U . W Ž U . q 14 W X Ž U .
2
s E0 .
A general solution of this equation is unknown but a cubic polynomial, W s P3 ŽU ., may be used to construct a solution Žsee also w12x.. Explicitly, we found that if E0 s 5r54, then W'
dU
ž / ds
2
2
s
5 3 '30 Ž 3 U y U .
is a solution. Now observe that, modulo scalable numerical factors, this equation is identical with Eq. Ž45.. Yet, due to the degeneracy of the first equation at u s 0, the resulting solutions are not identical. This can be only seen from the original Eqs. Ž44.. The third-order equation has a degeneracy at u s 0
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which isolates every half cycle of the periodic solution from its neighbors, turning it into a compacton that does not interact with its neighbors. In the quintic case this solution is a periodic wave which extends over the whole line. This is not overly surprising: we expect similar scaling to imply an affinity between solutions - but certainly not an identity. Indeed, quintic equations also tend to support pulsating travelling structures w13x, a` la breathers, which typically are not observed in third order dispersive equations. Thus, if one has to choose between cubic and quintic equations, enough difference is left to make this an easy decision. Note that combining the two equations in Ž44. which share the same scaling, leads to a more versatile model: u t q Ž u 3 . x q uu x x q 12 Ž u x .
d
2
q x
10
u 5 x s 0.
Ž 46 . Eq. Ž46. has the same scaling as Eq. Ž44. and for d s 1 it is also recognized as the quintic member of the KdV hierarchy. We note in passing that occasionally even a nonmonomial scaling may be represented by a simple equation. For instance, the dynamics of thin layer of water is described fairly well via the Naghdi–Green equations that support the solitary wave Žcf. w14x.;
z s acoshy2 Ž kh . , where
k2s
3a 4 Ž 1 q a.
2 ny 1
of the front u ; x H Ž x ., n ) 1, where H Ž x . is the unit Heaviside function. The leading counterbalancing terms yield 3y n n
u t ; Ž u . x x x ; x ny1 H Ž x . . Thus for n ) 3 the compacton solution cannot satisfy Eq. Ž12. in the classical sense. Indeed, all our previous numerical studies of the K Ž m,n. were limited to the n s 2,3 cases. Similar conclusions apply to Eq. Ž20. as well. Now consider this: if a given function F which vanishes at, say x s 0, and has a weak discontinuity there of a certain order, say re-ex F Ž x . ; x j, j ) 0, then F k where k ) 1, is better behaved than F. Rather than following the evolution of u, we shall utilize this observation to transform the K Ž m,n. equation and follow the evolution of u k . Note that unlike conventional weak solutions, here the regularization is obtained by multiplying the weak solution by its own kind! It is convenient to start with u t q Ž u m . x q q Ž v ,u . where
l2 s 1 q a.
Ž 47 .
Ž 48 .
Eq. Ž48. shares with Eq. Ž47. the same scaling and the same solitary waves! To the extent that examples can be used to infer a rule, the presented ones clearly indicate the fundamental role of scaling.
5. Strong formulation for the flow of compactons The compact solutions generated by the K Ž m,n. Eqs. Ž12. are nonanalytical functions. In the vicinity
Ž 49 .
Ž u s . t q A Ž u sqmy1 . x q s Ž u s u x x . x s 0, ss1q2v ,
Ž 50 .
and A s m srŽ m q 2 v .. For m s 2, Eq. Ž50. may be cast into Õt q Ž ÕL w u x . x s 0 where
Reading the scaling off Eq. Ž47. we consider u t t s u x x q 32 Ž u 2 . x x q 13 u t t x x .
2
q Ž v ,u . ' uu x x q v Ž u x . .
Let u s be a conserved quantity, then Eq. Ž49. may be cast into;
where and
x s 0,
Õ s us ,
and L is a linear operator. This is another twist on the issue of ’hidden’ linear features of nonlinear equations. Clearly, if v ) 0 then s ) 1 and the evolving objects in Ž50. have a higher degree of smoothness. For instance, for m s 2 and v s 1, Eq. Ž49. is recognized as the K Ž2,2. equation, where on the front u ; x 2 H Ž x . and thus the second derivative is discontinuous. Eq. Ž50. is then an alternative description of K Ž2,2.; 2 3
Ž u 3 . t q Ž u 4 . x q Ž u 4 . x x y 3 Ž Ex u 2 .
2 x
s 0.
Ž 51 .
P. Rosenaur Physics Letters A 275 (2000) 193–203
Here, Ž u 3 . t ; Ž u 4 . x x x ; x 5 H Ž x .. More generally: A. We define: S s u nq 1 , a conserved quantity of the K Ž m,n. equation, and R s u n , to obtain 2 nq1
St q
where
2 l
ls
Ž R l . x q Ž R2 . x x y 3Ž R x .
mqn
2 x
s 0,
.
n
Ž 52 .
On the front 2n
R;x
ny1
ny1
thus Ž R . x x x ; x
HŽ x. ,
HŽ x. .
Thus, in terms of R and S, we have a strong solution. After each time step, R has to be re-calculated from S via R s S , therefore this procedure becomes impractical for odd n’s if, due to numerical oscillations, S takes negative values in the vicinity of the front. To overcome this difficulty we use an alternative description; B. Let Z s u ny 1 and re-express the K Ž m,n.-equation as n nq 1
1 n
nq my2
ž
Z t q B0 Z
ny 1
/ q q Ž a ,Z . x
x
3
s B1 Ž Z 2r3 . x .
Ž 53 .
Here, B0 s and
m Ž n y 1. n Ž n q m y 2.
as
4yn 2 Ž n y 1.
,
B1 s
27 Ž n y 2 . 8 Ž n y 1.
2
,
Ž 56 .
Eq. Ž56. is the Lagrange image of the m-KdV Žsee Eqs. Ž10. and Ž14... Near the front u ; x 2r3 H Ž x . and we encounter the same difficulty as for the K Ž m,n s 4.. In terms of Z s u 3 we obtain 3
Zt q 2 Ž Z 2 . x q w ZZ x x x y Z x Z x x x s y1.5 Ž Z 2r3 . x . Note that the second term due to dispersion acts like a nonlinear diffusion. In the K Ž2,2. case the positive sign in front of this term acts like a backward diffusion that counteracts the tendency of the compacton to develop oscillations on its rear part. The need for this kind of stabilization is perhaps best manifested noting that if in Ž49. you take v s y1r2, the resulting equation has only a purely dispersive term uu x x x and does not seem to support the propagation of compactons w5–7x. In our case the negative sign in front of Z x Z x x , will hasten the decomposition of the compacton into a train of waves. However, if instead of Z we introduce Y s u 3r2 , then 3
Yt q 4Y 2 Yx q Y Ž YYx x . x s 23 Ž Yx . .
.
Now near the front any K Ž m,n. solution yields; Z ; x 2 H Ž x .. The right hand side of Ž53. is the ’penalty’ for using a variable which is not a conserved quantity. To appreciate the benefits of using Z consider the K Ž m s n,n. case. Now the compactons take the form Z s l Z0 cos 2 Ž v 0 s . ,
Numerical experiments reveal that Eq. Ž55.is very easy to integrate and the interaction seems very close to be elastic. In contrast, the singular behavior at the edge of its ’raw version’, the K Ž4,4., poses a considerable difficulty and the interaction is not nearly as clean w15x. Numerical stability of the compacton poses another limitation on what is an acceptable representation. Consider, for instance, u t q Ž u 4 . x q Ž u 3 u x x . x s 0.
3qn 2
201
Z0 s const.
for <) < F pr2, Ž 54 .
and vanish elsewhere. For n s 3,4; Z0 s 3r2,8r5, and v 0 s 1r3,3r2. Thus, for instance, K Ž4,4. reads; 3
Zt q 2 Ž Z 2 . x q 4 w ZZ x x x x s 3 Ž Z 2r3 . x .
Ž 55 .
Ž 57 .
In Ž57. the diffusive part, YYx Yx x , has a favorable sign for stability. Although in terms of Y the solution is not as smooth as in terms of Z, the last equation is better suited to describe a stable evolution of compactons. In fact, its numerical evolution is straight forward w15x. A final comment is in order: the fact that in one representation the problem is numerically stable while in another it is not, makes one wonder whether these are merely numerical difficulties, or do they represent a deeper issue related to the fact that we are dealing with mapping of weak solutions. Although in general the mapping does not seem to have an effect on the emerging structures, a subtle differences emerge when a change of parity is involved. For
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instance, in the K Ž3,3. equation all terms are odd and thus it is invariant under u ™ yu. Thus both compactons and anti-compactons Žyu. move in the same direction. However in Z-representation Žsee Eq. Ž53.. the solution is invariant under Z ™ yZ and t ™ yt, which means that the anti-compactons move in opposite direction. C. In the case of Eq. Ž20. and m s n q 1, we use its conserved quantity Z s u n to obtain; Zt q
nq1
where
2
Ž Z 2 . x q n q Ž v ,Z . x s 0 ,
v s Ž 1 y n . r2 n,
Ž 58 .
which is recognized as Eq. Ž49.. Since Z ; H Ž x . x 2 on the front, Z satisfies Ž58. in the conventional sense. D. Finally, we comment upon the difference between admissible and non-admissible singularities. Let u t q 2 u m u x q u x x x s 0.
Ž 59 .
Integrating twice Ž s s x y l t, and A and B are consts.., yB y Au y l u 2 q
4
Ž m q 1. Ž m q 2.
u mq2 q u 2s s 0.
Ž 60 . For m s 2 and m s 4 there is a special trigonometric solution, m y KdV ; where
us"
4k cos 2 Ž k s .
,
3 y 2cos 2 Ž k s .
l s 4k 2 ,
Ž 61 .
and m s 4;
us"
10 1r4 'k cos Ž k s .
(3 y 2cos Ž k s . 2
,
now l s k 2 .
To find Ž61. set B s 0,Y s Y0 q 1ru and let the constants A and Y0 be chosen such that in Y units the new potential retains only cubic and quadratic parts. Then, in terms of Z 2 s 1rY, Ž61. is obtained. The second case follows from the first, if one takes A s 0, B / 0 and u ´ u 2 .
We continue with the m-KdV. Let Õ s u 2 , then in terms of Õ Eq. Ž59. reads Õt q Ž Õ 2 . x q Õ x x x y
3Õ x2
ž / 4Õ
s 0.
Ž 62 .
x
For which solution Ž61. becomes Õs
16 k 2 cos 4 Ž k s . 3 y 2cos 2 Ž k s .
2
.
Ž 63 .
Now suppose one tries to generate a compacton from the periodic solution Ž61.. To this end, restrict the solution to the < s < F pr2 k interval and set it to zero elsewhere. Since u ; x 2 H Ž x . near the edge of the compactified solution, the third derivative will generate a d Ž x . function which cannot be counterbalanced anywhere in the equation. Next, consider Eq. Ž62. and the the compactification of Õ in Ž63.. Now; Õ ; x 4 H Ž x . at the front and the third derivative is continuous! Does this mean that the compactified form of Ž63. is an accepted solution of Eq. Ž62.? It would be very nice if that was the case – unfortunately it is not because, A: since the m-KdV is integrable one can in principle, use the compactified version of Ž61. as an initial datum to solve the inverse scattering problem. The non-analyticity has to generate some radiation. Remapping this solution in terms of Õ, will not eliminate the radiative part. B: Unlike the K Ž4,4. equation where u ; x 2r3 H Ž x . near the edges of its compacton, but the singularity due to the dispersion is exactly counterbalanced by the singularity of u t , in Eq. Ž62. the singularity due to dispersion is not balanced by any other term in the equation. The use of Õ in Ž62. only postpones the appearance of the unbalanced singularity by two orders but does not change the effect. Taking two more derivatives in Ž62. will uncover it. Thus, the ’compactified solution’ of Ž63. cannot persist and has to decompose.
Acknowledgements I thank R. Camassa and P. Olver for a number of enjoyable and illuminative discussions. This work
P. Rosenaur Physics Letters A 275 (2000) 193–203
was supported in part by a grant from the Israel Science Foundation and carried out in part during a stay at The Center of Nonlinear Studies at the LosAlamos National Laboratory and visit to The Courant Institute of Mathematical Sciences supported via Air-Force grant a F49620-98-1-0256.
203
front of this kink is a paraboloid, but unlike the purely dispersive case, it is not followed by a receeding front. In the overdamped case the kink has a monotone profile; U n'Vs
l 4
½
1 y exp Ž m nh .
cosh dn Ž h q h 0 . cosh Ž dnh 0 .
5
, q
Appendix A. A dispersive-dissipative interaction
h F 0,
We consider travelling wave solutions of Eq. Ž26. Žsee Ref. w16x for another approach to dispersive-dissipative interactions.. We have to solve
and vanishes elsewhere. Here dn s Dn r2 and h 0 is found via dnh 0 s ytanhy1 Ž m nrdn .. Here apart from the front, dispersion plays a secondary role.
Ž A.3 .
(
Vhh y m nVh q 4V s l , where V s U n
and
If Vn s expŽgh . then m2n y 16. The modes pending on whether underdamped case let
mn s
ž
nq1 n
/
m.
Ž A.1 .
(
2g "s m n " Dn where Dn s are over Žunder. damped dem ) Ž-. 4 nrŽ n q 1.. In the v n ' y Dn r2 and set
(
l Vs
4
1 y aexp Ž hm n . cos
vn Ž h q h0 .
4q.
The constants a and h 0 are found demanding that at, say, h s 0 both V and V X vanish. This determines a, while h 0 is found via v nh 0 s tany1 Ž m nrv n .. The resulting solution reads; U n'Vs
h F 0,
l 4
½
1 y exp Ž m nh .
cos v n Ž h q h 0 . cos Ž v nh 0 .
5
, q
Ž A.2 .
and vanishes elsewhere. This is a semi-compact oscillating kink. In the original x y y y z space the
References w1x P. Pelce´ ŽEd.., Dynamics of curved fronts, Academic Press, 1988. w2x R.E. Goldstein, D.M. Petrich, Phys. Rev. Lett. 67 Ž1991. 3203. w3x K. Nakayama, M. Wadati, J. Phys. Soc. Jpn. 62 Ž1992. 473. w4x K. Nakayama, H. Segur, M. Wadati, Phys. Rev. Lett. 69 Ž1992. 3425. w5x P. Rosenau, J.M. Hyman, Phys. Rev. Lett. 70 Ž1993. 564. w6x P. Rosenau, Phys. Rev. Lett. 73 Ž1994. 1737. w7x P. Rosenau, Phys. Lett. A 230 Ž1997. 305. w8x P. Rosenau, Phys. Lett. A 211 Ž1996. 265. w9x R. Camassa, D.D. Holm, Phys. Rev. Lett. 71 Ž1993. 1661. w10x J. Rubinstein, personal communication. w11x P.G. Drazin, R.S. Johnson, Solitons: an Introduction, Cambridge University Press, Cambridge 1990. w12x S. Kichenassamy, P.J. Olver, Siam J. Math. Anal. 23 Ž1992. 1141. w13x J.M. Hyman, P. Rosenau, Phys. D 123 Ž1998. 502. w14x W. Choi, R. Camassa, to appear in J. Fluid Mech. w15x J.M. Hyman, P. Rosenau, in progress. w16x P. Rosenau, Phys. D 123 Ž1998. 525.