Solitons in a system of coupled Korteweg-de Vries equations

WAVE MOTION 11 (1989) 261-269 NORTH-HOLLAND

261

SOLITONS IN A SYSTEM OF COUPLED KORTEWEG-DE VRIES EQUATIONS Yuri S. K I V S H A R Institute for Low Temperature Physics and Engineering, UkrSSR Academy of Sciences, 47 Lenin Avenue, Kharkov 310164, U.S.S.R.

Boris A. M A L O M E D P.P. Shirshov Institute o f Oceanology, 23 Krasikov Street, Moscow 117218, U.S.S.R.*

Received 3 February 1988

A system of two Korteweg-de Vries equations coupled by small linear and nonlinear terms, which is a model of a resonant interaction between two internal gravity-wave modes in a shallow stratified liquid, is considered. The present paper is a continuation of a preceding one, where only the linear coupling was dealt with. Various dynamical processes involving one-mode solitons are investigated. It is demonstrated that two solitons belonging to the different wave modes may form an oscillatory bound state (a bi-soliton) which provides an explanation for the numerical results of Gear & Grimshaw demonstrating leapfrogging motion of the two interacting solitons. In the framework of a perturbation theory based on the inverse scattering transform, the frequency of the bi-soliton's internal oscillations is found, and emission of radiation by a weakly excited bi-soliton is studied. Phase shifts and radiative energy losses accompanying a collision between two free solitons belonging to the different modes are calculated. In addition, a collision between a free soliton and a bi-soliton is considered, and it is demonstrated that the collision may result in break-up of the bi-soliton.

1. Introduction The p r e s e n t p a p e r is the c o n t i n u a t i o n of a preceding p a p e r o f the second a u t h o r [1] devoted to the investigation o f soliton d y n a m i c s in the f r a m e w o r k o f the G e a r - G r i m s h a w m o d e l [2, 3]: Ult - - 6 U l U l x + Ulxxx --- --elu2Uzx -- ez( UlU2)x -- eau2 . . . .

(1)

U2t -- 6flU2U2x q-/3U2xxx -- Vou2x = -or [8l(Ul u2) x -I- EzUl t/ix d- E3Ulxxx ].

(2)

These e q u a t i o n s describe a r e s o n a n t i n t e r a c t i o n of two transverse i n t e r n a l gravity-wave m o d e s in a shallow stratified liquid. I n (1), (2), the f u n c t i o n s Ul a n d u2 stand for wave variables in the two wave modes, e l , e2 a n d e 3 are i n t e r - m o d e c o u p l i n g constants which in the present p a p e r will be regarded as small parameters, while the p a r a m e t e r s / 3 , a a n d Vo are arbitrary. As has b e e n revealed in the n u m e r i c a l e x p e r i m e n t s by G e a r a n d G r i m s h a w [2, 3], two solitons b e l o n g i n g to different wave m o d e s u n d e r g o small. This i n t e r a c t i o n m a y result m o t i o n looks like " l e a p f r o g g i n g " . has b e e n p u t forward in p a p e r [1]

strong interaction p r o v i d e d that their relative velocity is sufficiently in f o r m i n g a b o u n d oscillatory state o f the two solitons, so that their A theoretical description of the " l e a p f r o g g i n g " t w o - s o l i t o n d y n a m i c s on the basis of a p e r t u r b a t i o n theory for solitons (see, e.g., [4-7]). I n

* Address for correspondence. 0165-2125/89/$3.50 O 1989, Elsevier Science Publishers B.V. (North-Holland)

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reference [ 1] both the adiabatic dynamical processes and radiative effects, i.e., emission of small-amplitude quasi-linear waves by the leapfrogging solitons, have been considered. However, reference [1] dealt with the particular case el = e2=0, when only the linear coupling - e 3 was present. According to numerical results by G e a r and Grimshaw [2, 3], the nonlinear couplings -8.], e2 may, however, also be important. The present paper is devoted to the theoretical investigation of the general case, i.e., 8.~, 8.2, 83 • 0. In Section 2 we study the bound two-soliton state (we call it a bi-soliton) in the adiabatic approximation, i.e., disregarding effects of the emission of small-amplitude waves by the bi-soliton (this bi-soliton had been revealed in the numerical experiments of [2, 3]). The calculation can be accomplished explicitly in two particular cases: when the amplitudes of the two solitons are equal, and when one amplitude is much greater than the other. In Section 3 we briefly consider weak adiabatic interaction of the two solitons for the case when their relative velocity is not small and we calculate collision-induced phase shifts of the solitons. In the same section we consider a collision between a free solition and a bi-soliton and we demonstrate that, under certain conditions, the collision may break the bi-soliton. In Section 4 we calculate an intensity of emission from a weakly excited bi-soliton, with the aid of a perturbation theory based on the inverse scattering transform (IST) (an analogous approach has been employed to describe analytically radiative effects for the s i n e - G o r d o n [8, 9], and the nonlinear Schr6dinger [10, 11] equations). The second part of Section 4 is devoted to the investigation of radiative effects accompanying a collision of the two free (unbound) solitons. According to the numerical results of G e a r and Grimshaw [2, 3], the radiative effects are especially important for the general case when 8.1, 8.2 # 0.

2. Adiabatic dynamics of a bi-soliton

2.1. Solitons of equal amplitudes In the unperturbed case, when the coupling terms - e ~ , e2, e3 are absent, (1) and (2) split into two exactly integrable Korteweg-de Vries (KdV) equations. The soliton solutions to these equations are

us = - 2 K 2 sech 2 Z~,

Zj = Kj(x - ~j),

j = 1, 2,

(3)

where K~.2 are the solitons' amplitudes, and the solitons' phases ¢1,2 evolve according to the equations d~', dt = 4K12'

d~'---3= 4flK~ - Vo. dt

(4)

In the presence of perturbations, the evolution of the solitons' parameters K1,2 and '~.2 is described by the following adiabatic equations (see, e.g., [5]) dK~_ dt d K 2 --

dt

e 2K1 e

dZl - - -cosh 2 Z1 '

(5)

dZ2

(6)

2K2 -4K1-~

cosh 2 Z2 ' dZ1P](ul u2)(tanhZl+

d t

4K 1

d~- =

- Vo-~K32

'

Z1 cosh 2 Z

'

dZ2 P2(u~, u2)(tanh Z2+ cosh 2 Z2 '

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where eP~.2(u~, u2) = ePL2[u](Z]), u2(Z2)] stands for the perturbation, i.e., for the r.h.s, o f (1) and (2) respectively. The same equations can be derived in a n o t h e r way as canonical H a m i l t o n i a n equations of motion (see the derivation o f the evolution equations for the particular case e~ = e2 = 0 in [1]). It is pertinent to recall that the perturbation-induced interaction between the two solitons (3) is strong provided their velocities coincide, i.e., according to (4), if 2

2

4K1 = 4ilK2-- V0.

(9)

If the solitons' amplitudes K~ and K2 are o f the same order, the strong coupling condition (9) implies that, u n d e r the action of a perturbation, K~ and K2z p e r f o r m small oscillations a r o u n d their m e a n values (KI2) and (K2), the m e a n values being related by (7). As was m e n t i o n e d in [1], the calculations cannot be a c c o m p l i s h e d explicitly for an arbitrary value o f (K2). In this subsection we consider the particular case (K 2) = (K 2) ~ K 2, i.e., according to (9), K2 = Vo/4(fl - 1). The evolution equations for a weakly excited b o u n d state of two solitons can be obtained from (5)-(8) if one substitutes KL2 = K + AL2 ,

(10)

where K is the just m e n t i o n e d constant parameter, and the variables AL2 are implied to be small in c o m p a r i s o n with K. Straightforward but rather lengthy calculations o f the integrals in the r.h.s, o f (5) and (6) yield dA] = 2K412e3io(~) _ (e I -{-e2+6e3)ii(~)], dt

(5')

dA2 = 2ctr412e3io(~) _ (e~ + e2 + 6e3)I1(~')], dt

(6')

d~', = 4K2+ 8KA, + eF(~), dt

(7')

d~'2 = 4K2/3 + 8K/3h2 -- Vo+ eF(~), dt

(8')

where --- K ( ~ , - ~ 2 ) ,

2 Io(~') = cosh 2 ~. tanh 4 ~. [3~" - 3 tanh ~ - ~"tanh 2 ~'], I1(~) =

cosh 2 ~. tanh 6 ~ [ - ~ tanh 3 ~ + 5 tanh ~ - ~(5 - t a n h 2 ~) cosh z ~

F(~') being a function which rapidly decreases as I~'l~ ~ (its explicit form is not used in this paper). F r o m (5')-(8') one can obtain the system of two equations for the relative p a r a m e t e r s o f the solitons,/t =- ;h -/3;t2, dA ~ - = 2(1 + afl )K4[ 2e3Io( ~ )-- ( e, + e2 + 6e3) Ii( (, ) ],

(11)

d.._~= 8K2A + eK(1 -- a)F(~). dt

(12)

For the case el

= e2=0,

the system of (11), (12) coincides with that from [1].

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It is easy to investigate small oscillations o f the bi-soliton in a vicinity o f the equilibrium solution ~"= O, A = A o ~ ( 1 - a)eF(O)K. In the linear approximation, oscillations are described by the equation ~'=

2 -Wo~',

(13a)

where 256r E3-1-~[ - - 2 r EI -t-o) 02 ~- -2T[

82)]K6(1 +/~a,)

(13b)

(we a s s u m e e 3 + ~ ( e I + e2)> O, otherwise a bi-soliton is absent). The system (11) and (12) significantly simplifies if a = 1. Then it can be reduced to one equation (cf. (16) from [13) d2~"_ dt 2

OU({) _ 16K6(1 +/3)[2e3Io(ff) - ( e , + e2+ 6e~)l,(~')], O~

(14)

which describes the m o t i o n o f a unit mass particle in the potential

d('[2e3Io((')-(el+e2+6e3)I~((')].

U(~) =--16K6(I + ~ )

(14')

-oc 5 This potential is depicted in Figs. 1 and 2. In the case el + e2> -~83, it is attractive for small ~ (see Figs. l(a) and 2(a)), and in the vicinity o f the potential m i n i m u m ~ = 0, small oscillations are described by 5 (13). In the opposite case, el + e2 < -~e3 the potential has a m a x i m u m at ~ = 0 (Figs. l(b) and 2(b)). 5 5 For el + e2 > - ~e3, E3 > 0 and el + e 2 < ~lea[, e 3 < 0, the potential U(~r) has additional extrema at ~"= ±~rm, 5 namely m a x i m a (Fig. l(a)) or minima (Fig. 2(b)). In the case el + e2> -~e3, e3> 0 the value brm is equal to the m a x i m u m amplitude o f the bi-soliton's internal oscillations. The value ~ m is determined by a solution o f the equation

I,(sr) f(sr) _=

Io(~')

5 tanh ~ ' - 133tanh3 ~ - ~ ' ( 5 - t a n h 2 ~ r ) / c o s h 2 =

tanh 2 ~'(3~"- 3 tanh ~"- ( tanh 2 ~')

v(~)

8

Y

0

Fig. 1. The effective potential U ( sr) describing the interaction of the two solitons with nearly equal amplitudes at a = 1, e 3 > 0:

(a) e , + ~ > -~e~; (b) ~+~2< - ~ .

Yu.S.

Kivshar, B.A. Malomed / Coupled Korteweg-de Vries equations

265

a

°

Fig. 2. The same as in Fig. 1, e3<0: (a) ~,+~2>~1~1; (b) e,+e2<~le3l.

2e3

- e l + e2+6e3

-3'.

The numerically calculated function f ( ~ ) is shown in Fig. 3 for ~ > 0 (the function is even). It is evident that ~m exists provided 0 < 3, < 4, i.e., el + e2 > --5E3, e3 > 0 or e 1+ e2 < ~-Ie31, e3 < 0. For the particular case el = e2 = 0, the effective potential U(ff) takes the form depicted in Fig. l(a) ( e 3 > 0) or Fig. 2(b) (e 3 < 0). Indeed, at el = e 2 = 0 one finds 3 ' = 1 < 4 and the extrema points +~m--~ +2.1 exist for any e3 [1].

2.2. Strongly differing amplitudes of the solitons In the preceding we have dealt with the case when the solitons' amplitudes /(l a n d / ( 2 were implied to be o f the same order o f magnitude. N o w we consider the other case when /(2
(15)

if/(2
0

~,,,

Fig. 3. The function f(~) for the determination of the maximum amplitude ~'~0; 3' -= 2e3/(el +

e2 +

6e3)-

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266

easily derive the set of equations for the solitons' parameters (cf. eqs. (20), (21) from [1]):

dA1 8K4 dt

--

dK2

K

r

2--1

2--

Z[EIK2-e~e2K n- 2e3K~],

(16a)

2 2 80IKK2Z[Elff2..~_I ~'E2K 2_F,'~ ZE'3K 21 2] ,

. . . .

dt

(16b)

dz

- - = 8KA~- 4K22/3+ O(e), dt

(16c)

where z = ( E l - ~2), K1 = ( - - Vo/4) 1/2+ AI ~ K + A1 (IA~I,~ K). Further analysis of the adiabatic equations of motion (16) is straightforward. The equilibrium solution is z =0, A~ ~ K~/3/2K+O(e), K2 being here an arbitrary small constant, and the frequency of small oscillations around this equilibrium position is ,o2=

-

3. Adiabatic interaction of colliding solitons

If the relative velocity V of the two solitons is not small, their interaction is weak (see [1-3]). In this case we deal with the collision problem. In the adiabatic approximation this collision is purely elastic, i.e., it results only in phase shifts of the solitons. This effect is nontrivial since the shifts are absent when the equations (1) and (2) are uncoupled. For the simplest case K1 = r2-= K and a = 1 one can obtain from (11) (in the lowest approximation one should insert the law of the motion of the uncoupled solitons, ~= ~Vt): A(t)=

J-_o~d

dt'-

1 U(r,=KVt), 8K3V

where U(ff) is defined by (14'), and, finally, A ~ ' ~ / ~_~ d~'(t) d----~d ' :

2 ( l +V2 f l ) K 3 [ 2e3 f ~

Uo(x) d x - ( e l + e : + 6 e 3 )

I7~ Ul(x) dx ] ,

where

Un(x)=-f~o d y I , ( y ) ,

n=0,1.

Collision between a free soliton and a bi-soliton may give rise to more interesting effects. Let, e.g., the free soliton with an amplitude K' belong to the first subsystem. Then, in the lowest approximation, its collision with the component of the bi-soliton belonging to the same subsystem results in the unperturbed phase shift of the latter [12] A~I = K-' logl(K + K')/(K -- K')].

(17)

Evidently, this is equivalent to a very quick shift of the relative coordinate ff of the bi-soliton from the equilibrium position ~"= ~q = 0, Figs. l(a), 2(a), or ff = ~req---±~rm, Fig. 2(b), to sr = ~'eq+ KAff~. In the case e3 > 0, el + e2 > - 2%3, Fig. 1(a), the latter shift causes decay of the bi-soliton into two free solitons, provided

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A~"-= rA~'~/> ~'m-According to (17), this condition is equivalent to the following limitation on the amplitude Kr:

K tanh(½~m)<~ K'~ < r coth(½~'m).

(17')

If K' lies outside the interval (1 7'), the collision results in excitation of internal oscillations of the bi-soliton. In the case e3 < 0, e] + e2 < ~[e31, Fig. 2(b), the collision breaks the bi-soliton if the shifted (after-collision) value of the bi-soliton's relative coordinate, ~"= q-~m "~ K A~, lies in the region where U ( ~ ) > 0. A similar analysis of the break-up of a bi-soliton due to its collision with a free soliton described by a pair of coupled sine-Gordon equations has been developed in [9].

4. Radiative effects

4.1. Emission of radiation by leapfrogging solitons The oscillatory motion of the bound solitons is not strictly periodic due to radiative losses, i.e., emission of small-amplitude waves by the oscillating bi-soliton. One can calculate the intensity of emission from a bi-soliton which performs small oscillations by means of a perturbation theory based on the inverse scattering transform. This approach had been used in a number of papers (see, e.g., [1, 6-8, 10] and, in particular, it was employed for the calculation of radiative effects accompanying interactions of solitons in a system of coupled KdV [1], sine-Gordon [8-9], and nonlinear SchrSdinger [10, 11], equations. In the framework of this approach the radiation wave-field is described by the complex amplitude b(~:), ~: being the real positive spectral parameter which is actually the radiation wavenumber (see [1] and references therein). The spectral density of the radiation energy E is dE

~ ( ~ ) -= ~

16

= ~ ~'[b(~)l 2,

(18)

provided Ib(s¢)12,~1. The radiation in a system of two coupled equations is described by two complex functions bl(~:) and b2(s¢), which denote the amplitudes pertaining to the first (ul) and second (u2) wave modes. In the presence of perturbations, the evolution equations for bl(~) and b2(s¢) may be presented as follows (j = 1, 2): Obj( ~,

at

t)

= 8i~:3bj (so'

e f~oo dxPi(ul ' u2) ~ ( ~ : , x ) , t)+~-~aj(¢)

(19)

where a(~) and qt(~:, x) are the so-called Jost coefficient and Jost function for the one-soliton solution of the KdV equation (see, e.g., [1, 5]): a~(s~) = st- iKJ ~:+iKj '

(20)

~ ( s ¢, x) = e iex ~:+iKj tanh Zj ~:+ iKj ,

Zi = Kj(x- ~),

(21)

where, as above, rj and ~ stand for the solitons' amplitudes and phases (see (3)), and e ~ stands for the r.h.s, of (1) ( j = 1) and (2) (j = 2). In the case of the oscillating bi-soliton our aim is to calculate the emission power spectral density, i.e.,

~.(e) ~ ~ ~(e) =

e~ Re b*(e) - - - g - j ,

(22)

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where the asterisk denotes the complex conjugate. Substituting eqs. (20) and (21) into (19), and a solution of eq. (19) into (22), one can obtain (see details of similar calculations performed in [1] for the particular case el = ee = 0, e 3 ~ 0) the following expression for the emission power spectral density pertaining to, e.g., the first wave mode:

~4/l(,) = (l~6ga~oK)2( el + e2 + lOg3)2~ ( ,---~K2) .

(23)

a and oJ being the amplitude and the frequency of the bi-soliton's oscillations (cf. eq. (32) from [1]). As one can see from (23), in the lowest approximation that is being considered, the emission is concentrated at the wave-number Go= OJ/8K2; the physical meaning of this result is quite clear: the oscillating bi-soliton emits small-amplitude quasilinear waves, whose frequency coincides, in the soliton's rest reference frame, with the oscillation frequency w. The total emission power in the first wave mode is 1 6 ~ 3 (l+a/3)(el+ez+lOes)e(el+e2+~es)K8a 2, W1---f0 '~ °W,(sC)dsC= ( 2 ) (\~-~]

(24)

where we have substituted expression (14) for w. According to reference [1], one can easily find the amplitude A1 of the first emitted wave mode:

2 A1=

W,/4K2£2.

(25)

Expressions for W2 and A2 pertaining to the second wave mode can be obtained in full analogy with (24) and (25).

4.2. Energy emission during collision of two free solitons Let us proceed to the consideration of radiative effects accompanying the collision of two free (unbound) solitons belonging to different wave modes. Their adiabatic interaction is trivial: it is characterized by the solitons' phase shifts only (see Section 3). In this case the relevant quantity is, instead of the emission power spectral density 7¢'(~), the spectral density g(~:) of the energy emitted during the collision. One can perform a calculation of this quantity using formula (.18). If the emission is absent prior to the collision, i.e., b(sc, t = - c o ) = 0, the spectral densities gj(~:) (j = 1, 2) are determined by the final amplitudes

(Bj)r=-Bj(~,t=+oo)=I~dtdBj(~'t) dt

'

Bj(~, t) ~- bj(~, t) exp(-8i~:3t), dBj/dt being determined by expression (19). We will confine our attention to the case K 1 = K 2 ~ - K . The quantities (Bj)f can be explicitly calculated for an arbitrary (non-small) value of the relative velocity V-= Vo--4K2(1--/3) of the colliding solitons. The result is as follows:

4rr2K3A

{ K2A 2[~el( A 2+ 4 ) + E3A2]

(B0f = ~:V(K2+ ~:2) sinh(½rrA) sinh(½~rC)

q.-~ C[llK3c3-{-2~K2C2--} - K(2~2 q-IK2)C -[-2~:3+ 2S~K2]}, where

C=-A-2£/K,

A ~ 8~(Ke+ SO2)/KV.

(26)

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269

The quantity (B2)f can be found analogously. For the case V~4K 2, the final expression (26) takes the simple form (cf. eq. (48) from [1]): ql-2K3A3 (B~)f- 6scV sinh2(½~rA) [(el + e2)(A2+ 4) + 24A2e3],

(27)

where A ~ 8~K / V. Inserting (27) into (18), one can readily find the total collision-induced change of the soliton's energy, 8E~, due to radiative losses:

~3VK3 f = =

2

I

Jo

xs d x ~

sh (~'rrx)

[(~:-'kea)2x4+8~:(e+e3)x2.-k16~:2]

212 ~--'T VK3[CI( ~ + e3) 2+ c2t~(e-F e3) q- c3~2], ,ff

where ~ - = ~ ( e l + e2),

cl ~ 1.20,

c2~2.95,

C3 ~ 2.78.

The quantities ~2(~:) and 8 E 2 c a n be found analogously.

Acknowledgment We are indebted to Professor L.M. Brekhovskikh for his interest in our work. References [1] B.A. Malomed, "Leapfrogging solitons in a system of coupled KdV equations", Wave Motion 9, 401-411 (1987). [2] J.A. Gear and R. Grimshaw, "Weak and strong interactions between internal solitary waves", Stud. Appl. Math. 70, 235-258 (1984). [3] J.A. Gear, "Strong interactions between solitary waves belonging to different wave modes", Stud. Appl. Math. 72, 95-124 (1985). [4] D.J. Kaup and A.C. Newell, "Solitons as particles, oscillators and in slowly changing media: A singular perturbation theory", Proc. Roy. Soc. London Set. A 361, 413-436 (1978). [5] V.L Karpman and E.M. Maslov, "Perturbation theory for solitons", Zhurn. Exp. Teor. Fiz. (Soy. Phys. JETP) 73, 537-549 (1977). [6] B.A. Malomed, "Emission from a small-amplitude sine-Gordon breather under the action of a conservative perturbation", Phys. Lett. A 102, 83-84 (1984). [7] Yu.S. Kivshar and B.A. Malomed, "Many-particle effects in nearly integrable systems", Physica D24, 125-154 (1987). [8] O.M. Braun, Yu.S. lGvshar and A.M. Kosevich, "Interaction between kinks in coupled chains of adatoms", J. Phys. C 21, 3881-3900 (1988). [9] Yu.S. Kivshar and B.A. Malomed, "Dynamics of fluxons in a system of coupled Josephson junctions", Phys. Rev. B 37, 9325-9330 (1988). [10] Yu.S. Kivshar and B.A. Malomed, "Perturbation-induced radiative losses in collision of NSE solitons", J. Phys. A 19, L967-L971 (1986). [11] Yu.S. Kivshar and B.A. Malomed, "Dynamics of solitons in a system of linearly coupled nonlinear SchrSdinger equations", Physica Scripta, submitted (1988). [12l V.E. Zakharov, S.V. Manakov, S.P. Novikov and L.P. Pitaevsky, Theory of Solitons. Inverse Scattering Transform, Nauka, Moscow, 1980.