Eur. Phys. J. B (2012) 85: 302DOI: 10.1140/epjb/e2012-30366-9

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Soliton propagation and collision in a variable-coefficient coupledKorteweg-de Vries equation

H.Q. Zhaoa

Business Information Management School, Shanghai Institute of Foreign Trade, 1900 Wenxiang Road, Shanghai 201620,P.R. China

Received 7 May 2012 / Received in final form 21 June 2012Published online 5 September 2012 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2012

Abstract. In this paper, a variable-coefficient coupled Korteweg-de Vries equation is presented and studiedby Hirota bilinear method. The multi-soliton solutions expressed in the form of Pfaffians are obtained. Wefurther analyze dynamic characters of these soliton solutions. The appearances of resonant soliton behaviorsinvolving some novel soliton fusion and fission phenomena have been reported.

1 Introduction

The classical solitons are nonlinear waves that maintainstheir shape and speed after a fully nonlinear interaction,except for a phase shift [1,2]. Recently, remarkable interestis turning to the investigation of the resonant interactionsof the solitons, which have been studied both theoreticallyand experimentally by many authors [3–7]. For instance,at a specific time, one soliton can be splitted into two ormore solitons; or on the contrarily, two or more solitonsmay merge into one soliton. Now an interesting questionis that the solitons fusion, fission and fusion-fission inter-actions reveal some novel characteristics?

As the inhomogeneities of media and non-uniformitiesof boundaries are taken into account, the variable-coefficient soliton equations are considered to be more re-alistic than their constant-coefficient counterparts in Bose-Einstein condensates, arterial mechanics and optical-fibercommunications, etc. [8–18]. For example, the variable-coefficient Korteweg-de Vries (KdV) equations, which canbe used to describe the large-amplitude internal waves inthe coastal waters of the oceans [15] and nonlinear periodicwaves over a gradual slope with bottom friction [17], havebeen extensively studied during the last three decades.On the other hand, some kinds of coupled KdV equationshave attracted more and more attention in recent years.Progress on these coupled KdV equations have been madein the study of the soliton solutions, soliton propagationproperties, infinite conservation laws, Lax pair, symme-tries and Hamiltonian structures etc. However, there hasnot been much work on the variable-coefficient coupledKorteweg-de Vries equations so far.

The purpose of the present paper is to investigatethe following generalized coupled variable-coefficient KdV

a e-mail: hqzhao@163.com

equation (cvcKdV)

ut + f(t)uux + g(t)uxxx + l(t)u + q(t)ux =

k(t)(

uxφx − uφxx

φ

)x

, (1.1a)

φt + g(t)φxxx +12f(t)uφx + q(t)φx = 0, (1.1b)

where f(t), g(t), l(t), q(t) and k(t) are analytic functionsof variable t. The system (1.1) with some special choicesfor f(t) = 6, g(t) = 1, l(t) = 0, q(t) = 0, k(t) = 3 wouldbe the classical part of one of supersymmetric extensionsof the KdV model [19,20] if the variable transformationv = u + (log φ)xx and U = (log φ)x were considered [21].The Lax pair associated with this system is obtained bythe use of prolongation technique [22]. Therefore, the sys-tem (1.1) is also called as superextensions of the variable-coefficient KdV equation, and may have potential applica-tions in two dimensional quantum gravity [23]. However,to the best of our knowledge, the multi-soliton solutionsfor (1.1) have not been reported and the features of thesoliton propagation and collision, caused by the variablecoefficient, have not been discussed.

The paper is organized as follows. In the next sec-tion, we will bilinearized the cvcKdV equation (1.1)and N -soliton solution is obtained by Pfaffian tech-nique [24,25]. In Section 3, we carry out an asymptoticanalysis of these soliton solutions. Our results show thatthe soliton propagation, fusion and fission properties un-der the variable coefficient reveal some novel characteris-tics. Finally, we conclude the paper in Section 4.

Page 2 of 6 Eur. Phys. J. B (2012) 85: 302

2 Bilinear form and N-soliton solutionfor the cvcKdV equation

By introducing the dependent transformation

u =12C0

e−∫

l(t)dt(ln F )xx, φ =G

F, (2.2)

where F and G are two real functions of the variablesx and t, and C0 is a nonzero constant, the bilinear equationof (1.1) can be represented into the following forms

[Dt + g(t)D3x + q(t)Dx]F · G = 0, (2.3a)

[DxDt + g(t)D4x + q(t)D2

x]F · G = 0, (2.3b)

with the constraint

f(t) = C0g(t)e∫

l(t)dt, k(t) = 3g(t),

where the bilinear operators Dx and Dt are defined by

Dmx Dn

t m · n = (∂x − ∂x′)m(∂t − ∂t′)n

× m(x, t)n(x′, t′)|x′=x,t′=t.

Using the perturbational method, we obtain the three soli-ton solution to the (2.3), which is expressed as follows

F = 1 + a1eη1 + a2e

η2 + a3eη3 + a12e

η1+η2

+ a13eη1+η3 + a23e

η2+η3 + a123eη1+η2+η3 , (2.4a)

G = 1 + b1eη1 + b2e

η2 + b3eη3 + b12e

η1+η2

+ b13eη1+η3 + b23e

η2+η3 + b123eη1+η2+η3 , (2.4b)

where

ηj = pjx − p3j

∫g(t)dt − pj

∫q(t)dt, (2.4c)

ajk =pj − pk

(pj + pk)2αjk, bjk =

pj − pk

(pj + pk)2βjk, (2.4d)

αjk = ajpjbk − akpkbj ,

βjk = bjpjak − bkpkaj (j, k = 1, 2, 3), (2.4e)

and

a123 =(p1 − p2)(p1 − p3)(p2 − p3)

(p1 + p2)2(p1 + p3)2(p2 + p3)2α123,

b123 =(p1 − p2)(p1 − p3)(p2 − p3)

(p1 + p2)2(p1 + p3)2(p2 + p3)2β123, (2.4f)

with

α123 = −[a1a2(p21 − p2

2)b3p3 − a1a3(p21 − p2

3)b2p2

+ a2a3(p22 − p2

3)b1p1],

β123 = −[b1b2(p21 − p2

2)a3p3 − b1b3(p21 − p2

3)a2p2

+ b2b3(p22 − p2

3)a1p1],

where aj , bj and pj (j = 1, 2, 3) are arbitrary parameters.

These expressions suggest that N -soliton solutionto (2.3) are expressed by Pfaffians

F = pf(d0, a, r1, r2, ..., rN , cN , ..., c2, c1)Δ= pf(d0, a, •),

(2.5a)

G = pf(a, b, r1, r2, ..., rN , cN , ..., c2, c1)Δ= pf(a, b, •),

(2.5b)

where the entries of the Pfaffians are defined as

pf(dm, rj) = pmj eηj , (m ≥ 0, j = 1, 2, ...N),

pf(d0, a) = 1, pf(a, b) = 1, pf(dm, a) = 0, (m ≥ 1),pf(a, rj) = −eηj , pf(a, cj) = −aj,

pf(b, cj) = bj, (j = 1, 2, ..., N),

pf(rj , rk) = aj,keηj+ηk , pf(rj , ck) = δj,k,

pf(cj, ck) = −cj,k, (j, k = 1, 2, ...N),pf(dm, cj) = pf(dm, b) = pf(dm, dn) = pf(b, rj) = 0,

(m, n ≥ 0, j = 1, 2, ..., N),

and

δj,k ={

1, j = k,

0, j �= k,

ηj = pjx − p3j

∫g(t)dt − pj

∫q(t)dt,

aj,k =pj − pk

pj + pk, cj,k =

ajpjbk − akpkbj

pj + pk.

In what follows, we will show that F and G given by (2.5)are N -soliton solution to the (2.3). By using of thesePfaffians, the differential formulae of F and G are ob-tained

Fx = pf(d1, a, •), (2.6a)Fxx = pf(d2, a, •), (2.6b)

Fxxx = pf(d3, a, •) + pf(d0, d1, d2, a, •), (2.6c)Ft = g(t)[−pf(d3, a, •) + 2pf(d0, d1, d2, a, •)]

− q(t)pf(d1, a, •), (2.6d)Gx = pf(d0, d1, a, b, •), (2.6e)

Gxx = pf(d0, d2, a, b, •), (2.6f)Gxxx = pf(d0, d3, a, b, •) + pf(d1, d2, a, b, •), (2.6g)

Gt = g(t)[−pf(d0, d3, a, b, •) + 2pf(d1, d2, a, b, •)]− q(t)pf(d0, d1, a, b, •). (2.6h)

Substituting these relations (2.5) and (2.6) into (2.3a), wefind that the bilinear equation is reduced to the Pfaffianidentify [25]

g(t)[pf(d0, d2, a, b, •)pf(d1, a, •)− pf(d1, d2, a, b, •)pf(d0, a, •)+ pf(d0, d1, d2, a, •)pf(a, b, •)− pf(d0, d1, a, b, •)pf(d2, a, •)] ≡ 0. (2.7)

Eur. Phys. J. B (2012) 85: 302 Page 3 of 6

Therefore, F and G satisfy the bilinear equation (2.3a).Furthermore, in order to prove that (2.5) also satis-fies (2.3b), we have the second expression for G whichis equal to (2.5b)

G = pf(e0, a, r1, r2, ..., rN , cN , ..., c2, c1)Δ= pf(e0, a, •),

(2.8)and its derivatives

Gx = pf(e1, a, •), (2.9a)Gxx = pf(d0, d1, e1, a, •), (2.9b)

Gxxx = pf(d0, d2, e1, a, •), (2.9c)Gxxxx = pf(d0, d3, e1, a, •) + pf(d1, d2, e1, a, •), (2.9d)

Gxt = g(t)[−pf(d0, d3, e1, a, •) + 2pf(d1, d2, e1, a, •)]+ q(t)pf(d0, d1, e1, a, •), (2.9e)

where new entries are defined by

pf(e0, a) = 1, pf(e1, a) = 0, pf(em, cj) = −bjpmj ,

(m = 0, 1, j = 1, 2, . . . , N),pf(em, rj) = pf(dm, e1) = 0,

(m ≥ 0, j = 1, 2, ..., N).

For the convenience of computation, we have torewrite (2.3b) in the equivalent form

GxtF −GxFt + g(t)GxxxxF −3g(t)GxxxFx +3g(t)GxxFxx

− g(t)GxFxxx + q(t)GxxF − q(t)GxFx = 0, (2.10)

which is derived by x-derivative of the (2.3a). Substitut-ing (2.5a), (2.6a)−(2.6d), (2.8) and (2.9) into (2.10), wefind that (2.10) is reduced to the pfaffian identity [25]

g(t)[pf(d0, d2, e1, a, •)pf(d1, a, •)− pf(d1, d2, e1, a, •)pf(d0, a, •)+ pf(d0, d1, d2, a, •)pf(e1, a, •)− pf(d0, d1, e1, a, •)pf(d2, a, •)] ≡ 0. (2.11)

Thus we have proved that F and G given by (2.5) is theN -soliton solution for (2.3).

3 Propagation characteristics of the solitons

In this section, we mainly concentrate on the interac-tions of resonant solution using two soliton solution u.The explicit form of the two soliton solution can be con-structed as

F = 1 + a1eη1 + a2e

η2 + a12eη1+η2 , (3.12a)

G = 1 + b1eη1 + b2e

η2 + b3eη3 + b12e

η1+η2 , (3.12b)

where ηj , a12, b12 defined by (2.4c), (2.4d). From the ex-pression of the two soliton solution, it is observed that theterms a12, b12 related to the phase shifts of the solitons,

depend on the parameters, aj , bj , (j = 1, 2), and the ve-locity of each soliton can be influenced by g(t), q(t). Thus,it is possible to form different types of resonant solitonbehaviors.

For the case of the a12 being finite, the asymptoticforms corresponding to (3.12) are given by

F ∼{

1 + a1eη1 (η1 ∼ O(1), η2 ∼ −∞)

a2 + a12eη1 (η1 ∼ O(1), η2 ∼ +∞)

F ∼{

1 + a2eη2 (η2 ∼ O(1), η1 ∼ −∞)

a1 + a12eη2 (η2 ∼ O(1), η1 ∼ +∞).

For a1 > 0, a2 > 0, a12 > 0, the solution u with (3.12)represents regular interaction of two solitons. The solitonsamplitude can be expressed as |3p2

j

C0e−

∫l(t)dt|, (j = 1, 2)

and the phase shift of the two solitons is given by 12 ln a12

a1a2.

Furthermore, the soliton velocity vj for u can be ob-tained by

vj = −dx

dt=

d(p2j

∫g(t)dt +

∫q(t)dt)

dt= p2

jg(t) + q(t).

(3.13)

This means that g(t), l(t) and pj influence simultane-ously to the soliton velocity. It is noted that the sign andabsolute value of the velocity vj determines the propa-gation direction and speed of the solitons, respectively.When v1v2 < 0 and g(t) and q(t) are chosen as periodictime function, the propagation trajectory of the solitonspresents the two bidirectional solitons interactions withperiodicity oscillation velocity (see Fig. 1a). Likewise, ifv1v2 > 0, the corresponding trajectory is to overtakesolitons collision with periodicity oscillation velocity (seeFig. 1b).

If a12 = 0, (3.12) becomes

F = 1 + a1eη1 + a2e

η2 ,

from which we have the asymptotic forms

F ∼{

1 + a1eη1 (η1 ∼ O(1), η2 ∼ −∞)

a2 (η1 ∼ O(1), η2 ∼ +∞)

F ∼{

1 + a2eη2 (η2 ∼ O(1), η1 ∼ −∞)

a1 (η2 ∼ O(1), η1 ∼ +∞)

u ∼

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

12C0

e−∫

l(t)dt a1a2(p1−p2)2

[a1e12 (η1−η2)+a2e

12 (η2−η1)]2

(η1 − η2 ∼ O(1), η1 ∼ η2 ∼ +∞)

0(η1 − η2 ∼ O(1), η1 ∼ η2 ∼ −∞).

Therefore, the solution (3.12) exhibits either the tworegular solitons (with forms 3p2

1C0

e−∫

l(t)dt sech2 12 (η1 +

ln a1) and 3p22

C0e−

∫l(t)dt sech2 1

2 (η2 + ln a2)) fused into one

Page 4 of 6 Eur. Phys. J. B (2012) 85: 302

(a) (b)

Fig. 1. (Color online) The figures show contour maps of two solitons in space-time coordinates. Parameters used are a1 = b1 = 3,a2 = b2 = 4, l(t) = 0, C0 = 6, (a) p1 = 1, p2 =

√3, g(t) = 2 sin( t

5), q(t) = −4 sin( t

5), (b) p1 = 1, p2 = 1.6, g(t) = 10 sin(2t),

q(t) = sin(2t).

(a) (b)

Fig. 2. (Color online) The plot of one soliton split into two overtaking solitons and then two overtaking solitons merges intoone soliton under periodic speed. Parameters are a2 = a1b2p1

b1p2, a1 = b1 = 1, b2 = 1.5, g(t) = 2 sin(2t), q(t) = 10 sin(2t), l(t) = 0,

C0 = 6, (a) p1 = −2, p2 = −1, (b) p1 = 2, p2 = 1.

regular soliton (with forms e−∫

l(t)dt 3√

a1a2(p1−p2)2

C0×sech2

[12 (η1 − η2) + η0] where eη0 =√

a1a2

) or one regular soli-ton broke into two regular solitons under the conditiona1 > 0, a2 > 0. Figure 2 shows that the solitons un-der periodicity oscillation velocity break into two over-taking solitons (v1v2 > 0) and then merge into one soli-ton. In the comparison, in Figure 3, a soliton splits intotwo bidirectional solitons (v1v2 < 0) and then mergesinto one soliton. We can also present more solitons tra-jectory with the different velocity by choosing the differ-

ent variable-coefficients g(t) and q(t). Figure 4a illustratesthe overtaking (v1v2 > 0) collision of two solitons with dif-ferent velocities. The bidirectional solitons with differentvelocities has been displayed in Figure 4b.

Under quasi-resonant condition, (a12 � 0), the result-ing solution u represents a higher soliton (first soliton)emits third soliton as it approaches to a lower soliton (sec-ond soliton), and then the third soliton moves after andinteraction with the lower soliton (second soliton). Thelower soliton (second soliton) exchanges the energy withthe higher soliton (first soliton) by absorbing the third

Eur. Phys. J. B (2012) 85: 302 Page 5 of 6

(a) (b)

Fig. 3. (Color online) The plot of one soliton split into two bidirectional solitons and then two bidirectional solitons merges intoone soliton under periodic speed. Parameters are a2 = a1b2p1

b1p2, a1 = b1 = 1, b2 = 2, g(t) = 2 sin( t

5), q(t) = −4 sin( t

5), l(t) = 0,

C0 = 6, (a) p1 = 1, p2 =√

3, (b) p1 = −1, p2 = −√3.

(a) (b)

Fig. 4. (Color online) The figures show contour maps of two solitons in space-time coordinates. Parameters used are a2 = a1b2p1b1p2

,

a1 = b1 = 1, b2 = 2, l(t) = 0, C0 = 6, (a) p1 = 2, p2 = 1, g(t) = t, q(t) = 5 sin(t), (b) p1 = 1, p2 =√

3, g(t) = 2 sin( t5) + t,

q(t) = −4 sin( t5) − 5t.

soliton. We would like to point out that due to the peri-odicity oscillation velocity, the process of fusion and fissionof solitons is operated in a cyclical manner. Figure 5a de-picts two overtaking solitons under periodicity oscillationvelocity interact with each other by emitting and then ab-sorbing a third soliton. Correspondingly, we can have twobidirectional solitons under periodicity oscillation velocityinteract with each other by emitting and then absorbinga third soliton in Figure 5b. We find that the solution

exhibit the cyclical spider-web properties, which have notbeen observed so far.

4 Conclusion

In this paper, a variable-coefficient coupled Korteweg-deVries equation is investigated. Based on the Hirota bi-linear method and Pfaffian technique, the multi-soliton

Page 6 of 6 Eur. Phys. J. B (2012) 85: 302

(a) (b)

Fig. 5. (Color online) The plot of two solitons fission and fusion together under periodic speed to the cvcKdV equation witha1 = p2

p1+10−16, b1 = 1, b2 = 1, a2 = 1, p1 = 2, p2 = 1, (a) g(t) = 6 sin(t), q(t) = −2 sin(t), (b) g(t) = 3 sin( t

2), q(t) = −10 sin( t

2).

solutions in the form of Pfaffians have been constructed.Furthermore, the propagation and collision behaviors ofthe two soliton solution are analyzed by the use of asymp-totic forms. Some novel properties of the resonant inter-actions have been illustrated.

The work is partially supported by the Shanghai 085 Projectand the National Natural Science Foundation of China un-der grant 10971136. We are greatly indebted to the refereefor his very valuable remarks and suggestions on the originalmanuscript.

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