9 April 2001

Physics Letters A 282 (2001) 1822www.elsevier.nl/locate/pla

Soliton solutions for a generalized HirotaSatsuma coupled KdVequation and a coupled MKdV equation

Engui FanInstitute of Mathematics, Fudan University, Shanghai 200433, PR China

Received 20 November 2000; received in revised form 22 February 2001; accepted 23 February 2001Communicated by A.R. Bishop

Abstract

We make use of an extended tanh-function method and symbolic computation to obtain respectively four kinds of solitonsolutions for a new generalized HirotaSatsuma coupled KdV equation and a new coupled MKdV equation, which wereintroduced recently by Wu et al. (Phys. Lett. A 255 (1999) 259). 2001 Elsevier Science B.V. All rights reserved.Keywords: Coupled MKdV equation; Soliton solution; Riccati equation; Symbolic computation

1. Introduction

Recently, by introducing a 4 4 matrix spectralproblem with three potentials, Wu et al. derived anew hierarchy of nonlinear evolution equations [1].Two typical equations in the hierarchy are a newgeneralized HirotaSatsuma coupled KdV equation

ut = 12uxxx 3uux + 3(vw)x,vt =vxxx + 3uvx,

(1)wt =wxxx + 3uwx,and a new coupled MKdV equation

ut = 12uxxx 3u2ux + 32vxx + 3(uv)x 3ux,

(2)vt =vxxx 3vvx 3uxvx + 3u2vx + 3vx.With w = v and w = v, Eq. (1) reduces respectivelyto a new complex coupled KdV equation [1] and

E-mail address: faneg@fudan.edu.cn (E. Fan).

the HirotaSatsuma equation [2,3]. Eq. (2) becomesa generalized KdV equation for u= 0 and the MKdVequation for v = 0, respectively. The soliton solutionsfor these two equations are still unknown. The aimof this Letter is to construct four kinds of solitonsolutions for them by using an extended tanh-functionmethod and symbolic computation [4]. The key ideaof this method is to take full advantages of a Riccatiequation involving a parameter and use its solutions toreplace the tanh-function in the tanh-function method[57], which simply proceeds as follows. For a givenpartial differential equation, say, in two independentvariables,

(3)H(u,ux,ut , uxx, . . .)= 0,we first consider its travelling solutions u(x, t) =u() = x + t , then Eq. (3) becomes an ordinarydifferential equation. The next crucial step is that thesolution we are looking for is expressed in the form

(4)u()=mi=0

aii,

0375-9601/01/$ see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0375-9601(01) 00 16 1- X

E. Fan / Physics Letters A 282 (2001) 1822 19

with

(5) = k(1 2),where k is a parameter, = (), = d/d . Theparameter m can be found by balancing the highest-order linear term with the nonlinear terms. Substitut-ing (4) and (5) into the relevant ordinary differentialequation will yield a set of algebraic equations withrespect to ai , k, because the coefficients of i haveto vanish. From these relations ai , k, can be deter-mined. The Riccati equation (5) has the general solu-tions

(6) = tanh(k), = coth(k).The algorithm presented here is also a computerizablemethod, in which generating an algebraic system fromEq. (3) and solving it are two key procedures and la-borious to do by hand. But they can be implementedon a computer with the help of Mathematica. The out-puts of solving the algebraic system from a computercomprise a list of the form {, k, a0, . . .}. In general, if or any of the parameters is left unspecified, then itis to be regard as arbitrary for the solution of Eq. (3).For simplification, singular coth-type soliton solutionsare omitted in this Letter, since they always appear inpairs with tanh-type solutions according to (6).

2. Soliton solutions for the generalizedHirotaSatsuma coupled KdV equation

To look for the travelling wave solution of Eq. (1),we make the transformation u(x, t)= u(), v(x, t) =v(), w(x, t) =w(), = x + t and change Eq. (1)into the form

u = 12u 3uu + 3(vw),

v = v + 3uv,(7)w = w + 3uw.

Balancing the highest-order linear terms and nonlinearterms in Eq. (7) gives the following two anstze:u= a0 + a1 + a22,v = b0 + b1 + b22,

(8)w = c0 + c1 + c22,

and

u= a0 + a1 + a22,v = b0 + b1,

(9)w= c0 + c1.Substituting (8) into Eq. (7) and using Mathematica

yields a set of algebraic system for ai, bi, ci (i = 0,1,2), k and , namely

ka1 + k3a1 + 3ka0a1 3kb1c0 3kb0c1 = 0,3ka21 + 2ka2 + 8k3a2 + 6ka0a2 6kb2c06kb1c1 6kb0c2 = 0,

ka1 4k3a1 3ka0a1 + 9ka1a2 + 3kb1c0+3kb0c1 9kb2c1 9kb1c2 = 0,

3ka21 2ka2 20k3a2 6ka0a2 + 6ka22+6kb2c0 + 6kb1c1 + 6kb0c2 12kb2c2 = 0,

3k3a1 9ka1a2 + 9kb2c1 + 9kb1c2 = 0,12k3a2 6ka22 + 12kb2c2 = 0,kb1 2k3b1 3ka0b1 = 0,3ka1b1 + 2kb2 16k3b2 6ka0b2 = 0,kb1 + 8k31 + 3ka0b1 3ka2b1 6ka1b2 = 0,3ka1b1 2kb2 + 40k3b2 + 6ka0b2 6ka2b2 = 0,6kb1 + 3ka2b1 + 6ka1b2 = 0,24k3b2 + 6ka2b2 = 0,kc1 2k3c1 3ka0c1 = 0,24k3c2 + 6ka2c2 = 0,3ka1c1 + 2kc2 16k3c2 6ka0c2 = 0,kc1 + 8k3c1 + 3ka0c1 3ka2c1 6ka1c2 = 0,3ka1c1 2kc2 + 40k3c2 + 6ka0c2 6ka2c26k3c1 + 3ka2c1 + 6ka1c2 = 0,

for which, with the aid of Mathematica, we find

a0 = 13( 8k2), a2 = 4k2,

b0 =4(3k4c0 2k2c2 + 4k4c2)

3c22, b2 = 4k

4

c2,

(10)a1 = b1 = c1 = 0,where c0, k, and c2 = 0 are arbitrary constants. Thenfrom (6), (8) and (10) we obtain the soliton solutions

20 E. Fan / Physics Letters A 282 (2001) 1822

of Eq. (1) of bell-type for all u, v and w:

u= 13( 8k2)+ 4k2 tanh2[k(x + t)],

v =4(3k4c0 2k2c2 + 4k4c2)

3c22

+ 4k2

c2tanh2

[k(x + t)],

w = c0 + c2 tanh2[k(x + t)].

Similarly, substituting ansatz (9) into (7) yieldska1 + k3a1 + 3ka0a1 3kb1c0 3kb0c1 = 0,3ka21 + 2ka2 + 8k3a2 + 6ka0a2 6kb1c1 = 0,ka1 4k3a1 3ka0a1 + 9ka1a2 + 3kb1c0+3kb0c1 = 0,

3ka21 2ka2 20k3a2 6ka0a2 + 6ka22+6kb1c1 = 0,

3k2a1 9ka1a2 = 0, 12k3a2 6ka22 = 0,kb1 2k3b1 3ka0b1 = 0,kb1 + 8k3b1 + 3ka0b1 3ka2b1 = 0,3ka1b1 = 0, 3ka1c1 = 0,6k3b1 + 3ka2b1 = 0,kc1 2k3c1 3ka0c1 = 0,kc1 + 8k3c1 + 3ka0c1 3ka2c1 = 0,6k3c1 + 3ka2c1 = 0,which has solutions

a0 = 13( 2k2), a1 = 0, a2 = 2k2,

b0 =4k2c0( + k2)

3c21, b1 = 4k

2( + k2)3c1

,

where c0, , k and c1 = 0 are arbitrary constants.In this way, we also find another soliton solution ofEq. (1) of bell-type for u but kink-type for v, w:

u= 13( 2k2)+ 2k2 tanh2[k(x + t)],

v =4k2c0( + k2)

3c21

+ 4k2( + k2)

3c1tanh

[k(x + t)],

w = c0 + c1 tanh[k(x + t)].

3. Soliton solutions for the coupled MKdVequation

Let u(x, t)= u(), v(x, t)= v(), = x+t , thenEq. (2) becomes

u = 12u 3u2u + 3

2v + 3(uv) 3u,

(11)v = v 3vv 3uv + 3u2v + 3v.Balancing the highest-order linear terms and nonlinearterms leads to the following two anstze:

(12)u= a0 + a1, v = b0 + b1,and

(13)u= a0 + a1, v = b0 + b1 + b22.Substituting (12) into (11) gives

ka1 + k3a1 + 3ka1 + 3ka20a1 3ka1b03ka0b1 = 0,

6ka0a21 + 3k2b1 6ka1b1 = 0,ka1 4k3a1 3ka1 3ka20a1 + 3ka31+3ka1b0 + 3ka0b1 = 0,

6ka0a21 3k2b1 + 6ka1b1 = 0,3k3a1 3ka31 = 0,3ka1 + kb1 2k3b1 3ka20b1 + 3k2a1b1+3kb0b1 = 0,

6ka0a1b1 + 3kb21 = 0, 6ka0a1b1 3kb21 = 0,3ka1 kb1 + 8k3b1 + 3k20b1 6k2a1b13ka21b1 3kb0b1 = 0,

6k3b1 + 3k2a1b1 + 3ka21b1 = 0,for which, with the aid of Mathematica, we find

a0 = b12k , a1 = k, b0 =

2

(1+ k

b1

),

(14) = 14

(4k2 6+ 6k

b1+ 3b

21

k2

),

where b0 = 0, k = 0 are arbitrary constants. By using(12) and (14), we get kink-type soliton solutions of

E. Fan / Physics Letters A 282 (2001) 1822 21

Eq. (2) for both u and v:

u= b12k

+ k tanh(k),

v = 2

(1+ k

b1

)+ b1 tanh(k),

with

= x + 14

(4k2 6+ 6k

b1+ 3b

21

k2

)t .

For ansatz (13), with help of Mathematica, we have

ka1 + k3a1 + 3ka1 + 3ka20a1 3ka1b03ka0b1 3k2b2 = 0,

6ka0a21 + 3k2b1 6ka1b1 6ka0b2 = 0,ka1 4k3a1 3ka1 3ka20a1 + 3ka31 + 3ka1b0+3ka0b1 + 12k2b2 9ka1b2 = 0,

6ka0a21 3k2b1 + 6ka1b1 + 6ka0b2 = 0,3k3a1 3ka31 9k2b2 + 9ka1b2 = 0,3ka1 + kb1 2k3b1 3ka20b1 + 3k2a1b1+3kb0b1 = 0,

6ka0a1b1 + 3kb21 + 2kb2 16k3b2 6ka20b2+6k2a1b2 + 6kb0b2 = 0,

3ka1 kb1 + 8k3b1 + 3ka20b1 6k2a1b1 3ka21b13kb0b1 12ka0a1b2 + 9kb1b2 = 0,

6ka0a1b1 3kb21 2kb2 + 40k3b2 + 6ka20b212k2a1b2 6ka21b2 6kb0b2 + 6kb22 = 0,

6k3b1 + 3k2a1b1 + 3ka21b1 + 12ka0a1b29kb1b2 = 0,

24k3b2 + 6k2a1b2 + 6ka2b2 6kb22 = 0.Solving these equations by means of Mathematicagives

a0 = 0, a1 = k,b0 = 12

(4k2 + ), b1 = 0, b2 =2k2,

= 12(2k2 3),

where k is an arbitrary constant. Then we find thatanother kind of soliton solution for Eq. (2) is of kink-

type for u but bell-type for v:

u= k tanh(k),v = 1

2(4k2 + ) 2k2 tanh2(k),

with

= x + 12(2k2 3)t .

Remark. We have found four kinds of soliton solu-tions for the new generalized HirotaSatsuma coupledKdV equation (1) and the new coupled KdV equa-tion (2) by using a Riccati equation and symbolic com-putation. Two kinds of them are singular soliton solu-tions. Such solutions develop a singularity at a finitepoint, i.e., for any fixed t = t0, there exist x0 at whichthese solutions blow up. There is much current interestin the formation of so-called hot spots or blow upof solutions [810]. It appears that these singular solu-tions will model this physical phenomena. The methodused in this Letter has some merits in contrast with thetanh-function method. It not only uses a more simplealgorithm to produce an algebraic system but also canpick up singular soliton solutions with no extra effort.In addition, this method is also computerizable, whichallow us to perform complicated and tedious algebraiccalculation on a computer.

Acknowledgements

I am grateful to Professor Gu Chaohao, ProfessorHu Hesheng and Professor Zhou Zixiang for theirenthusiastic guidance and help. I also would like toexpress my sincere thanks to the referees for theirhelpful suggestions. This work has been supported byChinese Basic Research Plan Mathematics Mecha-nization and a Platform for Automated Reasoning,the Postdoctoral Science Foundation of China and theShanghai Postdoctoral Science Foundation of China.

References

[1] Y.T. Wu, X.G. Geng, X.B. Hu, S.M. Zhu, Phys. Lett. A 255(1999) 259.

[2] R. Hirota, J. Satsuma, Phys. Lett. A 85 (1981) 407.[3] J. Satsuma, R. Hirota, J. Phys. Soc. Jpn. 51 (1982) 332.

22 E. Fan / Physics Letters A 282 (2001) 1822

[4] E.G. Fan, H.Q. Zhang, Phys. Lett. A 246 (1998) 403.[5] W. Malfliet, Am. J. Phys. 60 (1992) 650.[6] E.J. Parkes, B.R. Duffy, Comput. Phys. Commun. 98 (1996)

288.

[7] E.J. Parkes, B.R. Duffy, Phys. Lett. A 229 (1997) 217.[8] N.F. Smyth, J. Austral. Math. Soc. Ser. B 33 (1992) 403.[9] P.A. Clarkson, E.L. Mansfield, Physica D 70 (1993) 250.

[10] N.A. Kudryashov, D. Zargayan, J. Phys. A 29 (1996) 8067.