A New Solution to the Hirota—Satsuma Coupled KdV Equations by the Dressing Method

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Commun. Theor. Phys. 60 (2013) 266–268 Vol. 60, No. 3, September 15, 2013

A New Solution to the Hirota–Satsuma Coupled KdV Equations by the Dressing

Method∗

ZHU Jun-Yi (Ádº),1,† ZHOU De-Wen (±©),2 and YANG Juan-Juan ( ïï)1

1School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

2College of Science, Zhongyuan University of Technology, Zhengzhou 451191, China

(Received Novembor 22, 2112; revised manuscript received March 2, 2013)

Abstract The Hirota–Satsuma coupled KdV equations associated 2 × 2 matrix spectral problem is discussed by the

dressing method, which is based on the factorization of integral operator on a line into a product of two Volterra integral

operators. A new solution is obtained by choosing special kernel of integral operator.

PACS numbers: 02.30.Ik, 02.30.Jr, 04.20.JbKey words: dressing method, Hirota–Satsuma coupled KdV equation, exact solution

1 Introduction

The dressing method proposed by Zakharov and

Shabat[1] played a crucial role in the development of soli-

ton theory. Since then, many versions of dressing method

were developed to solve the soliton equations.[2−9] The

dressing method has been proven to be a powerful tool

to construct the integrable equations and simultaneously

give their solutions.

In this article, we will study the Hirota–Satsuma cou-

pled KdV (HS-KdV) equations[10] by virtue of the dressing

method,

ut −1

2uxxx + 3uux − 3(vw)x = 0 ,

vt + vxxx − 3uvx = 0 ,

wt + wxxx − 3uwx = 0 . (1)

The equations describe an interaction of two long waves

with different dispersion relations. A lot of research on

Eq. (1) has been conducted, for example, Darboux trans-

formation, Backlund transformation, Miura transforma-

tions, double periodic solutions, solitary wave solutions

and other properties.[10−19] A type of HS equation was

investigated in Ref. [12] through dressing scalar differen-

tial operators. The HS-KdV equations (1) associated with

4 × 4 matrix spectral problem were studied in Refs. [13-

15,17–19]. In the present paper, we will apply the dressing

method based on a set of 2×2 matrix operators to the HS-

KdV equations and obtain some new explicit solutions.

2 The Spectral Transform and Lax Pair

In this section, we consider the 2 × 2 matrix initial

differential operators

M 1 = σ3∂2x , M2 = I∂t − 2I∂2

x , (2)

where I is identity matrix. The associated dressed op-

erators M1 and M2 can be obtained from the following

dressing relation

M j(I + K) − (I + K)M j = 0 , j = 1, 2 , (3)

where K is a matrix Volterra operator. It is noted that

for any vector function ϕ(x) we have

Kϕ(x) =

∫ ∞

x

K(x, z, t)ϕ(z)dz ,

where the kernel K(x, z, t) of the Volterra operator satis-

fied the symmetry condition

KT(z, x) = −JK(x, z)J , J =

(0 1

−1 0

), (4)

here T denotes the transpose of a matrix. Here and after

we shall suppress the variable t dependence in K for the

sake of convenience.

It is readily verified that, in view of Eqs. (2)–(4), the

dressed operators M j , j = 1, 2 in terms of the kernel of

the Volterra operator take the forms

M1 = M 1 + 2σ3Kx + [σ3, [Kx]z=x] ,

M2 = M 2 − 6Kx∂x − 6KxK − 6([Kx]z=x)x , (5)

where K = K|z=x = K(x, x) and [A, B] = AB − BA.

From the constraint condition (4), we know that K re-

duces to a scale function. In order to derive the HS-KdV

equations, we introduce the potential function Q as follow

Q = 2σ3Kx + [σ3, [Kx]z=x] , (6)

and set Q = Qd + Qo, where Qd and Qo denote the diag-

onal part and off-diagonal part of the matrix Q, respec-

tively. Note that K is a scale function, Eq. (6) implies

∗Supported by the National Natural Science Foundation of China under Grant No. 11001250†E-mail: jyzhu@zzu.edu.cn

c© 2013 Chinese Physical Society and IOP Publishing Ltd

http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn

No. 3 Communications in Theoretical Physics 267

that

Q = −

(u v

w −u

), (7)

where

u = −2Kx , v = −2([Kx]z=x)12 ,

w = −2([Kx]z=x)21 . (8)

or

2σ3Kx = −Qd , [σ3, [Kx]z=x] = −Qo . (9)

Then the compatibility condition of the dressed oper-

ators (5) gives

Qt = 3Qxxx + σ3Pxx − SQx + [Q, P ] ,

σ3Sxx + 2σ3Px + 6Qxx = 0 ,

P = −3σ3QdK − 6([Kx]z=x)x, S = −3σ3Q

d , (10)

which can be further reduced to

Qdt −

1

2Qd

xxx − 3σ3QdQd

x − 3σ3(Qo)2x = 0 ,

Qot + Qo

xxx + 3σ3QdQo

x = 0 . (11)

The components of Eq. (11) give the HS-KdV equations

(1).

3 The Explicit Solutions

In the following, we shall construct the explicit solution

of the HS-KdV equations (1). To this end, we consider the

following commutation relations

[M j , F ] = 0 , j = 1, 2 , (12)

where the matrix integral operator F has the definition

Fϕ(x) =

∫ ∞

−∞

F (x, z, t)ϕ(z)dz , (13)

with the kernel F (x, z, t) satisfies the same symmetry con-

dition (4) as that of K(x, z).

Substitution Eq. (2) into Eq. (12) we find

σ3Fxx − Fzzσ3 = 0 , Ft = 2(Fxx + Fzz) . (14)

Let the kernel F (x, z, t) takes the form

F (x, z, t) = −JΩT(x, t)C−1Ω(z, t) , (15)

where Ω(x, t) is a N × 2 matrix, C satisfied CT = −C is

N × N matrix. It is easy to see that if Ω admits

Ωxx = C−1Ωσ3 , Ωt = 2Ωxxx , (16)

then F defined by Eq. (15) is the solution of Eq. (14). It

is remarked that the rank of the matrix C must be even.

In addition, by virtue of the Gel’fand–Levitan–Mar-

chenko equation

K(x, z, t)+F (x, z, t)+

∫ ∞

x

K(x, s, t)F (s, z, t)ds = 0, (17)

and the expression (15) we know that

K(x, z, t) = JΩT(x, t)L−1(x, t)Ω(z, t) , (18)

where

L(x, t) = C +

∫ ∞

x

Ω(s, t)JΩT(s, t)ds . (19)

It is noted that in order to give the explicit solution of

HS-KdV equation (1), one need to solve the system (16).

As an example, we consider the case N = 2 and let C = J

and Ω = (g, f) with

g(x, t) = (g1(x, t), g2(x, t))T ,

f(x, t) = (f1(x, t), f2(x, t))T ,

Then Eq. (16) reduce to

gxx = −Jg , fxx = Jf ,

gt = 2gxxx , ft = 2fxxx . (20)

The solutions of this system can be taken as

g1 = A cosh ξ cos η + B sinh ξ sin η ,

g2 = A sinh ξ sin η − B cosh ξ cos η , (21)

and

f1 = C cosh ξ sin η + D sinh ξ cos η ,

f2 = −C sinh ξ cos η + D cosh ξ sin η , (22)

where A, B, C, D are arbitrary constants and

ξ = kx − 4k3t , η = kx + 4k3t , 4k4 = 1 . (23)

In this case of N = 2, the kernel K(x, z, t) has the

form

K(x, z, t)

= ∆−1

(fT(x, t)Jg(z, t) fT(x, t)Jf(z, t)

gT(x, t)Jg(z, t) −gT(x, t)Jf(z, t)

), (24)

where

∆(x, t) = 1 +

∫ ∞

x

gT(s, t)Jf(s, t)ds . (25)

Note that

fT(x, t)Jf(x, t) = 0 = gT(x, t)Jg(x, t) ,

fT(x, t)Jg(x, t) = −gT(x, t)Jf(x, t) = ∆x , (26)

then from Eqs. (8) and (20) we know that

u = −2d2

dx2ln ∆ ,

v = −2γ∆−1(sinh 2ξ − sin 2η) ,

w = 2δ∆−1(sinh 2ξ + sin 2η) , (27)

where

∆ = ε(t) + α cosh 2ξ cos 2η + β sinh 2ξ sin 2η , (28)

with ε(t) is some function and

α = −1

8k(AC + BD + AD − BC) ,

γ =k

2(C2 + D2) ,

β = −1

8k(AC + BD − AD + BC) ,

δ = −k

2(A2 + B2) .

268 Communications in Theoretical Physics Vol. 60

Fig. 1 Solution (27) with k =√

2/2, A = B = C = D = 2, ε(t) = 0. The figure of w is rotated, since the figuresof w and v are similar.

It is remarked that the case of N = 3 is not suitable for the present scheme. While for N = 4, one may choose

C =(

0 J

J 0

)and f , g are four dimensional vector functions. The same procedure as above can be applied to obtain the

solutions of HS-KdV equations (1). Figure 1 describe the shape and motion of the new solution (27) in a certain case.

4 Conclusion

We use the dressing method based on a set of 2 × 2 matrix operators to study the HS-KdV equation. A suitable

symmetry condition for the kernel of Volterra operator is introduced to establish the relation between the kernel and

the potential function, and then the associated Lax pair and equation are derived. We find an interesting way to

construct the explicit solutions of the HS-KdV equation, in which a new solution is obtained.

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