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: [email protected]
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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