Abstract and Applied Analysis

Volume 2014 (2014), Article ID 295068, 6 pages

http://dx.doi.org/10.1155/2014/295068

Research Article

The Semidirect Sum of Lie Algebras and Its Applications to C-KdV Hierarchy

Xia Dong,1,2 Tiecheng Xia,² and Desheng Li¹

¹Department of Mathematics, Shenyang Normal University, Shenyang 110034, China

²Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 18 January 2014; Accepted 8 May 2014; Published 27 May 2014

Academic Editor: Yufeng Zhang

Copyright © 2014 Xia Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By use of the loop algebra , integrable coupling of C-KdV hierarchy and its bi-Hamiltonian structures are obtained by Tu scheme and the quadratic-form identity. The method can be used to produce the integrable coupling and its Hamiltonian structures to the other integrable systems.

1. Introduction

Integrable coupling is a new topic of the Soliton theory; especially, looking for the new Hamiltonian structure of integrable coupling is more important. The integrable coupling of some known integrable hierarchies is obtained. But their Hamiltonian structure has not been presented because there exists a limitation in trace identity till the quadratic-form identity [1] and the variational identity [2] are proposed. In this paper, a higher-dimensional Lie algebra and the loop algebra are constructed [3, 4]. With the help of Tu scheme [5] and the quadratic-form identity, the integrable coupling of C-KdV hierarchy as well as its bi-Hamiltonian structures is produced.

2. Basic Principle of the Semidirect Sum of Lie Algebras

Let be a linear space over real or complex number field together with multiplication, for any , if satisfy

(1)distributive law

(2)multiplication commutativity

Then, is called algebra.

Lie algebra is an algebra over number field , if its multiplication satisfies the following:

(1)bilinearity

(2)anticommutative

(3)the Jacobi identity

where denote the multiplication of ,  . The multiplication of Lie algebra is called Lie product. One kind of the most important Lie algebras on integrable systems is , where denote matrix order over number field .

satisfies for arbitrary Lie algebra ; then is called Lie ideal.

Lie algebra is called simple Lie algebra if has and 0 as Lie ideal and without other Lie ideal. Semisimple Lie algebra can be written as

where is simple Lie algebra. We have already known that , , and are all semisimple Lie algebras which has been studied by Cartan long ago [5]. We also know that Lie algebra can be written as

where is semisimple Lie algebras and is solvable Lie algebras [3, 6, 7] and denote the semidirect sum. So we can apply the above basic principle to integrable coupling systems.

3. C-KdV Hierarchy

Firstly, let us recall the construction of the C-KdV hierarchy [8, 9]. Consider the basis of

The loop algebra is presented as .

The C-KdV spectral problem reads as

Upon setting , solving the stationary zero curvature equation,

engenders

The compatibility conditions of the spectral problems

determine the C-KdV hierarchy of Soliton equations

where

4. A New Integrable Coupling of the C-KdV Hierarchy

In what follows, we expand Lie algebra into a bigger one as the following Lie algebra :

We do this along with the following commutative relations:

Taking and , it is easy to verify that

where is semisimple Lie algebras and is solvable Lie algebras [3, 6, 7].

In terms of the Lie algebra , we constructed the loop algebra as follows [4, 10], with the following commutative relations:

, , when , and , when . With the help of above equations, we consider an isospectral problem:

Set

Solving the stationary zero curvature equation (10) permits that

where , , , and are nonzero constants.

Assume that ; then (10) may be written as

A direct calculation reads

Take ; then the zero curvature equation

is equivalent to

where and are Hamiltonian operators.

From (22), a recurrence operator is obtained, which satisfies

where

It is easy to verify that

Therefore, the hierarchy (26) is Liouville integrable. Taking , , and , (26) reduces to (13). According to the integrable theory, the hierarchy (26) is the integrable coupling of the C-KdV hierarchy. Furthermore, in the following part we will point out that there exist bi-Hamiltonian structures from constructing of Lie loop algebras.

5. The Bi-Hamiltonian Structures of the Hierarchy (26)

Let

We have =.

In what follows, from , we get

Solving the matrix equation for gives rise to

So we have .

A direct calculation reads

where and .

Substituting the above formulas into the quadratic-form identity yields

Comparison of coefficients of of both sides of the above equations leads to

To fix the we take into the above equation and find .

So

Comparison of coefficients of of both sides of the above equations gives

In this situation, we have .

So

Thus the bi-Hamiltonian structures of the system (26) are given by

From the system (26), we easily give the following equations:

6. Conclusion

On the one hand, we obtain a new integrable coupling of C-KdV hierarchy by expanding a bigger Lie algebra. On the other hand, the bi-Hamiltonian structures of the integrable coupling of C-KdV hierarchy are observed by use of the quadratic-form identity.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The project is in part supported by the Natural Science Foundation of China (Grant no. 11271008), the First-class Discipline of University in Shanghai, and the Shanghai University.

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