Advances in Mathematical Physics

Volume 2013 (2013), Article ID 520765, 8 pages

http://dx.doi.org/10.1155/2013/520765

Research Article

Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy and Its Super Hamiltonian Structures

Sixing Tao¹ and Tiecheng Xia²

¹School of Mathematics and Information Science, Shangqiu Normal University, Shangqiu 476000, China

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

Received 25 February 2013; Revised 26 July 2013; Accepted 9 August 2013

Academic Editor: Xing Biao Hu

Copyright © 2013 Sixing Tao and Tiecheng Xia. 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.

Nonlinear integrable couplings of super Broer-Kaup-Kupershmidt hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained.

1. Introduction

With the development of soliton theory, super integrable systems associated with Lie super algebra have aroused growing attentions by many mathematicians and physicists. It was known that super integrable systems contained the odd variables, which would provide more prolific fields for mathematical researchers and physical ones. Several super integrable systems, including super AKNS hierarchy, super KdV hierarchy, super NLS-MKdV hierarchy, super Tu hierarchy, super Broer-Kaup-Kupershmidt hierarchy, have been studied in [1–8]. There are some interesting results on the super integrable systems, such as Darboux transformation in [9], super Hamiltonian structures in [10–12] binary nonlinearization in [13], and reciprocal transformation in [14].

The research of integrable couplings of the well-known integrable hierarchy has received considerable attention in [15–23] A few approaches to construct linear integrable couplings of the classical soliton equation are presented by permutation, enlarging spectral problem, using matrix Lie algebra [24] constructing new loop Lie algebra and creating semidirect sums of Lie algebra. Recently, Ma [25] and Ma and Zhu [26] presented a scheme for constructing nonlinear continuous and discrete integrable couplings using the block type matrix algebra. However, there is one interesting question for us is how to generate nonlinear super integrable couplings for the super integrable hierarchy.

In this paper, we would like to construct nonlinear super integrable couplings of the super soliton equations through enlarging matrix Lie super algebra. We take the Lie algebra as an example to illustrate the approach for extending Lie super algebras. Based on the enlarged Lie super algebra , we work out nonlinear super integrable Hamiltonian couplings of the super Broer-Kaup-Kupershmidt hierarchy. Finally, we will reduce the nonlinear super Broer-Kaup-Kupershmidt integrable Hamiltonian couplings to some special cases.

2. Enlargement of Lie Super Algebra

Consider the Lie super algebra . Its basis is

where , , and are even elements and , and are odd elements. Their nonzero (anti)commutation relations are

Let us enlarged the Lie super algebra to the Lie super algebra with a basis

where , , , , , and are even, and , and are odd.

The generator of Lie super algebra , , , satisfy the following (anti)commutation relations:

Define a loop super algebra corresponding to the Lie superalgebra , denote by

The corresponding (anti)commutative relations are given as

3. Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy

Let us start from an enlarged spectral problem associated with :

where , , , and are even potentials but and are odd ones.

In ordr to obtain super integrable couplings of super integrable hierarchy, we first solve the adjoint representation of (7),

with

where , , , , , and are commuting fields and , and are anticommuting fields. Then we obtain

Substituting

into the previous equation gives the following recursive formulas:

From these equations, we can successively deduce

Equations (12) can be written as

where

with

Then, let us consider the spectral problem (7) with the following auxiliary spectral problem:

with

From the compatible condition, , according to (7) and (17), we get the zero curvature equation:

which gives a nonlinear Lax super integrable hierarchy:

The super integrable hierarchy (20) is a nonlinear super integrable couplings for the super BKK hierarchy:

4. Super Hamiltonian Structures

A direct calculation reads

Substituting the above results into the super trace identity in [11, 12],

yields that

Comparing the coefficients of on both side of (24),

From the initial values in (12), we obtain . Thus we have

It then follows that the nonlinear super integrable couplings (20) possess the following super Hamiltonian form

where

is a super Hamiltonian operator and () are Hamiltonian functions.

It can be verified that

where and are conjugate operators of and , respectively.

If we define the following Poisson bracket

where denotes the inner product, as , we can obtain

From (14) and (31), we have

Suppose are arbitrary nonnegative integers consider

By using the properties of and in (29), we can obtain

Ascending the subscript of left factor , then we have

By using the antisymmetric law of Poisson bracket (31), we have

So the conserved functionals are in involution in pairs under the Poisson bracket (30).

5. Reductions

Taking , the hierarchy (27) reduces to a nonlinear integrable couplings of the Broer-Kaup-Kupershmidt hierarchy.

When in (27), we obtain the nonlinear super couplings of the second super Broer-Kaup-Kupershmidt equations:

Particularly, taking in (37), we can get the nonlinear integrable couplings of the second order Broer-Kaup-Kupershmidt equations:

Let in (37); we have

If setting , , and in (37), we obtain the second order super Broer-Kaup-Kupershmidt equations:

6. Remarks

In this paper, we introduced an approach for constructing nonlinear integrable couplings of super integrable hierarchy. Zhang [27] once employed two kinds of explicit Lie algebra and to obtain the nonlinear integrable couplings of the GJ hierarchy and Yang hierarchy, respectively. It is easy to see that Lie algebra given in [27] is isomorphic to the Lie algebra span in . So we can obtain nonlinear integrable couplings of super GJ and Yang hierarchy easily. The method in this paper can be applied to other super integrable systems for constructing their integrable couplings.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61072147), the Natural Science Foundation of Henan Province (no. 132300410202), the Science and Technology Key Research Foundation of the Education Department of Henan Province (no. 12A110017) and the Youth Research Foundation of Shangqiu Normal University (no. 2011QN12).

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