Abstract

Nonlinear integrable couplings of super classical-Boussinesq hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then, its super Hamiltonian structures were established by using super trace identity. 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 C-KdV hierarchy, and super classical-Boussinesq hierarchy have been studied [1–4]. There are some interesting results on the super integrable systems, such as Darboux transformation [5], super Hamiltonian structures [6, 7], binary nonlinearization [8], and reciprocal transformation [9].

The research of integrable couplings of the well-known integrable hierarchy has received considerable attentions [10–15]. A few approaches to construct linear integrable couplings of the classical soliton equation are presented by permutation, enlarging spectral problem, using matrix Lie algebra constructing new loop Lie algebra [16], and creating semidirect sums of Lie algebra. Recently, Ma and Zhu [17, 18] 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 which 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 algebra. Based on the enlarged Lie super algebra , we work out nonlinear super integrable Hamiltonian couplings of the super classical-Boussinesq hierarchy. Finally, we will reduce the nonlinear super classical-Boussinesq integrable Hamiltonian couplings to some special cases.

2. Enlargement of Lie Superalgebra

Consider the Lie superalgebra . Its basis is where , , are even elements and , are odd ones. Their nonzero (anti)commutation relations are

Let us enlarge the Lie superalgebra to the Lie superalgebra with a basis where , , , , , are even and , are odd. The generators of the Lie superalgebra , , , satisfy the following (anti)commutation relations: Define a loop superalgebra corresponding to the Lie superalgebra , denoted by The corresponding (anti)commutative relations are given as

3. Nonlinear Super Integrable Couplings of the Super Classical-Boussinesq Hierarchy

Let us start from an enlarged spectral problem associated with where where , , , and are even potentials but and are odd ones. In order to obtain super integrable couplings of super integrable hierarchy, we first solve the adjoint representation (8) with where , , , , , and are commuting fields, and are anticommuting fields, and , are anticommuting fields.

Substituting into the above equation gives the following recursive formulas: From these equations, we can successively deduce Equations (12) can be written as whereThen, let us consider the spectral problem (8) with the following auxiliary problem: with From the compatible condition according to (8) and (17), we get the zero equation: which gives a nonlinear Lax super integrable hierarchy The super integrable hierarchy (19) is a nonlinear super integrable coupling for the super classical-Boussinesq hierarchy

4. Super Hamiltonian Structures

A direct calculation reads Substituting the above results into the super trace identity in [6, 7] yields that Comparing the coefficients of on both sides of (23), From the initial values in (12), we obtain . Thus, we have It then follows that the nonlinear super integrable couplings (22) possess the following super Hamiltonian form: where is a super Hamiltonian operator and are Hamiltonian functions.

5. Reductions

Taking , the hierarchy (26) reduces to a nonlinear integrable coupling of the super classical-Boussinesq hierarchy.

When in (26), we obtain the nonlinear super integrable couplings of the second-order super classical-Boussinesq equations Taking in (28), we obtain the nonlinear super integrable couplings of the second-order super classical-Boussinesq equation

Letting in (28), we have

When setting , , in (28), we obtain the second-order super classical-Boussinesq equations

Conflict of Interests

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