Nonlinear Super Integrable Couplings of Super Yang Hierarchy and Its Super Hamiltonian Structures

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1. Introduction

With the development of soliton theory, super integrable systems associated with Lie superalgebra 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 KP hierarchy etc., have been studied in [1] [2] [3] [4]. There are some interesting results on the super integrable systems, such as Darboux transformation [5], super Hamiltonian structures in [6] [7], binary nonlinearization [8] and reciprocal transformation [9] and so on.

There search of integrable couplings of the well known integrable hierarchy has received considerable attention [10] [11] [12]. A few approaches to construct linear integrable couplings of the classical soliton equation are presented by permutation, enlarging spectral problem, using matrix Lie algebra [12] constructing new loop Lie algebra and creating semi-direct sums of Lie algebra. Zhang [13] once employed two kinds of explicit Lie algebra F and G to obtain the nonlinear integrable couplings of the GJ hierarchy and Yang hierarchy, respectively. Recently, You [14] presented a scheme for constructing nonlinear super integrable couplings for the super integrable hierarchy.

Inspired by Zhang [13] and You [14], we hope to construct nonlinear super integrable couplings of the super Yang hierarchy 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 Yang hierarchy. Finally, we will reduce the nonlinear super Yang integrable Hamiltonian couplings to some special cases.

2. Enlargement of Lie Super Algebra

Consider the Lie super algebra. Its basis is

(1)

where are even element and are odd elements. Their non- zero (anti) commutation relations are

(2)

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

(3)

where are even, and are odd.

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

(4)

Define a loop super algebra corresponding to the Lie super algebra gl(6, 2), denote by

(5)

The corresponding (anti) commutative relations are given as

(6)

3. Nonlinear Super Integrable Couplings of Super Yang Hierarchy

If Let us start from an enlarged spectral problem associated with,

(7)

where are even potentials, but are odd ones.

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

(8)

with

(9)

where and are commuting fields, and are anti-com- muting fields.

Substituting

(10)

into previous equation gives the following recursive formulas

(11)

From previous equations, we can successively deduce

Equation (11) can be written as

(12)

where

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

(13)

with

(14)

From the compatible condition according to (7) and (13), we get the zero curvature equation

(15)

which gives a nonlinear Lax super integrable hierarchy

(16)

The super integrable hierarchy (16) is a nonlinear super integrable couplings for the Yang hierarchy in [15]

(17)

4. Super Hamiltonian Structures

A direct calculation reads

(18)

Substituting above results into the super trace identity [7]

(19)

Comparing the coefficients of on both side of (19). From the initial values in (11), we obtain. Thus we have

(20)

It then follows that the nonlinear super integrable couplings (16) possess the following super Hamiltonian form

(21)

where

(22)

is a super Hamiltonian operator and are Hamiltonian functions.

5. Reductions

Taking (16) reduces to a nonlinear integrable couplings of the Yang hierarchy in [13].

When in (16), we obtain the nonlinear super integrable couplings of the second order super Yang equations

(23)

Especially, taking in (23), we can obtain the nonlinear integrable couplings of the second order Yang equations in [13]

(24)

If setting in (23), we obtain the second order super Yang equations of (17) in [15]

(25)

6. Remarks

In this paper, we introduced an approach for constructing nonlinear integrable couplings of super integrable hierarchy. The method in this paper can be applied to other super integrable systems for constructing their integrable couplings. How to obtain the soliton solutions about equations deduced in this paper is worth considering for our future work.

Acknowledgements

This work work was supported by the Natural Science Foundation of Henan Province (No. 162300410075), the Science and Technology Key Research Foundation of the Education Department of Henan Province (No.14A110010).

References

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