Nonlinear Super Integrable Couplings of a Super Integrable Hierarchy

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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 KP hierarchy, etc., have been studied in [1]-[4]. There are some interesting results on the super integrable systems, such as Darboux transformation in [5], super Hamiltonian structures in [6] [7], binary nonlinearization [8] and reciprocal transformation [9] and so on.

The research of integrable couplings of the well known integrable hierarchy has received considerable attention [10]-[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 [13] constructing new loop Lie algebra and creating semi-direct sums of Lie algebra. Recently, You [14] presented a scheme for constructing the nonlinear super integrable couplings for the super integrable hierarchy. Zhang [15] once constructed an integrable hierarchy and discussed Lax representation, Darboux transformation for its constrained flows. Shi [16] constructed the super extension of this hierarchy.

In this paper, we hope to construct nonlinear super integrable couplings of this super integrable hierarchy which was constructed in [16] 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 this super integrable hierarchy. Finally, we will reduce the nonlinear super integrable couplings to some special cases.

2. Enlargement of Lie Super Algebra B(0, 1)

Consider the Lie super algebra B(0, 1). 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 B(0, 1) to the Lie super algebra gl(6, 2) with a basis

(3)

where are even, and are odd.

The generator of Lie super algebra gl(6, 2), 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 a Super Integrable Hierarchy

If Let us start from an enlarged spectral problem associated with gl(6, 2),

(7)

where are even potentials, but are odd ones.

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

(8)

with

(9)

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

Substituting

(10)

into previous equation gives the following recursive formulas

(11)

From previous equations, we can successively deduce

Equations (11) can be written as

(12)

where

(13)

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

(14)

From the compatible condition according to (7) and (14), 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 integrable hierarchy in [16]

(17)

4. Super Hamiltonian Structure

A direct calculation reads

(18)

Substituting above results into the super trace identity [7]

(19)

and comparing the coefficients of on both side of (19)

(20)

From the initial values in (11), we obtain. Thus we have

(21)

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

(22)

where

(23)

is a super Hamiltonian operator and are Hamiltonian functions.

5. Reductions

Taking (16) reduces to a nonlinear integrable couplings of the integrable hierarchy in [15].

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

(24)

Let in (24), we have

(25)

Especially, taking in (24), we can obtain the nonlinear integrable couplings of the second order integrable equations

(26)

If setting in (24), we obtain the second order super integrable equations of (17)

(27)

6. Conclusion

In this paper, we introduced an approach for constructing nonlinear integrable couplings of super integrable hierarchy. Zhang [17] 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 [17] is isomorphic to the Lie algebra span in gl(6, 2). So we can obtain nonlinear integr-

able couplings of super GJ and Yang hierarchy easily. The method in this paper can be applied to other super integrable systems for constructing their super integrable couplings.

Acknowledgements

This work was supported by the Natural Science Foundation of Henan Province (No.132300410202), the Science and Technology Key Research Foundation of the Education Department of Henan Province (No. 14A110010), the Youth Backbone Teacher Foundation of Shangqiu Normal University(No. 2013GGJS02).

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