_{1}

Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. And its super Hamiltonian structures were established by using super trace identity. As its reduction, special cases of this nonlinear super integrable coupling were obtained.

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 [

The research of integrable couplings of the well known integrable hierarchy has received considerable attention [

In this paper, we hope to construct nonlinear super integrable couplings of this super integrable hierarchy which was constructed in [

Consider the Lie super algebra B(0, 1). Its basis is

where

Let us enlarge the Lie super algebra B(0, 1) to the Lie super algebra gl(6, 2) with a basis

where

The generator of Lie super algebra gl(6, 2),

(4)

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

The corresponding (anti)commutative relations are given as

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

where

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

with

where

Substituting

into previous equation gives the following recursive formulas

From previous equations, we can successively deduce

Equations (11) can be written as

where

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

From the compatible condition

which gives a nonlinear Lax super integrable hierarchy

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

A direct calculation reads

Substituting above results into the super trace identity [

and comparing the coefficients of

From the initial values in (11), we obtain

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

where

is a super Hamiltonian operator and

Taking

When

Let

Especially, taking

If setting

In this paper, we introduced an approach for constructing nonlinear integrable couplings of super integrable hierarchy. Zhang [

in [

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.

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).

Sixing Tao, (2016) Nonlinear Super Integrable Couplings of a Super Integrable Hierarchy. Journal of Applied Mathematics and Physics,04,648-654. doi: 10.4236/jamp.2016.44074