Nonlinear Super Integrable Couplings of a Super Integrable Hierarchy

Sixing Tao

doi:10.4236/jamp.2016.44074

Abstract

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.

Highlights

With the development of soliton theory, super integrable systems associated with Lie super algebra have aroused growing attentions by many mathematicians and physicists
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
You [14] presented a scheme for constructing the nonlinear super integrable couplings for the super integrable hierarchy

Summary

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. You [14] presented a scheme for constructing the nonlinear super integrable couplings for the super integrable hierarchy. We hope to construct nonlinear super integrable couplings of this super integrable hierarchy which was constructed in [16] through enlarging matrix Lie super algebra. Based on the enlarged Lie super algebra gl(6, 2) , we work out nonlinear super integrable Hamiltonian couplings of this super integrable hierarchy. The generator of Lie super algebra gl(6, 2), ei (1 ≤ i ≤ 8) satisfy the following (anti) commutation relations:. The corresponding (anti)commutative relations are given as [eiλ m , e= jλ n ] [ei , ej ]λ m+n , ∀ei , ej ∈ gl(6, 2)

Nonlinear Super Integrable Couplings of a Super Integrable Hierarchy

Super Hamiltonian Structure

Reductions

Conclusion