Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3)

Jian Zhang,Chiping Zhang,Yunan Cui

doi:10.1155/2017/9743475

Abstract

Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.

Highlights

Among the well-known soliton hierarchies are the KdV hierarchy, the AKNS hierarchy, and the Kaup-Newell hierarchy [1]
The trace identity is used for constructing Hamiltonian structures of soliton equations, which is proposed by Tu [2, 3]
In the case of non-semi-simple Lie algebras, integrable couplings of soliton equations are generated by zero curvature equations [4, 5] and the corresponding Hamiltonian structures are obtained by the variational identity [6,7,8]

Summary

Introduction

Among the well-known soliton hierarchies are the KdV hierarchy, the AKNS hierarchy, and the Kaup-Newell hierarchy [1]. In the case of non-semi-simple Lie algebras, integrable couplings of soliton equations are generated by zero curvature equations [4, 5] and the corresponding Hamiltonian structures are obtained by the variational identity [6,7,8]. It is always interesting to explore any new procedure for generating integrable couplings for different soliton hierarchies, even from existing non-semisimple Lie algebras. Seeking new integrable systems including soliton hierarchies and integrable couplings forms a pretty important and interesting area of research in mathematical physics. Bi-integrable couplings and tri-integrable couplings of soliton hierarchies, Ma proposed a new way to generate integrable couplings through a few classes of matrix Lie algebras consisting of block matrices [10]. Our work is essentially motivated by [17,18,19]

Bi-Integrable Couplings and Hamiltonian Structures

Tri-Integrable Couplings and Hamiltonian Structures

Conclusion