Nonlinear Bi-Integrable Couplings of Multicomponent Guo Hierarchy with Self-Consistent Sources

Hongwei Yang,Zhenyu Liu,Baoshu Yin,Huanhe Dong

doi:10.1155/2012/272904

Abstract

Based on a well-known Lie algebra, the multicomponent Guo hierarchy with self-consistent sources is proposed. With the help of a set of non-semisimple Lie algebra, the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy with self-consistent sources are obtained. It enriches the content of the integrable couplings of hierarchies with self-consistent sources. Finally, the Hamiltonian structures are worked out by employing the variational identity.

Highlights

Since the notion of integrable couplings was proposed in view of Virasoro symmetric algebras 1, 2 and the soliton theory, considerable research has been reported on the integrable couplings and associated properties of some known interesting integrable hierarchies, such as the AKNS hierarchy, the KN hierarchy, and the Burger hierarchy 3–6
In order to produce multicomponent integrable systems, a type of multicomponent loop algebra GM was structured, and it followed that some multicomponent integrable systems were given and Hamiltonian structures of multicomponent systems were constructed through the component-trace identity in 7–13 and a general structure between Lie algebras and integrable coupling was given in 14, 15
Ma proposed the notion of nonlinear integrable couplings and gave the general scheme to construct the nonlinear integrable couplings of Advances in Mathematical Physics hierarchies 16

Summary

Introduction

Since the notion of integrable couplings was proposed in view of Virasoro symmetric algebras 1, 2 and the soliton theory, considerable research has been reported on the integrable couplings and associated properties of some known interesting integrable hierarchies, such as the AKNS hierarchy, the KN hierarchy, and the Burger hierarchy 3–6. In 24–26 , Yu and Xia discussed the integrable couplings of hierarchies with self-consistent sources and obtained a series of integrable hierarchies with self-consistent sources and their couplings systems as well as Hamiltonian structures In these papers, the integrable couplings are linear for supplementary variables. We will derive the nonlinear bi-integrable couplings of multicomponent Guo hierarchy with self-consistent sources and their Hamiltonian structures. It will enrich the content of the integrable couplings of hierarchies with self-consistent sources.

Multicomponent Guo Hierarchy with Self-Consistent Sources

Conclusions