Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with

Advances in Mathematical Physics, Jun 2017

Based on the three-dimensional real special orthogonal Lie algebra , by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with 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.

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Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with

Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with Jian Zhang,1,2 Chiping Zhang,1 and Yunan Cui3 1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2Harbin University of Science and Technology, Rongcheng Campus, Rongcheng 264300, China 3Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China Correspondence should be addressed to Jian Zhang; nc.ude.tsubrh@866naijgnahz Received 29 November 2016; Accepted 10 May 2017; Published 4 June 2017 Academic Editor: Andrei D. Mironov Copyright © 2017 Jian Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Based on the three-dimensional real special orthogonal Lie algebra , by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with 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. 1. Introduction 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 [6–8]. An integrable coupling equationis a triangular integrable system of the following form [9]: where is a function of variables and , , and . If is nonlinear with respect to the second dependent variable , the integrable coupling is called nonlinear. An integrable system of the following form [10] is called a bi-integrable of (1). Similarly, an integrable system of the following form [10] is called a tri-integrable of (1). Integrable couplings correspond to non-semi-simple Lie algebras , and such Lie algebras can be written as semidirect sums [11]: The notion of semidirect sums means that and satisfy , where , with denoting the Lie bracket of . Obviously, is an ideal of . The subscript indicates a contribution to the construction of coupling systems. We also require the closure property between and under the matrix multiplication: , where . Integrable couplings are useful tools for describing and explaining nonlinear phenomena of new evaluation equations. There are very rich mathematical structures behind integrable couplings. In particular, integrable couplings generalize the symmetry problem and describe other integrable properties of integrable equations. In order to enrich multicomponent integrable equations, it has been an important task to explore more integrable properties from multi-integrable couplings. For example, one can find work on the integrable couplings [12, 13]. It is always interesting to explore any new procedure for generating integrable couplings for different soliton hierarchies, even from existing non-semi-simple Lie algebras. Recently, seeking new integrable systems including soliton hierarchies and integrable couplings forms a pretty important and interesting area of research in mathematical physics. To generate integrable couplings, 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]. Recently, bi-integrable couplings and tri-integrable couplings for the KdV hierarchy and the AKNS hierarchy have been studied considerably [14, 15]. From [16, 17], bi-integrable couplings of a new soliton hierarchy associated with and bi-integrable couplings of a new soliton hierarchy associated with have been studied. In this paper, we will construct bi-integrable and tri-integrable couplings associated with for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Our work is essentially motivated by [17–19]. 2. Bi-Integrable Couplings and Hamiltonian Structures2.1. Bi-Integrable Couplings Associated with So as to generate bi-integrable couplings, we introduce a kind of block matrices:where is an arbitrary nonzero constant and , , and are square matrices of the same order. In the following, we define the corresponding non-semi-simple Lie algebra by a semidirect sum: withwhere the loop algebra is defined by Obviously, we have the matrix commutator relation: with , , and being defined by Let us consider the Lie algebra . It has a basis with which the structure equations of are , ,  . The soliton hierarchy introduced in [18] has a spectral problem with the spectr (...truncated)


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Jian Zhang, Chiping Zhang, Yunan Cui. Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with, Advances in Mathematical Physics, 2017, 2017, DOI: 10.1155/2017/9743475