文章基本信息

标题：Generating integrable lattice hierarchies by some matrix and operator Lie algebras
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作者：Yu-Feng Zhang ; Yan Wang
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2016
卷号：2016
期号：1
页码：313
DOI：10.1186/s13662-016-1039-4
语种：English
出版社：Hindawi Publishing Corporation
摘要：Two types of matrix Lie algebras are presented. We make use of the first loop algebra to obtain a new ( 1 + 1 ) $(1+1)$ -dimensional integrable discrete hierarchy, which generalizes a result given by Gordoa et al., whose reduction is a discrete modified KdV system. Then we produce another new ( 2 + 1 ) $(2+1)$ -dimensional integrable discrete hierarchy with three fields under a ( 2 + 1 ) $(2+1)$ -dimensional non-isospectral linear problem. We again generalize the ( 1 + 1 ) $(1+1)$ - and ( 2 + 1 ) $(2+1)$ -dimensional discrete hierarchies to obtain a positive and negative integrable discrete hierarchy. In addition, we obtain a discrete integrable coupling system of the ( 1 + 1 ) $(1+1)$ -dimensional discrete hierarchy presented in the paper by enlarging such the loop algebras. Next, we apply the second matrix loop algebra to introduce an isospectral problem and deduce a new integrable discrete hierarchy, whose quasi-Hamiltonian structure is derived from the trace identity proposed by Tu Guizhang, which can be reduced to some modified Toda lattice equations. A type of Darboux transformation of a reduced discrete system of the latter integrable discrete hierarchy is obtained as well. We introduce two types of operator-Lie algebras according to a given spectral problem by a matrix Lie algebra and apply the r-matrix theory to obtain a few lattice integrable systems, including two ( 2 + 1 ) $(2+1)$ -dimensional lattice systems.
关键词：integrable discrete hierarchy ; discrete integrable coupling ; Lie algebra