Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation
type: Item
instance of: scholarly article
title: Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation
zbMATH Open document ID: 1184.37055
author: Xin-Yue Li
author: Yuan-Qing Zhang
author: Qiu-Lan Zhao
published in: Journal of Computational and Applied Mathematics
publication date: 2009-11-06T00:00:00Z
review text: There are very few completely integrable lattice differential equations. The list includes the Ablowitz--Ladik lattice, the Toda lattice, the Blaszak--Marciniak lattice, and their modifications. The authors construct two hierarchies of new lattice differential equations, which are related to the same discrete isospectral problem \[ \psi_{n+1} = U_n(u,\lambda) \psi_n, \quad U_n(u,\lambda) = \left( \begin{matrix} \lambda & 1 + r_n \\ \lambda s_n & 1 + s_n + r_n s_n \end{matrix} \right). \] The hierarchies are built in positive and negative powers of \(\lambda\) in the power series expansions. Not only integrable lattice equations are computed in the explicit form but also the Hamiltonian structure, the conservation laws, and the Darboux transformations are computed for equations of positive hierarchy.
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 5627006
zbMATH Keywords: discrete integrable system
zbMATH Keywords: discrete zero curvature equation
zbMATH Keywords: Hamiltonian structure
zbMATH Keywords: conservation laws
zbMATH Keywords: Darboux transformation
reviewed by: Dmitry Pelinovsky
MaRDI profile type: Publication
full work available at URL: j.cam.2009.09.009
OpenAlex ID: W2135382488
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DOI: 10.1016/J.CAM.2009.09.009
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description: scientific article; zbMATH DE number 5627006