On the Lie-Poisson structure of the nonlinearized discrete eigenvalue problem
type: Item
instance of: scholarly article
zbMATH Open document ID: 1026.37057
DOI: 10.1063/1.533440
author: Dianlou Du
author: Y. Wu
published in: Journal of Mathematical Physics
publication date: 2001-08-30T00:00:00Z
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 22E70
zbMATH DE Number: 1639321
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1063/1.533440
OpenAlex ID: W2013866380
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title: On the Lie-Poisson structure of the nonlinearized discrete eigenvalue problem
review text: It is well-known that the Lie-Poisson structure associated with a Lie algebra constitutes a generalized Hamiltonian structure on a Poisson manifold. In the paper under review a discrete version of the generalized finite-dimensional integrable Hamiltonian system is constructed. First a Lie group homomorphism from \(\text{SL}(2,\mathbb R)\) into \(\text{SL}(3,\mathbb R)\) is introduced by means of the adjoint action of \(\text{SL}(2, \mathbb R)\) onto its Lie algebra. Next a \(3\times 3\) discrete matrix eigenvalue problem, which has the same isospectral evolution equation as the \(2\times 2\) matrix eigenvalue problem, is given. Under a constraint between potentials and eigenfunctions a Poisson map, which is the nonlinearization of the \(3\times 3\) matrix spectral problem, is obtained and the integrability of this map is investigated. Finally a reduction on the symplectic foliation induced by the corresponding Lie-Poisson structure is established, and the resulting integrable symplectic map is exactly the nonlinearized \(2\times 2\) eigenvalue problem.
reviewed by: Tomasz Rybicki
description: scientific article; zbMATH DE number 1639321