Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications
type: Item
instance of: scholarly article
title: Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications
zbMATH Open document ID: 0602.58017
DOI: 10.1007/BF01466596
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: In the introduction (Section 1) the authors start with a very short review of methods of investigation of integrability properties of some special classes of infinite-dimensional Hamiltonian systems, expressed by nonlinear evolution equations. To describe these properties different approaches have been proposed and are known now. The authors claim that the approach proposed in the paper does not belong to either of the known types. The main idea, of fairly geometrical nature, is based on the direct investigation of the integrability structures defined on some differentiable manifolds in order to give criteria which guarantee the reducibility of such structures on certain regular submanifolds. By using this method we can recover the integrability structure of more relevant hierarchies of evolution equations discussed in the literature. In order to provide an overview of the contents of the whole paper it seems to be enough to list the titles of Sections 2-11: (2) Poisson- Nijenhuis manifolds; (3) The basic PN manifold; (4) The theory of restriction; (5) Some elements of the geometry of PN manifolds; (6) A reduction method for Nijenhuis manifolds; (7) The integrability structure of the AKNS hierarchy; (8) The recursion operator for the Heisenberg chain; (9) Integrability structure for KN hierarchy; (10) The recursion operator for WKI hierarchy; (11) A further example of reduction of a Nijenhuis structure. The KdV hierarchy.
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 37J15
Mathematics Subject Classification ID: 37K60
Mathematics Subject Classification ID: 37L60
zbMATH DE Number: 3971811
zbMATH Keywords: Poisson-Nijenhuis manifolds
zbMATH Keywords: basic PN manifold
zbMATH Keywords: reduction method for Nijenhuis manifolds
zbMATH Keywords: AKNS hierarchy
zbMATH Keywords: Heisenberg chain
zbMATH Keywords: KN hierarchy
zbMATH Keywords: recursion operator for WKI hierarchy
zbMATH Keywords: Nijenhuis structure
zbMATH Keywords: KdV hierarchy
author: Franco Magri
author: Carlo Morosi
author: Orlando Ragnisco
reviewed by: Andrzej Pelczar
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01466596
OpenAlex ID: W2092690254
description: scientific article; zbMATH DE number 3971811