Commuting flows and infinite-dimensional tori: sine-Gordon
type: Item
instance of: scholarly article
title: Commuting flows and infinite-dimensional tori: sine-Gordon
zbMATH Open document ID: 1386.35372
author: Martin jun. Schwarz
published in: Journal of Nonlinear Science
publication date: 2017-12-06T00:00:00Z
review text: The author considers in this paper the periodic sine-Gordon equation \[ u_{tt}-u_{xx}=\sin u, \] where \(u(x,t)\) is a real valued function in the class of smooth one functions. He explains why the complete set of conserved functionals for sine-Gordon is an infinite-dimensional torus and identifies the generic flow (generated by this equation) with straight-line almost periodic motion on an infinite-dimensional torus. The paper is organized as follows: the first section is an introduction to the subject. Section 2 contains preliminaries and covers complete integrability of nonlinear evolution equations. Moreover, the author gives an infinite-dimensional version of the well known Arnold-Liouville theorem [\textit{V. I. Arnol'd}, Sib. Mat. Zh. 4, 471--474 (1963; Zbl 0131.39102)] or [\textit{V. I. Arnold}, Mathematical methods of classical mechanics. New York - Heidelberg - Berlin: Springer-Verlag. (1978; Zbl 0386.70001)]. Finally, section 3 is devoted to applying the results obtained in section 2 to the study of the sine-Gordon equation.
reviewed by: Ahmed Lesfari
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35R15
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 6815940
zbMATH Keywords: sine-Gordon
zbMATH Keywords: almost periodic flow
zbMATH Keywords: infinite-dimensional tori
MaRDI profile type: Publication
full work available at URL: s00332-017-9369-2
OpenAlex ID: W2590455540
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DOI: 10.1007/S00332-017-9369-2
description: scientific article; zbMATH DE number 6815940