Chaos and integrability in a nonlinear wave equation
type: Item
instance of: scholarly article
title: Chaos and integrability in a nonlinear wave equation
zbMATH Open document ID: 0802.34044
DOI: 10.1007/BF02219194
author: V. Pereyra
published in: Journal of Dynamics and Differential Equations
publication date: 1994-04-19T00:00:00Z
review text: This paper analyses the integrability of the equation \[ u_{tt}- u_{xx}- au+ u\left({2\over \pi L}\int^{\pi L}_ 0 u^ 2(x)dx\right)^ \alpha= 0,\tag{1} \] where \(a\), \(\alpha\) are positive constants, on the interval \([0,\pi L]\) with Dirichlet boundary conditions. The term integrability is taken to mean Hamiltonian (or Liouville) integrability. The main tool used is the Fourier-sine expansion of the solution of (1) in \(x\). The coefficients of such an expansion satisfy a Hamiltonian system of ODEs in time. The author studies this system for the space of solutions that have a finite Fourier decomposition. For \(\alpha= 2,3,\dots\), the author proves that the 2-mode system possesses chaotic solutions for \(L> 1/\sqrt a\). (A heuristic argument that this holds for any \(\alpha\neq 1,1/2,1/3,\dots\) is also given.) However, for the case \(\alpha= 1\), it is shown that the system of ODEs is integrable for any (finite) number of modes. (An explicit set of commuting first integrals is presented and it is shown that the solution \(u\) are quasi-periodic in time, i.e. functions of periodic functions of \(t\).) (1) is therefore termed integrable in this case. The proof of nonintegrability for \(\alpha= 2,3,\dots\) relies on a nonintegrability result of \textit{L. M. Lerman} [Sel. Math. Sov. 10, 297- 306 (1991; Zbl 0743.58017)], and the proof of integrability in the case \(\alpha= 1\) uses the construction of integrals by \textit{T. Cazenave}, \textit{A. Haraux} and \textit{F. B. Weissler} [J. Dyn. Differ. Equations 5, 129-154 (1993; Zbl 0783.35003)]. The results are interesting because for the case \(\alpha= 1\), (1) approximates the Sine-Gordon equation which is well-known to be integrable.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 34C28
zbMATH DE Number: 553858
zbMATH Keywords: integrability
zbMATH Keywords: Fourier-sine expansion
zbMATH Keywords: Hamiltonian system of ODEs
zbMATH Keywords: chaotic solutions
zbMATH Keywords: set of commuting first integrals
zbMATH Keywords: quasi-periodic
zbMATH Keywords: nonintegrability
zbMATH Keywords: Sine-Gordon equation
Wikidata QID: Q115392684
reviewed by: Nalini Joshi
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02219194
OpenAlex ID: W2141143082
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description: scientific article; zbMATH DE number 553858