An invariant measure for the equation utt-uxx+u3=0

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13 Scopus citations

Abstract

Numerical studies of the initial boundary-value problem of the semilinear wave equation utt-uxx+u3=0 subject to periodic boundary conditions u(t, 0)=u(t, 2 π), ut(t, 0)=ut(t, 2 π) and initial conditions u(0, x)=u0(x), ut(0, x)=v0(x), where u0(x) and v0(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial state u0(x), v0(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above "returning" property.

Original languageEnglish (US)
Pages (from-to)1-16
Number of pages16
JournalCommunications in Mathematical Physics
Volume98
Issue number1
DOIs
StatePublished - Mar 1 1985
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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