The Peierls-Nabarro and Benjamin-Ono equations

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Abstract

An intimate connection between the Peierls-Nabarro equation in crystal-dislocation theory and the travelling-wave form of the Benjamin-Ono equation in hydro-dynamics is uncovered. It is used to prove the essential uniqueness of Peierls' solution of the Peierls-Nabarro equation and to give, in closed form, all solutions of the analogous periodic problem. The latter problem is shown to be an example of global bifurcation with no secondary, symmetry-breaking, bifurcations for a nonlinear Neumann boundary-value problem or, equivalently, for an equation involving the conjugate operator, which is the Hilbert transform of functions on the unit circle.

Original languageEnglish
Pages (from-to)136-150
Number of pages15
JournalJournal of Functional Analysis
Volume145
Issue number1
DOIs
Publication statusPublished - 1 Apr 1997

ASJC Scopus subject areas

  • Analysis

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