On the wave length of smooth periodic traveling waves of the Camassa-Holm equation

A. Geyer, J. Villadelprat

Research output: Contribution to journalArticleResearchpeer-review

24 Citations (Scopus)

Abstract

© 2015 The Authors. This paper is concerned with the wave length λ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height a (or "peak-to-peak amplitude"). Our main result establishes monotonicity properties of the map a λ(a), i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated with the equation, which distinguish between the two possible qualitative behaviors of λ(a), namely monotonicity and unimodality. The key point is to relate λ(a) to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.
Original languageEnglish
Pages (from-to)2317-2332
JournalJournal of Differential Equations
Volume259
Issue number6
DOIs
Publication statusPublished - 1 Jan 2015

Keywords

  • Camassa-Holm equation
  • Center
  • Critical period
  • Traveling wave solution
  • Wave height
  • Wave length

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