Undular bore solution of the Camassa-Holm equation

T R Marchant, N F Smyth

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

Modulation theory is developed for a periodic peakon solution of the Camassa-Holm equation. An explicit simple wave solution of these modulation equations is then derived; this simple wave describing the evolution into an undular bore of an initial step. The characteristic on which the expansion fan occurs (propagating at a nonlinear group velocity) has a turning point, illustrating the fact that there is a minimum nonlinear group velocity at which the waves can propagate. A linear analytical solution, based on an integral of the Airy function, is then derived to describe the evanescent portion of the undular bore behind the turning point. Good agreement is found between the modulation theory plus Airy integral solution and numerical solutions.

Original languageEnglish
Article number057602
Pages (from-to)-
Number of pages4
JournalPhysical Review E
Volume73
Issue number5
DOIs
Publication statusPublished - May 2006

Keywords / Materials (for Non-textual outputs)

  • DE-VRIES EQUATION
  • PEAKED SOLITONS
  • WAVE

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