Periodic and solitary-wave solutions of the Degasperis-Procesi equation

V.O. Vakhnenko, E.J. Parkes

Research output: Contribution to journalArticle

114 Citations (Scopus)

Abstract

Travelling-wave solutions of the Degasperis-Procesi equation are investigated. The solutions are characterized by two parameters. For propagation in the positive x-direction, hump-like, inverted loop-like and coshoidal periodic-wave solutions are found; hump-like, inverted loop-like and peakon solitary-wave solutions are obtained as well. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. A transformed version of the Degasperis-Procesi equation, which is a generalization of the Vakhnenko equation, is also considered. For propagation in the positive x-direction, hump-like, loop-like, inverted loop-like, bell-like and coshoidal periodic-wave solutions are found; loop-like, inverted loop-like and kink-like solitary-wave solutions are obtained as well. For propagation in the negative x-direction, well-like and inverted coshoidal periodic-wave solutions are found; well-like and inverted peakon solitary-wave solutions are obtained as well. In an appropriate limit, the previously known solutions of the Vakhnenko equation are recovered.
Original languageEnglish
Pages (from-to)1059-1073
Number of pages14
JournalChaos, Solitons and Fractals
Volume20
Issue number5
DOIs
Publication statusPublished - Jun 2004

Keywords

  • Degasperis-Procesi equation
  • Vakhnenko equation
  • wave solutions
  • coshoidal wave

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