On weakly singular and fully nonlinear travelling shallow capillary–gravity waves in the critical regime

Dimitrios Mitsotakis, Denys Dutykh, Aydar Assylbekuly, Dauren Zhakebayev

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8 Scopus citations

Abstract

In this Letter we consider long capillary–gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially focus on the critical regime, where the surface tension is exactly balanced by the gravity force. We show that our long wave model with a critical Bond number admits stable travelling wave solutions with a singular crest. These solutions are usually referred to in the literature as peakons or peaked solitary waves. They satisfy the usual speed-amplitude relation, which coincides with Scott–Russel's empirical formula for solitary waves, while their decay rate is the same regardless their amplitude. Moreover, they can be of depression or elevation type independent of their speed. The dynamics of these solutions are studied as well.

Original languageBritish English
Pages (from-to)1719-1726
Number of pages8
JournalPhysics Letters, Section A: General, Atomic and Solid State Physics
Volume381
Issue number20
DOIs
StatePublished - 25 May 2017

Keywords

  • Capillary–gravity waves
  • Nonlinear dispersive waves
  • Peakons

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