Model equations for gravity-capillary waves in deep water

Benjamin Akers, Paul A. Milewski

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model is derived directly from the Euler equations. Two further simpler models are proposed, both having the full gravity-capillary dispersion relation, but preserving exactly either a quadratic energy or a momentum. Solitary wavepacket waves are calculated for each model. Each model supports the elevation and depression waves known to exist in the Euler equations. The stability of these waves is discussed, as is the dynamics resulting from instabilities and solitary wave collisions.

Original languageEnglish (US)
Pages (from-to)49-69
Number of pages21
JournalStudies in Applied Mathematics
Issue number1
StatePublished - Jul 2008

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

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