Dynamics of three-dimensional gravity-capillary solitary waves in deep water

Benjamin Akers, Paul A. Milewski

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30 Scopus citations


A model equation for gravity-capillary waves in deep water is proposed. This model is a quadratic approximation of the deep water potential flow equations and has wavepacket-type solitary wave solutions. The model equation supports line solitary waves which are spatially localized in the direction of propagation and constant in the transverse direction, and lump solitary waves which are spatially localized in both directions. Branches of both line and lump solitary waves are computed via a numerical continuation method. The stability of each type of wave is examined. The transverse instability of line solitary waves is predicted by a similar instability of line solitary waves in the nonlinear Schrödinger equation. The spectral stability of lumps is predicted using the waves' speed energy relation. The role of wave collapse in the stability of these waves is also examined. Numerical time evolution is used to confirm stability predictions and observe dynamics, including instabilities and solitary wave collisions.

Original languageEnglish (US)
Pages (from-to)2390-2408
Number of pages19
JournalSIAM Journal on Applied Mathematics
Issue number7
StatePublished - 2010

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

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