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Dynamics of gravity–capillary solitary waves in deep water

Published online by Cambridge University Press:  15 August 2012

Zhan Wang  and
Paul A. Milewski
Zhan Wang
Affiliation:
Department of Mathematics, University of Wisconsin–Madison, Madison WI, 53706, USA
Paul A. Milewski*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
*
†Email address for correspondence: [email protected]
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Abstract

The dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonlinear focusing of wavepackets, quasi-elastic collisions, and the generation of propagating, spatially localized, time-periodic structures akin to breathers.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 708 , 10 October 2012 , pp. 480 - 501
Copyright
Copyright © Cambridge University Press 2012

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