Abstract
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 Schrodinger 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 colli...
Highlights
Gravity-capillary waves are surface waves in the regime where the restoring effects of both gravity and surface tension are similar in magnitude
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
The transverse instability of line solitary waves is predicted by a similar instability of line solitary waves in the nonlinear Schrodinger equation
Summary
Gravity-capillary waves are surface waves in the regime where the restoring effects of both gravity and surface tension are similar in magnitude. For an air-water interface, this implies a free-surface length scale of approximately 1 cm. At this length scale, the phase speed has a minimum about which waves are locally nondispersive. Gravity-capillary solitary waves are localized, traveling, nonlinear waves whose Fourier transform decays rapidly away from this minimum. On a one-dimensional (1D) free surface of a two-dimensional (2D) fluid domain, gravitycapillary solitary waves resemble traveling wavepackets. In this work we ignore the effects of viscosity and surface dissipation which, at this length scale, need to be included
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