Nonlinear dynamics of three-dimensional solitary waves

Abstract

In problems of dispersive wave propagation governed by two distinct restoring-force mechanisms, the phase speed of linear sinusoidal wavetrains may feature a minimum, cmin, at non-zero wavenumber, kmin. Examples include waves on the surface of a liquid in the presence of both gravity and surface tension, flexural waves on a floating ice sheet, in which case capillarity is replaced by the flexural rigidity of the ice, and internal gravity waves in layered flows in the presence of interfacial tension. The focus here is on deep-water gravity-capillary waves, where cmin = 23 cm/s with corresponding wavelength Amin = 27r/kmin = 1.71 cm. In this instance, ignoring viscous dissipation, cmin is known to be the bifurcation point of two-dimensional (plane) and three-dimensional (fully localized) solitary waves, often referred to as "lumps"; these are nonlinear disturbances that propagate at speeds below cmin without change of shape owing to a perfect balance between the opposing effects of wave dispersion and nonlinear steepening. Moreover, Cmin is a critical forcing speed, as the linear inviscid response to external forcing moving at Cmin grows unbounded in time, and nonlinear effects as well as viscous dissipation are expected to play important parts near this resonance. In the present thesis, various aspects of the dynamics of gravity-capillary lumps are investigated theoretically. Specifically, it is shown that steep gravity-capillary lumps of depression can propagate stably and they are prominent nonlinear features of the forced response near resonant conditions, in agreement with companion experiment for the generation of gravity-capillary lumps on deep water. These findings are relevant to the generation of ripples by wind and to the wave drag associated with the motion of small bodies on a free surface.