Three-dimensional solitary waves in dispersive wave systems

Abstract

Fully localized three-dimensional solitary waves, commonly referred to as 'lumps', have received far less attention than two-dimensional solitary waves in dispersive wave systems. Prior studies have focused in the long-wave limit, where lumps exist if the long-wave speed is a minimum of the phase speed and are described by the Kadomtsev-Petviashvili (KP) equation. In the water-wave problem, in particular, lumps of the KP type are possible only in the strong-surface-tension regime (Bond number, B > 1/3), a condition that limits the water depth to a few mm. In the present thesis, a new class of lumps is found that is possible under less restrictive physical conditions. Rather than long waves, these lumps bifurcate from infinitesimal sinusoidal waves of finite wavenumber at an extremum of the phase speed. As the group and phase velocities are equal there, small-amplitude lumps resemble fully localized wavepackets with envelope and crests moving at the same speed, and the wave envelope along with the induced mean-flow component are governed by a coupled Davey-Stewartson equation system of elliptic-elliptic type. The lump profiles feature algebraically decaying tails at infinity owing to this mean flow. In the case of water waves, lumps of the wavepacket type are possible when both gravity and surface tension are present on water of finite or infinite depth for B < 1/3.

(cont.) The asymptotic analysis of these lumps in the vicinity of their bifurcation point at the minimum gravity-capillary phase speed, is in agreement with recent fully numerical computations by Parau, Cooker & Vanden-Broeck (2005) as well as a formal existence proof by Groves & Sun (2005). A linear stability analysis of the gravity-capillary solitary waves that also bifurcate at the minimum gravity-capillary phase speed, reveals that they are always unstable to transverse perturbations, suggesting a mechanism for the generation of lumps. This generation mechanism is explored in the context of the two-dimensional Benjamin (2-DB) equation, a generalization to two horizontal spatial dimensions of the model equation derived by Benjamin (1992) for uni-directional, small-amplitude, long interfacial waves in a two-fluid system with strong interfacial tension. The 2-DB equation admits solitary waves and lumps of the wavepacket type analogous to those bifurcating at the minimum gravity-capillary phase speed in the water-wave problem. Based on unsteady numerical simulations, it is demonstrated that the transverse instability of solitary waves of the 2-DB equation results in the formation of lumps, which propagate stably and are thus expected to be the asymptotic states of the initial-value problem for fully localized initial conditions.