Dynamics and stability of gravity-capillary solitary waves

Abstract

Over the past several years, it has been recognized that a new class of solitary waves can propagate in nonlinear dispersive wave systems if the phase speed of linear waves attains a local extremum at some finite wavenumber. Near such a point, solitary waves in the form of small-amplitude wavepackets can be obtained for which the phase speed of the carrier oscillations matches the group speed of their envelope. Such an extremum is found in the analysis of water waves when the restoring forces of both gravity and surface tension are taken into account, and certain kinds of these gravity-capillary solitary waves have been observed in experiments. While past theoretical studies have focussed mainly on determining steady solitary wave profiles, very little work has been done on examining their stability properties which is the thrust of this thesis. Beginning in the weakly nonlinear regime, an asymptotic analysis of linear stability is presented and comparison is made with numerical computations. Contrary to predictions of the nonlinear Schrbdinger (NLS) equation, some free solitary wave types are found to be unstable owing to exponentially effects terms that lie beyond standard two-scale perturba- tion theory. Moreover, numerical simulations show that unstable gravity-capillary solitary waves may decompose into stable solitary waves that have soliton properties. Stability results are then extended to the fully nonlinear regime to treat both free and forced situa- tions using numerical techniques to solve the full hydrodynamic equations in steady form. A dramatic difference is found between the linear stability of free and forced waves in both weakly and fully nonlinear cases, and results obtained here are compared with laboratory experiments.

(cont.) The analysis followed in the free-surface problem is then generalized to examine the dynamics of gravity-capillary interfacial solitary waves in a layered two-fluid system. Here, the linear stability and limiting wave forms of free solitary waves are determined over a range of system parameters using the full hydrodynamic equations. Finally, a related problem of gravity-capillary envelope solitons is considered under the general situation of unequal phase and group speeds. By asymptotic and numerical techniques it is found that envelope solitons are generally nonlocal-tails are radiated owing to a resonance mechanism that is beyond the NLS equation.