On coupled differential equations which model the evolution of interacting capillary-gravity wave modes and related questions of stability

摘要

The author considers the small-amplitude waves which may arise on the surface of a perfect fluid due to the interaction of theMth andNth harmonics of a fundamental wave. A pair of coupled nonlinear partial differential equations for the complex potential amplitudes are derived. These model the evolution of the wavetrain up to third order. Solutions to these equations are obtained and some of the corresponding wave profiles are sketched. The question of the stability of these waves to small perturbations is then addressed. Longitudinal, transverse, and oblique perturbations are considered. It turns out that the stability of these waves is dependent on the ratioM/N. Regions of stability are identified