Asymptotic Expansions for Solitary Gravity-Capillary Waves in Two and Three Dimensions

Asymptotic Expansions for Solitary Gravity-Capillary Waves in Two and Three Dimensions

Terry Haut
Department of Applied Mathematics
University of Colorado at Boulder

In this talk, we discuss high-order asymptotic series for two and three dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known solitary wave solution of the KdV equation; in three dimensions, the first term is the rational lump solution of the KP-I equation. We discuss how the two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large amplitude waves, and waves close to the solitary version of the Stokes' extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two- dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension.