Two-dimensional solitary waves for a Benney–Luke equation

Abstract

We prove the existence of finite-energy solitary waves for isotropic Benney–Luke equations that arise in the study of the evolution of small amplitude, three-dimensional water waves when the horizontal length scale is long compared with the depth. The family of Benney–Luke equations discussed in this paper includes the effect of surface tension and a variety of equivalent forms of dispersion. These equations reduce formally to the Korteweg–de Vries (KdV) equation and to the Kadomtsev–Petviashvili (KP-I or KP-II) equation in the appropriate limits. Existence of finite-energy solitary waves or lumps is proved via the concentration-compactness method. When surface tension is sufficiently strong (Bond number larger than 1/3), we prove that a suitable family of Benney–Luke lump solutions converges to a nontrivial lump solution for the KP-I equation.