Two-dimensional solitary waves for a Benney-Luke equation

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 co mpared with the depth. The family of Benney-Luke equations discussed in thi s paper includes the effect of surface tension and a variety of equivalent forms of dispersion. These equations reduce formally to the Korteweg-de Vri es (KdV) equation and to the Kadomtsev-Petviashvili (KP-I or KP-II) equatio n in the appropriate limits. Existence of finite-energy solitary waves or l umps is proved via the concentration-compactness method. When surface tensi on is sufficiently strong (Bond number larger than 1/3), we prove that a su itable family of Benney-Luke lump solutions converges to a nontrivial lump solution for the KP-I equation. (C)1999 Elsevier Science B.V. All rights re served.