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Two-dimensional solitary waves for a Benney-Luke equation. (English) Zbl 0935.35139
Summary: 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 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.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
76B25Solitary waves (inviscid fluids)
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction