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Existence and analyticity of lump solutions for generalized Benney-Luke equations. (English) Zbl 1051.35045
Summary: We prove the existence and analyticity of lump solutions (finite energy solitary waves) for generalized Benney-Luke equations that arise in the study of the evolution of small amplitude, three-dimensional water waves. The family of generalized Benney-Luke equations reduce formally to the generalized Korteweg-de Vries (GKdV) equation and to the generalized Kadomtsev-Petviashvili (GKP-I or GKP-II) equation in the appropriate limits. Existence lumps are 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 generalized Benney-Luke lump solutions converges to a nontrivial lump solution for the GKP-I equation.
MSC:
35L75Nonlinear hyperbolic PDE of higher (>2) order
35Q53KdV-like (Korteweg-de Vries) equations
35Q51Soliton-like equations