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Zbl 0935.35139
Pego, Robert L.; Quintero, José Raúl
Two-dimensional solitary waves for a Benney-Luke equation. (English)
[J] Physica D 132, No.4, 476-496 (1999). ISSN 0167-2789

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 2000:: *35Q53 KdV-like equations
76B25 Solitary waves, etc. (inviscid fluids)
76B15 Wave motions (fluid mechanics)

Keywords: weakly nonlinear waves; traveling waves; concentration-compactness

Cited in: Zbl 1137.35424 Zbl 1130.76324 Zbl 1136.76325

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