Dynamics of Partial Differential Equations

Volume 10 (2013)

Number 4

Existence, physical sense and analyticity of solitons for a 2D Boussinesq-Benney-Luke system

Pages: 313 – 342

DOI: http://dx.doi.org/10.4310/DPDE.2013.v10.n4.a1

Authors

Alex M. Montes (Departamento Matemáticas, Universidad del Cauca, Popayá, Colombia)

José R. Quintero (Departamento Matemáticas, Universidad del Valle, Cali, Colombia)

Abstract

We show the existence and the analyticity of solitons (solitary waves of finite energy) for a 2D-Boussinesq-Benney-Luke type system that emerges in the study of the evolution of long water waves with small amplitude in the presence of surface tension. We follow a variational approach by characterizing travelling waves as minimizers of some functional under a suitable constrain. Using Lion’s concentration-compactness principle, we prove that any minimizing sequences converges strongly, after an appropriate translation, to a minimizer. The Boussinesq-Benney-Luke system is formally close to the Benney-Luke equation and to the Kadomtsev-Petviashivili (KP) equation. For wave speed small and surface tension large, we assure some physical sense for this water wave system by establishing that a suitable (renormailized) family of solitons of the Boussinesq-Benney-Luke system converges to a nontrivial soliton for the KP-I equation.

Keywords

solitons, concentration-compactness principle, variational methods, analyticity

2010 Mathematics Subject Classification

35Q51, 37K05, 76B15, 76B25

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