Methods and Applications of Analysis

Volume 21 (2014)

Number 2

From periodic travelling waves to solitons of a 2D water wave system

Pages: 241 – 264

DOI: http://dx.doi.org/10.4310/MAA.2014.v21.n2.a4

Author

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

Abstract

We use a variational approach to establish the existence of $x$-periodic travelling waves and its interrelation with solitons for a 2D water wave system for three-dimensional water wave dynamics in the weakly nonlinear long-wave regime. As common in many 1D water wave models, we show that a special sequence of the $x$-periodic 2D travelling wave solutions parametrized by the period $k$ is uniformly bounded in norm and converges to a soliton in $\mathbb{R}^2$ (solitary wave of finite energy) in an appropriate sense, indicating that the shape of $x$-periodic 2D travelling waves of period $k$ and solitons are almost the same, as the period $k$ is big enough.

Keywords

2D Boussinesq water wave system, Periodic travelling waves, solitons, variational methods

2010 Mathematics Subject Classification

35A15, 35B10, 35Q35, 37K40

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