Center manifold application: existence of periodic travelling waves for the 2D $abcd$-Boussinesq system

TL;DR

This paper proves the existence of periodic travelling wave solutions for a 2D Boussinesq system related to water waves, using center manifold theory to analyze infinite-dimensional solution families and their global behavior.

Contribution

It introduces a novel application of center manifold reduction to establish existence and properties of periodic travelling waves in a 2D water-wave model.

Findings

Infinite-dimensional family of small periodic solutions identified.

Global solutions exist for small initial data in the bottom velocity.

Spatial evolution governed by a dispersive, nonlocal nonlinear wave equation.

Abstract

In this paper we study the existence of periodic travelling waves for the 2D $abcd$ Boussinesq type system related with the three-dimensional water-wave dynamics in the weakly nonlinear long-wave regime. We show that small solutions that are periodic in the direction of translation (or orthogonal to it) form an infinite-dimensional family, by characterizing these solutions through spatial dynamics when we reduce to a center manifold of infinite dimension and codimension the linearly ill-posed mixed-type initial-value problem. As happens for the Benney-Luke model and the KP II model for wave speed large enough and large surface tension, we show that a unique global solution exists for arbitrary small initial data for the two-component bottom velocity, specified along a single line in the direction of translation (or orthogonal to it). As a consequence of this fact, we show that the spatial evolution of bottom velocity is governed by a dispersive, nonlocal, nonlinear wave equation