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

Content Provider	Semantic Scholar
Author	Quintero, Jose R. Montes, Alex M.
Copyright Year	2015
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
Starting Page	641
Ending Page	667
Page Count	27
File Format	PDF HTM / HTML
DOI	10.12988/astp.2017.7837
Alternate Webpage(s)	http://www.m-hikari.com/astp/astp2017/astp9-12-2017/p/quinteroASTP9-12-2017.pdf
Alternate Webpage(s)	https://arxiv.org/pdf/1511.08489v1.pdf
Alternate Webpage(s)	https://doi.org/10.12988/astp.2017.7837
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article