Solitary wave solutions of a Whitham–Boussinesq system. (August 2021)

• Record Type: Journal Article
• Authors: Dinvay, E.; Nilsson, D.
• Is Part Of: Nonlinear analysis. Volume 60(2021)
• Publication Date: 2021-08
• Subjects: Nonlinear Dispersive Equations -- Solitary waves -- Concentration compactness -- Boussinesq system; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2020.103280
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in Dinvay et al. (2019), where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers (Dinvay, 2019; Dinvay et al., 2019) the initial-value problem was studied and well-posedness in classical Sobolev spaces was proved. Here we prove existence of solitary wave solutions and provide their asymptotic description. Our proof relies on a variational approach and a concentration-compactness argument. The main difficulties stem from the fact that in the considered Euler–Lagrange equation we have a non-local operator of positive order appearing both in the linear and non-linear parts. Our approach allows us to obtain solitary waves for a particular Boussinesq system as well.