Mathematical properties of Klein-Gordon-Boussinesq systems

TL;DR

This paper investigates the mathematical properties of the Klein-Gordon-Boussinesq system, including well-posedness, traveling wave solutions, and the validity of KdV approximation, providing both theoretical analysis and numerical insights.

Contribution

It offers a comprehensive mathematical analysis of the KGB system, including well-posedness, solitary wave existence, and validation of KdV approximation, which were not previously fully explored.

Findings

Established conditions for local and global solutions.

Derived existence of various solitary wave solutions.

Validated KdV approximation through computational methods.

Abstract

The Klein-Gordon-Boussinesq (KGB) system is proposed in the literature as a model problem to study the validity of approximations in the long wave limit provided by simpler equations such as KdV, nonlinear Schr\"{o}dinger or Whitham equations. In this paper, the KGB system is analyzed as a mathematical model in three specific points. The first one concerns well-posedness of the initial-value problem with the study of local existence and uniqueness of solution and the conditions under which the local solution is global or blows up at finite time. The second point is focused on traveling wave solutions of the KGB system. The existence of different types of solitary waves is derived from two classical approaches, while from their numerical generation several properties of the solitary wave profiles are studied. In addition, the validity of the KdV approximation is analyzed by computational means and from the corresponding KdV soliton solutions.