The validity of Whitham's approximation for a Klein-Gordon-Boussinesq model

TL;DR

This paper proves the validity of Whitham's approximation for a coupled Boussinesq-Klein-Gordon system, using normal form transformations and energy estimates, contributing to the broader theory of dispersive wave modulation.

Contribution

It introduces a rigorous proof of Whitham's approximation validity for a specific coupled dispersive system, expanding the theoretical framework for wave modulation analysis.

Findings

Validation of Whitham's approximation for the system

Development of a proof based on normal form transformations

Establishment of energy estimates for the approximation

Abstract

In this paper we prove the validity of a long wave Whitham approximation for a system consisting of a Boussinesq equation coupled with a Klein-Gordon equation. The proof is based on an infinite series of normal form transformations and an energy estimate. We expect that the concepts of this paper will be a part of a general approximation theory for Whitham's equations which are especially used in the description of slow modulations in time and space of periodic wave trains in general dispersive wave systems.