Travelling waves in the Boussinesq type systems

TL;DR

This paper investigates the existence of solitary wave solutions in various Boussinesq type systems modeling surface water waves, using a unified approach based on the implicit function theorem to simplify the analysis.

Contribution

It introduces a new, more straightforward method for proving the existence of solitary waves in Boussinesq systems, replacing complex variational techniques.

Findings

Existence of solitary waves established for multiple Boussinesq variants

Unified proof technique simplifies previous complex methods

Potential for broader application to similar wave models

Abstract

Considered herein are a number of variants of the Boussinesq type systems modeling surface water waves. Such equations were derived by different authors to describe the two-way propagation of long gravity waves. A question of existence of special solutions, the so called solitary waves, is of a particular interest. There are a number of studies relying on a variational approach and a concentration-compactness argument. These proofs are technically very demanding and may vary significantly from one system to another. Our approach is based on the implicit function theorem, which makes the treatment easier and more unified.