Stability of periodic travelling shallow-water waves determined by Newton's equation

TL;DR

This paper investigates the existence and stability of periodic travelling-wave solutions in certain shallow-water wave equations, employing advanced mathematical frameworks like Grillakis-Shatah-Strauss theory and Floquet analysis.

Contribution

It applies and extends stability analysis methods to generalized Benjamin-Bona-Mahony and Camassa-Holm equations for the first time.

Findings

Established conditions for the stability of periodic waves.

Identified parameter regimes where solutions are stable.

Demonstrated the effectiveness of Floquet theory in this context.

Abstract

We study the existence and stability of periodic travelling-wave solutions for generalized Benjamin-Bona-Mahony and Camassa-Holm equations. To prove stability, we use the abstract results of Grillakis-Shatah-Strauss and the Floquet theory for periodic eigenvalue problems.