Modulational instability in the full-dispersion Camassa-Holm equation

TL;DR

This paper analyzes the stability of small periodic traveling waves in a nonlinear dispersive water wave model extending the Camassa-Holm equation, incorporating full dispersion and nonlinear effects, with results aligning with classical water wave instabilities.

Contribution

It provides a detailed stability analysis of small periodic waves in a full-dispersion Camassa-Holm type equation, including effects of surface tension, improving understanding over previous models.

Findings

Without surface tension, results align with Benjamin-Feir instability.

With surface tension, the model predicts stability limits consistent with physical asymptotics.

The analysis extends stability results to a broader class of water wave models.

Abstract

We determine the stability and instability of a sufficiently small and periodic traveling wave to long wavelength perturbations, for a nonlinear dispersive equation which extends a Camassa-Holm equation to include all the dispersion of water waves and the Whitham equation to include nonlinearities of medium amplitude waves. In the absence of the effects of surface tension, the result qualitatively agrees with the Benjamin-Feir instability of a Stokes wave. In the presence of the effects of surface tension, it qualitatively agrees with those from formal asymptotic expansions of the physical problem and it improves upon that for the Whitham equation, correctly predicting the limit of strong surface tension. We discuss the modulational stability and instability in the Camassa-Holm equation and related models.