Stability of Periodic Travelling Wave Solutions to the Kawahara Equation

TL;DR

This paper investigates the stability of periodic travelling-wave solutions to the Kawahara equation, identifying resonance conditions and high-frequency instabilities through analytical and numerical methods.

Contribution

It provides a comprehensive stability analysis of Kawahara equation solutions, including resonance regimes and asymptotic instability growth, supported by numerical validation.

Findings

Resonance occurs in specific parameter regimes.

Small-amplitude solutions can exhibit high-frequency instabilities.

Numerical results confirm asymptotic predictions and highlight regimes beyond analytical scope.

Abstract

We analyse the stability of periodic, travelling-wave solutions to the Kawahara equation and some of its generalizations. We determine the parameter regime for which these solutions can exhibit resonance. By examining perturbations of small-amplitude solutions, we show that generalised resonance is a mechanism for high-frequency instabilities. We derive a quadratic equation which fully determines the stability region for these solutions. Focussing on perturbations of the small-amplitude solutions, we obtain asymptotic results for how their instabilities develop and grow. Numerical computation is used to confirm these asymptotic results and illustrate regimes where our asymptotic analysis does not apply.