Stability of Periodic Travelling Wave Solutions to the Kawahara Equation

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.