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We analyse the stability of periodic, travelling-wave solutions to theKawahara equation and some of its generalizations. We determine the parameterregime for which these solutions can exhibit resonance. By examiningperturbations of small-amplitude solutions, we show that generalised resonanceis a mechanism for high-frequency instabilities. We derive a quadratic equationwhich fully determines the stability region for these solutions. Focussing onperturbations of the small-amplitude solutions, we obtain asymptotic resultsfor how their instabilities develop and grow. Numerical computation is used toconfirm these asymptotic results and illustrate regimes where our asymptoticanalysis does not apply.

Stability of Periodic Travelling Wave Solutions to the Kawahara Equation