  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.
