On the stability of large-amplitude gravity-capillary surface waves

We consider the stability of periodic gravity-capillary waves of finite amplitude for small values of the surface tension. Linear stability with respect to both superharmonic and subharmonic perturbations is calculated for each solution, and our methodology obtains the full eigenvalue spectrum consisting of growth rates and temporal frequencies. For small surface tension, the gravity-capillary wave solution space consists of a countably-infinite number of solution branches that coalesce in the small-surface-tension limit, which forms one of the main complications of our study. When the energy is fixed as an amplitude constraint, we find that the superharmonic instability associated with near-limiting gravity waves emerges at smaller amplitudes in the presence of surface tension. Further, the modulational (long-wave) instability is seen to be stabilised for finite-amplitude solutions in the presence of surface tension. This occurs at surface tension values well below that previously obtained via weakly-nonlinear theory, and the stabilisation is nonmonotonic as very small fluctuations in the surface tension of solutions produce large changes in their stability properties.