On the instability of water waves with surface tension.

Abstract

We analyze the stability of solutions to Euler's equations in the presence of surface tension. First we compute stationary solutions to periodic Euler's equations in a travelling frame of reference and then we analyze their spectral stability. Depending on the coefficient of surface tension, we see resonant effects in the solutions. This results in a myriad of instabilities for gravity-capillary waves. Since the theory for analyzing the stability of water waves is general to all Hamiltonian systems, we extend the results to other equations, mainly ones that are used to model water waves in different asymptotic regimes. We compare the stability results for the model equations to those we obtain for the full water wave system and comment on the applicability of these models.