Computing time-periodic solutions of a model for the vortex sheet with surface tension

Abstract

We compute time-periodic solutions of a simple model for the vortex sheet with surface tension. The model has the same dispersion relation as the full system of evolution equations, and it also has the same destabilizing nonlinearity (if the surface tension parameter were to be set to zero, then this nonlinearity would cause an analogue of the Kelvin-Helmholtz instability). The numerical method uses a gradient descent algorithm to minimize a functional which measures whether a solution of the system is time periodic. We find continua of genuinely time-periodic solutions bifurcating from equilibrium.