Pattern formation in weakly damped parametric surface waves

We present a theoretical study of nonlinear pattern formation in parametric surface waves for fluids of low viscosity, and in the limit of large aspect ratio. The analysis is based on a quasi-potential approximation to the equations governing fluid motion, followed by a multiscale asymptotic expansion in the distance away from threshold. Close to onset, the asymptotic expansion yields an amplitude equation which is of gradient form, and allows the explicit calculation of the functional form of the cubic nonlinearities. In particular, we find that three-wave resonant interactions contribute significantly to the nonlinear terms, and therefore are important for pattern selection. Minimization of the associated Lyapunov functional predicts a primary bifurcation to a standing wave pattern of square symmetry for capillary-dominated surface waves, in agreement with experiments. In addition, we find that patterns of hexagonal and quasi-crystalline symmetry can be stabilized in certain mixed capillary-gravity waves, even in this case of single-frequency forcing. Quasi-crystalline patterns are predicted in a region of parameters readily accessible experimentally.