Pattern formation in weakly dampedparametric surface wavesBy

We present a theoretical study of nonlinear pattern formation in parametric surface waves for uids of low viscosity, and in the limit of large aspect ratio. The analysis is based on a quasi-potential approximation to the equations governing uid 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 nonlineari-ties. In particular, we nd that three-wave resonant interactions contribute signiicantly 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 nd 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.