  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.
