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.