Square patterns and quasipatterns in weakly damped Faraday waves

Pattern formation in parametric surface waves is studied in the limit of weak viscous dissipation. A set of quasipotential equations (QPEs) is introduced that admits a closed representation in terms of surface variables alone. A multiscale expansion of the QPEs reveals the importance of triad resonant interactions, and the saturating effect of the driving force leading to a gradient amplitude equation. Minimization of the associated Lyapunov function yields standing wave patterns of square symmetry for capillary waves, and hexagonal patterns and a sequence of quasipatterns for mixed capillary-gravity waves. Numerical integration of the QPEs reveals a quasipattern of eightfold symmetry in the range of parameters predicted by the multiscale expansion.