Discreteness and quasiresonances in weak turbulence of capillary waves

A numerical study is presented which deals with the kinematics of quasiresonant energy transfer in a system of capillary waves with a discrete wave number space in a periodic box. For a given set of initially excited modes and a given level of resonance broadening, the modes of the system are partitioned into two classes, one active, the other forbidden. For very weak nonlinearity the active modes are very sparse. It is possible that this sparsity explains discrepancies between the values of the Kolmogorov constant measured in numerical simulations of weakly turbulent cascades and the theoretical values obtained from the continuum theory. There is a critical level of nonlinearity below which the set of active modes has finite radius in wave number space. In this regime, an energy cascade to dissipative scales may not be possible and the usual Kolmogorov spectrum predicted by the continuum theory not realized.