On the interaction of four water-waves

The mathematical and statistical properties of the evolution of a system of four interacting surface gravity waves are investigated in detail. Any deterministic quartet of waves is shown to evolve recurrently, but the ensemble averages taken over many realizations with random initial conditions reach constant asymptotic values. The characteristic time-scale for which such asymptotic values are approached is extremely large when randomness is introduced through the initial phases. The characteristic time-scale becomes of an order comparable to that of the recurrence periods when beside the random initial phases, the initial amplitudes are taken to be Rayleigh-distributed. The ensemble-averaged results in the second case resemble, to a certain extent, those derived from the kinetic equation.