Deterministic and stochastic theory for a resonant triad

We study a simple model of three resonantly-interacting nonlinear waves. Linear stability analysis allows for an easy identification of stable and unstable initial conditions. Subsequently, a reformulation of the problem allows us to establish the onset of phase coherence, demonstrating its critical role in the growth of instabilities. Furthermore, we are able to provide explicit expressions for the time-varying moments of the modal amplitudes, as well as for the bispectrum which measures phase coherence. We provide direct insight into the long-time statistics of the system without Monte-Carlo simulation. This includes long-time asymptotics, which show that the system desynchronises, leading to the convergence of the second order moments and decay in the bispectrum.