A straightforward derivation of the four-wave kinetic equation in action-angle variables

Starting from action-angle variables and using a standard asymptotic expansion, we present an original and coincise derivation of the Wave Kinetic equation for a resonant process of the type 2 ↔ 2. Despite not being more rigorous than others, our procedure has the merit of being straightforward; it allows for a direct control of the random phases and random action of the initial wave field. We show that the Wave Kinetic equation can be derived assuming only initial random phases. The random action approximation has to be taken only after the weak nonlinearity and large box limits are taken. The reason is that the oscillating terms in the evolution equation for the action contain, as an argument, the action-dependent nonlinear corrections which is dropped, using the large box limit. We also show that a discrete version of the Wave Kinetic Equation can be obtained for the Nonlinear Schrödinger equation; this is because the nonlinear frequency correction terms give a zero contribution and the large box limit is not needed. In our calculation we do not make an explicitly use of the Wick selection rule.