On the wave turbulence theory for stochastic and random multidimensional KdV type equations

In this work we consider a multidimensional KdV type equation, the Zakharov-Kuznetsov (ZK) equation. We derive the 3-wave kinetic equation from both the stochastic ZK equation and the deterministic ZK equation with random initial condition. The equation is given on a hypercubic lattice of size $L$. In the case of the stochastic ZK equation, we show that the two point correlation function can be asymptotically expressed as the solution of the 3-wave kinetic equation at the kinetic limit under very general assumptions, in which the initial condition is out of equilibrium and the size $L$ of the domain is fixed. In the case of the deterministic ZK equation with random initial condition, the kinetic equation can also be derived at the kinetic limit, but under more restrictive assumptions.