On the wave turbulence theory for a stochastic KdV type equation

Abstract: Starting from the stochastic Zakharov-Kuznetsov equation, a multidimensional KdV type equation on a hypercubic lattice, we provide a derivation of the 3-wave kinetic 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: the initial condition is out of equilibrium, the dimension is $d\ge 2$, the smallness of the nonlinearity $\lambda$ is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation. Unlike the cubic nonlinear Schr\"odinger equation, for which such a general result is commonly expected without the noise, the kinetic description of the deterministic lattice ZK equation is unlikely to happen. One of the key reasons is that the dispersion relation of the lattice ZK equation leads to a singular manifold, on which not only $3$-wave interactions but also all $m$-wave interactions ($m\ge3$) are allowed to happen. This phenomenon has been first observed by Lukkarinen \cite{lukkarinen2007asymptotics} as a counterexample for which one of the main tools to derive kinetic equations from wave equations (the suppression of crossings) fails to hold true. To the best of our knowledge, the work provides the first rigorous derivation of nonlinear 3-wave kinetic equations. Also this is the first derivation for wave kinetic equations in the lattice setting and out-of-equilibrium.