Almost sharp wave kinetic theory of multidimensional KdV type equations with $d\ge 3$

Abstract

In this work, we study the random series expansion of a multidimensional KdV type equation with a diffusion term, the so-called Zakharov-Kuznetsov (ZK) equation. We impose random initial data and periodic boundary condition with period $L$ on this equation. Using the random series expansion, we derive the $3$-wave kinetic equation on the inertial range for $t\lesssim L^{-\varepsilon}T_{\text{kin}}$. Our result reaches kinetic time scale up to $\varepsilon$ loss.