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

TL;DR

This paper develops a kinetic theory for multidimensional KdV-type equations, specifically the Zakharov-Kuznetsov equation, using random series expansion to derive a 3-wave kinetic equation valid up to the kinetic time scale.

Contribution

It introduces a novel approach to derive the 3-wave kinetic equation for multidimensional KdV equations with random initial data.

Findings

Derived the 3-wave kinetic equation for ZK equations in higher dimensions.

Extended the validity of kinetic approximation up to the kinetic time scale with minimal loss.

Provided a rigorous mathematical framework for wave kinetic theory in multidimensional settings.

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.