From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov

TL;DR

This paper derives the Korteweg-De Vries and Zakharov-Kuznetsov equations from the Vlasov-Poisson system using a long wave scaling and the relative entropy method, connecting kinetic models to fluid-like equations.

Contribution

It introduces a novel long wave scaling for the Vlasov-Poisson equation and rigorously derives the KdV and ZK equations in the cold ions limit.

Findings

Derivation of KdV equation in 1D from Vlasov-Poisson

Derivation of Zakharov-Kuznetsov equation in higher dimensions

Use of relative entropy method for rigorous proof

Abstract

We introduce a long wave scaling for the Vlasov-Poisson equation and derive, in the cold ions limit, the Korteweg-De Vries equation (in 1D) and the Zakharov-Kuznetsov equation (in higher dimensions, in the presence of an external magnetic field). The proofs are based on the relative entropy method.