Dispersive Limit of the Euler-Poisson System in Higher Dimensions

TL;DR

This paper rigorously proves that solutions of the Euler-Poisson system for ion-acoustic waves converge to the Kadomtsev-Petviashvili II and Zakharov-Kuznetsov equations in higher dimensions under specific transformations, justifying their use as limits.

Contribution

It establishes the dispersive limit of the Euler-Poisson system in higher dimensions, providing a rigorous mathematical justification for the KP-II and ZK equations as asymptotic limits.

Findings

Global convergence to KP-II in 2D

Global convergence to ZK in 3D

Rigorous justification of dispersive limits

Abstract

In this paper, we consider the dispersive limit of the Euler-Poisson system for ion-acoustic waves. We establish that under the Gardner-Morikawa type transformations, the solutions of the Euler-Poisson system converge globally to the Kadomtsev-Petviashvili II equation in $\Bbb R^2$ and the Zakharov-Kuznetsov equation in $\Bbb R^3$ for well-prepared initial data, under different scalings. This justifies rigorously the KP-II limit and the ZKE limit of the Euler-Poisson equation.