Dispersive Limit of the Euler--Poisson System in Higher Dimensions

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 (KP-II) equation in $\Bbb R^2$ and the Zakharov--Kuznetsov equation (ZKE) 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.