KdV limit of the Euler-Poisson system

TL;DR

This paper proves that under a specific scaling, solutions of the Euler-Poisson system for ion-acoustic waves converge to solutions of the Korteweg-de Vries equation as the scaling parameter approaches zero.

Contribution

It rigorously establishes the KdV limit of the Euler-Poisson system for ion-acoustic waves under a particular scaling regime.

Findings

Solutions converge to KdV solutions as epsilon approaches zero.

Global in time convergence is proven.

The scaling regime is explicitly characterized.

Abstract

Consider the scaling $\varepsilon^{1/2}(x-Vt)\to x,\ \varepsilon^{3/2}t\to t$ in the Euler-Poisson system for ion-acoustic waves \eqref{equ1}. We establish that as $\varepsilon\to 0$, the solutions to such Euler-Poisson system converge globally in time to the solutions of the Korteweg-de Vries equation.