Small amplitude limit of solitary waves for the Euler-Poisson system

TL;DR

This paper proves that solitary wave solutions of the Euler-Poisson system converge to those of the KdV equation in the small amplitude limit, validating formal approximations used in plasma physics.

Contribution

It establishes the existence and convergence of solitary waves for the Euler-Poisson system to KdV solitons as amplitude diminishes, extending results to the isothermal case.

Findings

Existence of solitary wave solutions for the Euler-Poisson system.

Convergence of these solutions to KdV solitons as amplitude approaches zero.

Mathematical validation of Sagdeev's formal approximation.

Abstract

The one-dimensional Euler-Poisson system arises in the study of phenomena of plasma such as plasma solitons, plasma sheaths, and double layers. When the system is rescaled by the Gardner-Morikawa transformation, the rescaled system is known to be formally approximated by the Korteweg-de Vries (KdV) equation. In light of this, we show existence of solitary wave solutions of the Euler-Poisson system in the stretched moving frame given by the transformation, and prove that they converge to the solitary wave solution of the associated KdV equation as the small amplitude parameter tends to zero. Our results assert that the formal expansion for the rescaled system is mathematically valid in the presence of solitary waves and justify Sagdeev's formal approximation for the solitary wave solutions of the pressureless Euler-Poisson system. Our work extends to the isothermal case.