Conditional asymptotic stability of solitary waves of the Euler-Poisson system on the line

TL;DR

This paper proves that solutions of the Euler-Poisson system on the line asymptotically stabilize to solitary waves by combining virial inequalities and Kato smoothing, extending methods from NLS and KdV equations.

Contribution

It introduces a novel approach to establish the asymptotic stability of solitary waves for the Euler-Poisson system using virial and smoothing techniques.

Findings

Solutions close to a soliton remain close for all time

Solutions converge to a soliton as time approaches infinity

Method extends stability analysis techniques to Euler-Poisson system

Abstract

We apply the idea of using a combination of virial inequalities and Kato smoothing, previously applied to NLS and generalized KdV pure power equations to Euler-Poisson: we assume that a solution remains very close for all times to a soliton in an appropriate space and then we prove an asymptotic convergence to a soliton for $t\to +\infty$.