Justification of the NLS approximation for ion Euler-Poisson equation

TL;DR

This paper rigorously justifies the use of the nonlinear Schrödinger equation as an approximation for solutions to the ion Euler-Poisson system, providing error estimates and addressing analytical challenges.

Contribution

It offers a rigorous mathematical proof of the NLS approximation for the ion Euler-Poisson equation, including error bounds and techniques to handle resonances and regularity issues.

Findings

Error estimates in Sobolev norms between solutions and approximation

Overcoming resonances and regularity loss via normal form and energy methods

Validation of NLS as an accurate approximation for ion Euler-Poisson solutions

Abstract

The nonlinear Schr\"{o}dinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler-Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler-Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation.