Spectral instability of small-amplitude periodic waves of the electronic Euler-Poisson system

TL;DR

This paper demonstrates that nearly all small-amplitude periodic waves in the electronic Euler-Poisson system are spectrally unstable, revealing a novel type of instability that is neither modulational nor co-periodic, with growth rates analyzed.

Contribution

It introduces a new spectral analysis approach to identify instability in small-amplitude periodic waves of the Euler-Poisson system, beyond existing methods.

Findings

Almost all small-amplitude waves are spectrally unstable

Instability is neither modulational nor co-periodic

Growth rates of instability are quantified

Abstract

The present work shows that essentially all small-amplitude periodic traveling waves of the electronic Euler-Poisson system are spectrally unstable. This instability is neither modulational nor co-periodic, and thus requires an unusual spectral analysis and, beyond specific computations, newly devised arguments. The growth rate with respect to the amplitude of the background waves is also provided when the instability occurs.