Traveling Waves of the Vlasov--Poisson System

TL;DR

This paper analyzes the Vlasov--Poisson system for a two-species plasma in one spatial dimension, establishing conditions for the existence and uniqueness of various traveling wave solutions such as solitary waves, shock waves, and wave trains.

Contribution

It provides a complete classification of the existence and uniqueness of traveling wave solutions in the Vlasov--Poisson system, including conditions based on ion trapping and wave type.

Findings

Solitary waves can be unique or nonunique depending on conditions.

Shock waves are always unique.

Wave trains are never unique.

Abstract

We consider the Vlasov--Poisson system describing a two-species plasma with spatial dimension $1$ and the velocity variable in $\mathbb{R}^n$. We find the necessary and sufficient conditions for the existence of solitary waves, shock waves, and wave trains of the system, respectively. To this end, we need to investigate the distribution of ions trapped by the electrostatic potential. Furthermore, we classify completely in all possible cases whether or not the traveling wave is unique. The uniqueness varies according to each traveling wave when we exclude the variant caused by translation. For the solitary wave, there are both cases that it is unique and nonunique. The shock wave is always unique. No wave train is unique.