Traveling Waves of the Vlasov--Poisson System

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.