Ions-electrons-states for the two-component Vlasov-Poisson equation

TL;DR

This paper analyzes bifurcation phenomena for traveling periodic solutions in the one-dimensional two-species Vlasov-Poisson system, revealing complex solution structures and connections to Euler-Poisson equations.

Contribution

It extends previous work by allowing both ion and electron species to evolve dynamically, uncovering new bifurcation structures and global solution branches.

Findings

Bifurcation diagrams exhibit two or four solution branches depending on parameters.

Traveling periodic solutions form strip-like regions in phase space.

Connections established between Vlasov-Poisson and Euler-Poisson systems.

Abstract

We establish both local and global bifurcation results for traveling periodic solutions of the one-dimensional two-species Vlasov-Poisson equation. These solutions consist of strip-like regions of ions and electrons in phase space that propagate coherently and emerge from spatially homogeneous, velocity-dependent equilibrium layers. Depending on the geometry of the underlying equilibrium and on the selected Fourier mode, the bifurcation diagram exhibits either two or four solution branches. In all cases, the bifurcation is of pitchfork type; in symmetric configurations, the local structure near the equilibrium has a hyperbolic geometry. We further show that these locally constructed branches extend globally. This work extends the previous study by the same author of the purely electronic case, where the ions were modeled as an immobile neutralizing background. Allowing both species to evolve dynamically leads to a more intricate, higher-dimensional analysis. Finally, by means of an affine change of variables, we reveal a connection with the one-dimensional two-component Euler-Poisson system, which in turn enables the construction of traveling periodic waves of both small and large amplitude for that model as well.