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

Emeric Roulley

TL;DR

This work analyzes traveling periodic patches (ions–electrons layers) in the 1D two-component Vlasov–Poisson system, proving local and global bifurcation results for $\mathbf{m}$-symmetric states near spatially homogeneous equilibria. By formulating the patch dynamics as a Hamiltonian active-scalar system and exploiting Crandall–Rabinowitz theory, the authors construct small-amplitude traveling waves that bifurcate from flat strips; they establish both generic four-branch and reduced two-branch local bifurcation scenarios, with pitchfork-type transitions and hyperbolic local geometry in symmetric configurations. The results are extended globally via an analytic global bifurcation framework, yielding traveling waves of large amplitude and, through an affine change of variables, connections to the one-dimensional two-component Euler–Poisson system that admit small- and large-amplitude traveling waves with cubic pressure. Overall, the paper advances understanding of nonlinear electrostatic waves in plasmas, highlighting the rich bifurcation structure of ion–electron patches and linking kinetic and fluid models through structural reformulations.

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.

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

TL;DR

This work analyzes traveling periodic patches (ions–electrons layers) in the 1D two-component Vlasov–Poisson system, proving local and global bifurcation results for -symmetric states near spatially homogeneous equilibria. By formulating the patch dynamics as a Hamiltonian active-scalar system and exploiting Crandall–Rabinowitz theory, the authors construct small-amplitude traveling waves that bifurcate from flat strips; they establish both generic four-branch and reduced two-branch local bifurcation scenarios, with pitchfork-type transitions and hyperbolic local geometry in symmetric configurations. The results are extended globally via an analytic global bifurcation framework, yielding traveling waves of large amplitude and, through an affine change of variables, connections to the one-dimensional two-component Euler–Poisson system that admit small- and large-amplitude traveling waves with cubic pressure. Overall, the paper advances understanding of nonlinear electrostatic waves in plasmas, highlighting the rich bifurcation structure of ion–electron patches and linking kinetic and fluid models through structural reformulations.

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.
Paper Structure (10 sections, 9 theorems, 184 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 184 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The two-component Vlasov-Poisson system
  3. Ions and electrons layers
  4. Main results
  5. Reformulations
  6. Hamiltonian structure
  7. Link with the two-component Euler-Poisson system
  8. Small amplitude solutions
  9. Large amplitude solutions
  10. Toolkit in bifurcation

Key Result

Theorem 1.1

(Small amplitude Ions-Electrons-states) Let $a\triangleq\left(a_+^{[1]},a_+^{[2]},a_-^{[1]},a_-^{[2]}\right)\in\mathbb{R}^4$ with

Figures (1)

  • Figure 1: Ion/electron layers

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • ...and 6 more