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Internal free boundary problem for cold plasma equations

Lidia Gargyants, Anna Konovalova, Olga Rozanova

TL;DR

This work analyzes a one-dimensional cold-plasma model with an internal free boundary between two media of different constant ion backgrounds, formulating a Riemann problem in which the interface position $\Phi(t)$ is determined by a generalized Rankine-Hugoniot condition and the stability of intersecting Lagrangian trajectories. A distributional framework yields a strongly singular solution featuring a delta-density at the interface, with a nonlinear second-order ODE for $\Phi(t)$ and a coupled evolution for the delta mass $e(t)$. In parallel, rarefaction waves are constructed with affine-in-$x$ profiles in the adjacent region, governed by a closed system for $a_\pm,c_\pm$ and connected to shock dynamics via switching-point curves $\Psi_1$ and $\Psi_2^{\pm}$; the resulting dynamics may exhibit periodic behavior when the density ratio is commensurate. The study reveals intricate alternating sequences of singular shocks and rarefactions, whose structure depends on the density ratio $r=\sqrt{n_-}/\sqrt{n_+}$ and involves challenging questions of existence, uniqueness, and numerical computation due to degeneracy at interface-velocity turning points, motivating further theoretical and computational investigation.

Abstract

For the system of cold plasma equations describing the motion of electrons in the field of stationary ions, we consider the Riemann problem posed at an impenetrable interface between two media. These media differ in the magnitude of the constant ion field. The interface between the media is assumed to be free. Its position is determined from the generalized Rankine-Hugoniot conditions and the stability condition, that is, the intersection of Lagrangian particle trajectories at the interface.

Internal free boundary problem for cold plasma equations

TL;DR

This work analyzes a one-dimensional cold-plasma model with an internal free boundary between two media of different constant ion backgrounds, formulating a Riemann problem in which the interface position is determined by a generalized Rankine-Hugoniot condition and the stability of intersecting Lagrangian trajectories. A distributional framework yields a strongly singular solution featuring a delta-density at the interface, with a nonlinear second-order ODE for and a coupled evolution for the delta mass . In parallel, rarefaction waves are constructed with affine-in- profiles in the adjacent region, governed by a closed system for and connected to shock dynamics via switching-point curves and ; the resulting dynamics may exhibit periodic behavior when the density ratio is commensurate. The study reveals intricate alternating sequences of singular shocks and rarefactions, whose structure depends on the density ratio and involves challenging questions of existence, uniqueness, and numerical computation due to degeneracy at interface-velocity turning points, motivating further theoretical and computational investigation.

Abstract

For the system of cold plasma equations describing the motion of electrons in the field of stationary ions, we consider the Riemann problem posed at an impenetrable interface between two media. These media differ in the magnitude of the constant ion field. The interface between the media is assumed to be free. Its position is determined from the generalized Rankine-Hugoniot conditions and the stability condition, that is, the intersection of Lagrangian particle trajectories at the interface.
Paper Structure (11 sections, 1 theorem, 53 equations, 4 figures)

This paper contains 11 sections, 1 theorem, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Lagrangian trajectories (characteristics)
  3. Construction of a singular shock wave
  4. Definition of a generalized strongly singular solution
  5. Case of the Riemann data
  6. Construction of a rarefaction wave
  7. Search for switching points
  8. Conjugation of the shock waves in the compression and rarefaction regions
  9. Examples
  10. Conclusion and discussion
  11. Acknowledgments

Key Result

Theorem 1

Let the domain $\Omega\subset {\mathbb R}^2$ be divided by a smooth curve $\gamma=\{(t,x): x=\Phi(t) \}$ into left and right parts $\Omega_\mp$. Let the triple of distributions $(V,E,n)$K19 -- K21 and the curve $\gamma$ be a strongly singular generalized solution for the system K4. Then this solutio

Figures (4)

  • Figure 1: Plane $(t,x)$. Characteristics $x_-(t), x_+(t)$ and the position of the interface $\Phi(t)$. Left: Example 1, $n_-=n_+=1, V_+^0=0, V_-^0=1, E_+^0=-1, E_-^0=0.$ Right: Example 2, $n_-=1, n_+=3, V_+^0=-1, V_-^0=1, E_+^0=-1, E_-^0=1.$
  • Figure 2: Plane $(t,x)$, Example 3: $n_-=1, n_+=4, V_+^0=0, V_-^0=1, E_+^0=-1, E_-^0=1.$ Left: a schematic position of the interface. Right: rarefaction region within one period, graphs of $\Psi_1(t)$ (dash) and $\Psi_2^+(t)$ (dot), switching points $t_1^*$, $t_2^*$, $t_3^*$
  • Figure 3: Plane $(t,x)$, Example 3, detailed picture of characteristics $x_-(t), x_+(t)$ and the position of the interface $\Phi(t)$. Left: $t\in (0, t_1^*)$, rarefaction region with two rarefaction waves. Right: $t\in (t_1^*, t_2^*)$, rarefaction region with one rarefaction wave; the characteristic $x_+$ is removed.
  • Figure 4: Plane $(t,x)$, Example 3, detailed picture of characteristics $x_-(t), x_+(t)$ and the position of the interface $\Phi(t)$. Left: $t\in (t_2^*, t_3^*)$, rarefaction region with two rarefaction waves. Right: $t\in (t_3^*, 2 \pi)$, rarefaction region with one rarefaction wave; the characteristic $x_+$ is removed.

Theorems & Definitions (2)

  • Definition 3.1
  • Theorem 1