Stationary Bernstein-Greene-Kruskal structures in a current carrying relativistic cold plasma

TL;DR

This paper analytically investigates stationary nonlinear structures, akin to BGK waves, in a relativistic cold plasma with a current beam, revealing how their profiles depend on the energy density ratio and identifying conditions for harmonic and discontinuous behaviors.

Contribution

It introduces a new analytical framework for relativistic BGK-like structures in cold plasmas, detailing their behavior across different energy density regimes.

Findings

Fluid variables vary harmonically in the linear limit.

Fluid variables become anharmonic as the energy ratio increases.

Electric field discontinuities indicate formation of charged planes.

Abstract

Nonlinear stationary structures formed in a cold plasma with immobile ions in the presence of a relativistic electron current beam have been investigated analytically in the collisionless limit. These are cold plasma version of the relativistic Berstein-Greene-Kruskal (BGK) waves. The structure profile is governed by the ratio of maximum electrostatic field energy density to the relativistic kinetic energy density of the electron beam, {\it i.e.}, $\kappa_{R} = E_{m}/ (8 \pi n_{0} (\gamma_{0}-1)m_{0} c^{2})^{1/2}$, where $E_{m}$ is the maximum electric field associated with the nonlinear structure and $\gamma_{0}$ is the Lorentz factor associated with the beam velocity. It is found that, in the linear limit, {\it i.e.}, $\kappa_{R} \ll 1/\sqrt{\gamma_{0}}$, the fluid variables, {\it viz}, density, electric field, and velocity vary harmonically in space. In the range $0 < \kappa_{R} \leq 1/\sqrt{\gamma_{0}}$, the fluid variables exhibit an-harmonic behavior. For values of $\kappa_{R} > 1/\sqrt{\gamma_{0}}$, the electric field shows finite discontinuities at specific spatial locations indicating the formation of negatively charged planes at these locations.