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

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.