Self-consistent nonlinear transverse waves in relativistic plasmas

Previous investigations of the relativistic Weibel instability provide motivation to consider the nonlinear domain because, for asymmetric particle distributions, there is only an isolated unstable Weibel mode-reminiscent of nonlinear wave-types of behavior. From the collisionless Boltzmann equation together with Maxwell's equations, a nonlinear, self-consistent wave equation is derived that is solvable for a broad range of distribution functions. For monochromatic electrons the nonlinear equation can be solved exactly, but results in an unphysical behavior of the magnetic field due to the compact support required of the distribution function. The general equation can be solved by asymptotic representation producing physically correct nonlinear wave solutions over bounded domains with varying internal structure of the electric and magnetic fields that range from nearly Gaussian to "sawtooth" in shape. A lower limit on the nonlinear wave amplitude is required in order that the nonlinear wave be of limited extent and so not represent a sinusoidal disturbance with no bounding domain. Limits for the nonlinear wave maximum magnetic field, and particle number density within the nonlinear wave, are given by considering the constraints on the nonlinear wave due to radiation processes, electron collision effects, and electron degeneracy pressure. The basic physical scale results are depicted mostly conducive for astrophysical applications involving relativistic flows and gamma-ray emission, for which detailed investigations will be given elsewhere.