Exact Nonlinear Electromagnetic Whistler Modes

Abstract

We solve exactly the nonlinear, relativistic equations of motion for an electron moving in a right circularly polarized wave which propagates along a static uniform magnetic field ${B}_{0}{\mathbf{e}}_{z}$. In the wave frame where the induction electric field disappears, we find two constants of the motion. With their aid, we examine the particle trajectories and obtain the periods and amplitudes of oscillation in the $z$ direction. In the absence of collisions, the exact solution of the relativistic Vlasov equation in the wave frame is an arbitrary function of these constants. The requirement of self-consistency imposed by Maxwell's equations is examined and, in particular, we show that sufficient arbitrariness remains that no dispersion relation exists for these waves. However, for less general distribution functions, one may still have a dispersion relation independent of wave amplitude. When we require that the moments of the distribution be correct to first order in the amplitude of the wave, in analogy with the electrostatic case, we recapture the linearized distribution function together with a principal-value prescription for treating the usual singularity, and we also obtain the transverse Van Kampen modes.