Nonlinear Stationary Waves in Relativistic Plasmas

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Abstract
Theory of nonlinear stationary longitudinal waves in relativistic plasmas is treated by solving a relativistic Boltzmann equation without the conventional approximation of linearization. It is proved here that in plasmas of physically reasonable electron velocity distributions, stationary waves of arbitrary amplitude can be propagated and obey a dispersion equation. If the velocity of propagation V is less than the velocity of light c, the amplitude of these waves will generally be limited; if the velocity of wave propagation is greater than c, however, there is no amplitude limitation. The nonlinearity in the Boltzmann equation causes a decrease of frequency of oscillation for a given wave velocity in addition to the distortion in waveform from pure sinusoidal waves. Detailed solutions for plasmas with Maxwellian electron velocity distribution and relativistic electron beams are given. In the former case, a dispersion equation including nonlinear effect is derived through a series expansion to an order higher than that used in the linearized theory. In the latter, a complete analytic solution in terms of elliptical integrals is given. In both cases dispersion characteristics for several parameters are computed and plotted. The agreement between these nonlinear dispersion equations in the limit of vanishing amplitudes and those of the linearized theory published in literature is indicated.

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