The multiscaling method of the perturbation theory was used to examine the nonlinear waves propagating in the Brillouin electron beam in vacuo between two plane parallel electrodes. A dispersion correlation was obtained, and the amplitude of the waves was shown to be described by the Korteweg-de Vries equation that admits soliton solutions. The equation coefficients were determined as a function of parameters of the equillibrium unperturbed beam. A kind of one-soliton solution is presented.
Research Institute of Nuclear Physics, Moscow State University, Leninskie Gory, Moscow, 119992, Russia
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