Generation of antitropic electron beams by self-generated electric   field. Kinetic description
  This paper makes use of a one-dimensional kinetic model to investigate the nonlinear longitudinal dynamics of electron beams generated in the plasma under the influence of a self-generated electric field. It is expressed as where is a wave potential, , and charge particle distribution functions satisfy the Vlasov equation. It is proved that its correct solution is characterized by sudden change in the resonant part of the distribution function. Hence, in particular, the incorrectness of the established in the literature point of view follows that the fast, with velocities over V, and slow, with velocities under V, trapped particles are described by the same distribution function. Also for the first time it is shown that the self-generated strong electric field always produces antitropic electron beams with the velocities much larger than the value V, including the cold plasma limit. The possibility of implementing a new class of self-consistent wave structures with a nonzero average potential is shown. Maximum possible amplitude of a self-generated electric field in a stationary collisionless plasma is determined. 
N. S. Stepanov, L. A. Zelekson
physics.plasm-ph