On the theory of quasilinear relaxation of an electron beam emitting under conditions of collective Cerenkov effect

Multi-mode beam instabilities which develop in the conditions of the collective Cerenkov effect have been studied by quasilinear theory methods. The quasilinear approach is shown to be equivalent to the method of characteristics of the Vlasov equation as applied to the description of relaxation of the multi-mode beam instabilities. The excitation of a wide spectrum of electromagnetic oscillations by a dense beam in a retardation system has been studied. Besides, an equation of the Fokker-Planck type has been derived which describes the diffusion of electrons in a space of velocities, and also the laws of conservation of energy and momentum in the particle-field system.