Nonlinear nonstationary equations of the interaction between an electron beam and an electromagnetic field near the Brillouin zone boundary

The interaction of a relativistic electron beam with an electromagnetic field in a periodic electrodynamic system is considered in the context of studying nonstationary pulse processes in traveling-wave tubes in series of connected resonators and in relativistic Cerenkov oscillators. Nonlinear nonstationary equations and the corresponding boundary conditions are formulated for the case of interaction near the Brillouin zone boundary. Finally, an implicit finite-difference method is employed to obtain some numerical solutions; a signal of finite magnitude was observed at both ends of the system.