Application of the method of characterictics to obtain a solution of the transport equation (Q5942495)

A nonlinear system of partial differential equations describing the interaction of an electrostatistic field with a relativistic electron beam is considered. In the general case the nonlinear system can be solved only by numerical methods. An initial stage position of evolution of the system is considered in the article. By assuming that the beam of particles has nonperturbed trajectories some analytical solutions are found. The analytical solutions are correctly enough describing the behavior of the system in an initial stage of process evolution and they can be used as a test for numerical methods. A numerical example is considered and the results of investigations are presented.