The Cauchy problem for the Vlasov kinetic equation and the method of integration over initial data

A method for determining the general solution to the Cauchy problem formulated for a first-order linear homogeneous partial differential equation is developed and justified. It involves integration over the initial data of the system of characteristic equations corresponding to the original problem. The general method is applied to the initial and boundary value problems for the Vlasov relativistic equation. The method is shown to be absolutely suitable for solving the initial problem. Conditions are formulated under which the method can be applied to the boundary value problem. Numerous examples are considered. The nonlinear equations of the relativistic electron-beam amplifier of microwave electromagnetic waves and nonlinear equations describing the evolution of an initial perturbation in homogeneous plasma are rigorously derived.