Acceleration and self-focused particle beam drivers

Here it is shown that the Vlasov equation is an adequate model in case of high-intensity charged-particle beams. Several instances are analyzed when it is possible to construct an integral basis of the operator, associated with the dynamic system under study. This is the case, in particular, for the two-dimensional dynamic systems, just such systems describing the longitudinal motion of a perturbed system. For systems of more general structure we advance a method of reduction of the quasilinear Vlasov equation to an integral Fredholm equation. The main cases are examined when it is possible to construct kernels of corresponding integral operators. In particular, a feasibility to employ Feier - Chesaro kernels is demonstrated. Using the universality (according to V. I. Zubov) of Maxwell equations the problem of a search for stabilizing and focusing fields is reduced to the construction of Toeplitz matrix. Also conditions are analyzed, ensuring initiating of a continuous spectrum points within the spectrum of a dynamic system. Physically, this phenomenon is related to the chaotic motion of the particles. Also, the dispersion equation, expressed in terms of solutions to the Fredholm equation, is deduced.