Steady-states solutions of the Vlasov-Maxwell-Fokker-Planck system of proton channeling in crystals

The paper studies special classes of the stationary solutions of the generalized Vlasov- Maxwell-Fokker-Planck (VMFP) system. We reduce the VMFP equations to a nonlinear elliptic system with exponential nonlinearities. For the Vlasov-Poisson-Fokker-Planck system a new form of stationary states is obtained which generalizes the known ones from the works of K. Dressler and R. Glassey. We consider the one-dimensional case of the elliptic equations, corresponding to the axial symmetry of a crystal. For the associated boundary value problem, the existence of at least one solution is proved by the lower-upper solution method. Besides, we propose an iterative algorithm and perform illustrative numerical calculations. The numerical results are compared with our upper-lower solutions.