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

Abstract

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.