Resonance-induced surfatron acceleration of a relativistic particle

We study motion of a relativistic charged particle in a plane slow electromagnetic wave and background uniform magnetic field. The wave propagates normally to the background field. The motion of the particle can be described by a Hamiltonian system with two degrees of freedom. Parameters of the problem are such that in this system one can identify slow and fast variables: three variables are changing slowly and one angular variable (the phase of the wave) is rotating fast everywhere except for a neighborhood of a certain surface in the space of the slow variables called a resonant surface. Far from the resonant surface dynamics of the slow variables may be approximately described by the averaging method. In the process of evolution of the slow variables the particle approaches this surface and may be captured into resonance with the wave. Capture into this resonance results in acceleration of the particle along the wave front (surfatron acceleration). We study the phenomenon of capture and show that a captured particle never leaves the resonance and its energy infinitely grows. Passage through the resonant surface without capture leads to scattering at the resonance, i.e. a small phase-sensitive deviation of actual motion from the motion predicted by the averaging method. We find that repeated scatterings result in diffusive growth of the particle energy. The considered problem is a representative of a wide class of problems concerning passages through resonances in nonlinear systems with fast rotating phases. Estimates of accuracy of the averaging method in this class of problems were for the first time obtained by V. I. Arnold