Solution of a boundary value problem for the interaction of charged particles with the field of a plane electromagnetic wave

A theoretical analysis is made of the energy state of an electron moving in the field of a slow electromagnetic wave propagating parallel to a stationary magnetic field. Exact solutions to the nonlinear differential equations or electron motion are obtained, with allowance for the boundary conditions at the entrance to the existence region of the electromagnetic fields. The solution of the energy equation is reduced to the analysis of a nonlinear quasi-conservative oscillatory system with one degree of freedom and is obtained as a small-parameter power series expansion. The algorithm for the simultaneous solution of the energy and coordinate equations is constructed on the basis of an iterative approach.