Coupled nonlinear stationary waves in a plasma

Abstract

Coupled nonlinear waves in a plasma can often be expressed in terms of two nonlinear coupled ordinary second order differential equations by making the plane wave ansatz. In general these are quite difficult to solve unless some further drastic assumptions are made, e.g. one of the waves is treated as a driven wave. However, considerable progress can be made if one constant of motion is found. In this case the coupled stationary wave problem can be reduced to the standard classical mechanics problem of a nonlinear Hamiltonian with two degrees of freedom. Such Hamiltonians are known to generate a rich variety of nonlinear solutions which may be periodic, quasiperiodic or sometimes even ‘stochastic’. We obtain this broad range of solutions for the specific problem of nonlinear propagation of relativistically coupled electromagnetic and Langmuir waves in a cold plasma. The solutions (or ‘particle orbits’) are displayed in a Poincaré surface of section plots where they are classified and appropriately related to various approximate solutions discussed before.