Nonlinear coupling of polarized plasma waves

Weak nonlinear coupling between two polarized transverse waves in a weakly relativistic plasma is studied including the effect of dispersion. The frequency of the coupled waves is of the same order of magnitude as the electron plasma frequency. By using the multiple scale method it is shown that the slow modulation of the complex amplitudes is described by simultaneous nonlinear Schrödinger equations. Travelling wave solutions are then obtained for this system of equations by the analytical and the numerical methods. As the result of the wave-wave interaction, two envelope waves are, in general, composed of dispersive shock waves or non-periodic nonlinear wave-trains, while the usual solitary waves or periodic nonlinear wave-trains can exist as a special case. case. The results of analysis can readily be applied to other coupled waves with different polarizations in a nonlinear dispersive medium.