Relativistic nonlinear waves in a magnetised plasma

Relativistic propagation of a nonlinearly-coupled circularly-polarised electromagnetic wave and a Langmuir wave along the applied magnetic field is considered. Analytical solutions are given or are indicated for some special cases like purely transverse waves and purely longitudinal waves. In the presence of an applied magnetic field, the incident circularly polarised electromagnetic wave is found to propagate further into denser regions of the plasma — a result which is in accord with the so-called inverse Faraday effect. Finally, we shall consider general coupled transverse and longitudinal waves for which we give an approximate solution. We investigate whether this system of coupled waves exhibits any internal resonances and consequent energy exchange between them.