Modulational instability of two transverse waves in a cold plasma

By Whitham's method of averaged Lagrangian developed by Dysthe, two coupled nonlinear Schrödiriger equations are obtained for the evolution of the amplitude of two plane polarized transverse waves in a cold plasma, both propagating in the same direction. Relativistic corrections are included and amplitude-dependent frequency shifts are derived. It is found that two transverse waves are modulationally unstable. The relativistic terms play an important part in this instability.