One-dimensional (1D) propagation of a relativistically intense circularly polarized electromagnetic (EM) wave in an over-critical density plasma is investigated. Cases of fast group velocity to which ions cannot follow the motion and of slow propagation in which ion dynamics plays an important role are discussed. However, electrons can be treated as in static force balance keeping local charge neutrality. It is shown that plane waves are always unstable in the overdense plasma. In particular, two types of modulational instability are found in the case of slow propagation and their growth rates are obtained. It is also shown that an envelope solitary wave solution can be obtained in an overdense region. Density limit for the solitary wave propagation is obtained as a function of its amplitude. The solitary wave is a rarefaction wave for the case of fast propagation, while it becomes of compressional character propagating with supersonic speed for the case of slow propagation. A general expression for the propagation speed as a function of the plasma density and the solitary wave amplitude is obtained for the compressional solitary wave, and the upper and lower limits of the density (or the amplitude) for given amplitude (or density) are obtained. A three-dimensional (3D) effect is briefly discussed and a boundary value problem is formulated for the case in which the plasma fills a half space with the other half space being in vacuum. For the case of an EM wave with ultrarelativistic intensity the transmission coefficient into an over-critical density plasma is found to be a universal function of the ratio of the incident wave amplitude to the plasma density.
Some specific features of surface-wave propagation in a non-uniform plasma