Nonlinear electron plasma waves driven by a finite-charged particle pulse or rigid object moving at relativistic speeds are investigated. Quasi-stationary smooth and spiky wake waves comoving with the object are found. Localized soliton-like solutions are also shown to exist. Relativistic effects tend to prevent their formation because of the electron mass increase. The application of the very-large-amplitude wake density waves as a source of ultrahigh-energy cosmic-ray events is discussed.

The propagation of a short laser beam in plasma is investigated analytically. Relativistic ponderomotive force and space charge effects are included, and an equation describing the evolution of the laser spot size is derived. It is shown that self-consistent electron cavitation can lead to self channelling and thus long-distance self-focusing of the laser. The condition for the latter to occur is given.