Stability of relativistic transverse cold plasma waves

The stability against small perturbations of a certain class of nonlinear wave solutions of the equations that describe a cold unmagnetized electron-ion plasma is considered. Part 1 refers to circularly polarized waves. In this case it is possible to integrate the equations and obtain an algebraic dispersion relation in the form of an eighth order polynomial equation in the frequency. the analytical and numerical solution of this equation, for a wide range of variation of the parameters involved, reveals the existence of a very unstable mode. Particular attention is given to very large wave amplitudes for which the plasma behaves like an electron-positron plasma; in this case the growth rate of the unstable mode can be as high as the frequency of the nonlinear wave. The propagation of large amplitude plasma waves is a problem of considerable theoretical importance and with immediate application to the study of the interaction of pulsars and high power laser radiation with plasmas. The stability of exact, nonlinear, relativistic, superluminous, travelling, plane, periodic waves in a cold unmagnetized plasma against small amplitude perturbations which propagate in the same direction as the nonlinear wave, is studied. (author)