Nonlinear travelling longitudinal waves in cold plasmas

A unified analysis of travelling wave solutions of longitudinal waves in cold electron plasmas is presented. The Lorentz transformation technique is applied to simplify the analysis. It is shown, for a stationary plasma, that the exact (relativistic, nonlinear) solution reduces to the linear and the nonrelativistic nonlinear solutions at the appropriate limits. In addition, it is established that the travelling wave solutions are capable of yielding a condition for the occurrence of wavebreaking. The relation between stationary and streaming plasmas is investigated. It is shown that the nonlinear dispersion relation for a streaming plasma can be obtained from the nonlinear wave relation for a stationary plasma by a Lorentz transformation.