Slowly varying nonlinear waves in a cold plasma stream

This paper is concerned with the propagation of one-dimensional nonlinear waves in the relatively simple physical situation of a cold, collisionless, field-free, slightly non-uniform plasma comprising a stream of oscillating electrons and a stream of ions whose oscillatory motion is neglected. The waves are considered as a small perturbation from the corresponding uniform periodic solution. The equations which govern the slow variation of certain gross quantities associated with the waves are obtained by applying the ‘two-timing’ technique to the basic plasma equations. The properties of the resulting modulation equations are discussed and an exact solution of an initial value problem for them is given.