INTRODUCTION

This article is concerned with the uniformly-valid perturbation of those solutions of the Boltzmann and Maxwell equations which represent the propagation of periodic waves in a uniform plasma. No restriction on the amplitude of the wave is required but it is assumed that the basic monochromatic wave train is slowly-varying in the sense that, for example, such parameters of the wave as its amplitude or wavelength may change significantly only over a large number of periods. The work is thus related to the well-known Whitham theory (Whitham 1965,1967,1974) for treating similar wave phenomena in other fields, especially those of water waves, but no averaged Lagrangian technique has been obtained in general for the present problem although that approach will be discussed for the special cases of a cold and warm plasma in Section 3.3. On the other hand here the perturbation method is applied directly to the basic equations. Although in the original presentation (Butler and Gribben 1968) the latter were expressed in a relativistically-invariant form leading to a rather neat mathematical appearance for the general results in the present article we consider from the outset the simplest form of the equations of interest. In particular the plasma is taken to vary only in the direction of propagation of the one-dimensional wave considered; furthermore, collisions are neglected and the magnetic field is zero. In these circumstances the basic equations reduce to the Vlasov and Poisson equations in one space variable and time and the direct interpretation of the equations which result from application of the theory is simpler.

We should perhaps emphasize that we are not considering here the effects of wave-wave interactions but concentrating on the interaction of particles with the wave. The general theory covers in detail the behaviour of particles in the trapped regions (see Section 3.1) but as yet no problems have been solved for such cases.

The main purpose of the paper is to review the current status of this theory of slowly varying waves and, in so doing, advantage has been taken of the opportunity to summarize in Section 2 some appropriate uniform wave solutions available.