INTRODUCTION

The basis of the theory of the development of wave motion in classical physics is to treat the wave as a perturbation on an equilibrium or steady state, and to linearize the governing equations in the perturbation variables. This familiar technique has been fully exploited in the theory of the propagation of electromagnetic waves in plasmas, where it has produced a wealth of detailed analysis which has application to a wide range of physical phenomena.

On the other hand, waves in plasmas are subject to nonlinear effects which manifest themselves quite readily. Typical is the ‘Luxembourg effect, observed in the early years of broadcasting, in which the field of a ‘disturbing’ wave so influences the ionospheric environment through which a ‘wanted’ wave is propagated that the latter is noticeably modulated. Many other kinds of ‘wave-interaction’ are possible, and a variety of studies of the interaction between two or more waves probably forms the bulk of the literature on nonlinear plasma waves. The emphasis in such work is mainly on obtaining the first order corrections to linear theory, and investigating the conditions most favourable for interaction.

Another corner of nonlinear theory, which may perhaps have more attraction for the mathematician, owes its stimulus to comparatively recent additions to our physical understanding. It is recognized that both the laser and the pulsar are capable of generating an electromagnetic wave that, at its specific frequency, is so strong that its passage through a plasma constitutes a finite-amplitude, maybe even a largeamplitude, wave, whose description is quite outside the scope of linear theory. This poses the problem of finding solutions of the exact (nonlinear) governing equations. Specifically, it is of interest to seek periodic solutions, with the reasonable expectation that there will be those that recover the familiar waves of linear theory in the small amplitude limit. The earliest work of this kind, Akhiezer & Polovin (1956), in fact predated the appreciation of the applications mentioned, and did not apparently attract much attention until comparatively recently. A comprehensive review of developments up to a few years ago is available in Decoster (1978), and some more recent work is covered in the present paper.