INTRODUCTION

The physics of nonlinear wave motions is an extremely rich subject with applications to many fields. Several excellent accounts of it (eog» Lighthill (1978), Whitham (1974) Yuen and Lake (1980) or Lonngren and Scott (1978)) are available in the literature. These describe both the classical foundations originating in continuum mechanics and electrodynamics and the more recent work on exact solutions by the inverse scattering transform” An extensive literature of numerical work including comparison with experiment also exists (see, for example, Yuen and Ferguson (1978), Martin and Yuen (1980) and Fornberg and Whitham (1978)). The aim of the present article is relatively modest: it is to describe recent results on the qualitative dynamics of nonlinear wave systems with particular emphasis on asymptotic or long time behaviour of the system motion. This area is the subject of active current research by many workers and hence no attempt will be made at a comprehensive presentation. Instead, the first principles will be emphasized and the standpoint restricted to elementary, analytical methods.

The main theme of the article is simply stated in the form of a question: “under what conditions does the state of a nonlinear wave system recur?” This question remains vague unless the class of systems of interest is precisely specified” Furthermore, one must be quite clear about what is meant by ‘state’ and ‘recurrence of states'.

It is useful at this point to review briefly the historical origins of qualitative (or, equivalently, topological) dynamics. Apart from commonsense pronouncements like “History repeats itself”, the concept of recurrence goes back to PoincarS's famous theorem on recurrent motions in bounded Hamiltonian systems (Poincare” (1899)). This theorem states that in a bounded Hamiltonian system with a finite number of degrees of freedom, almost all (in the sense of the invariant phase volume) initial phase points will evolve in time such that they will return to an arbitrarily small neighbourhood of the initial point after a finite time interval. The ‘recurrence time’ will depend, in general, on the initial state and the pre-assigned smallness of the neighbourhood,,