INTRODUCTION.

It is well known that for many different physical systems weakly nonlinear long waves are described by the Korteweg-de Vries equation when the wave propagation is uni-directional and the background medium is homogeneous (e.g. Wtutham, 1974, Chapter 17). In some circumstances, the canonical evolution equation is modified from the K-dV, either in the nonlinear term, or more commonly, in the dispersive term; an example of the latter case is the deep fluid equation (BDA) derived by Benjamin (1967) and Davis and Acrivos (1967). Most existing theories discuss these equations for the uni-directional homogeneous case, when it is well known that they possess N—soliton solutions, and are exactly integrable through the inverse scattering transform technique (e.g. Ablowitz and Segur, 1981, Chapter 4).

On heuristic grounds we claim that the canonical equation to describe weakly nonlinear uni-directional long waves in an inhomogeneous medium is

Here the coefficients y and A are functions of T alone. In (1.1) if e is ε2 a small parameter measuring the amplitude of the wave, then T is ε3x, and ζ, is the convected coordinat where x is distance, t is time and c0 is the linear long wave phase speed; the medium is assumed to be inhomogeneous in the x-direction on a scale ε-3, and so c0 is a function of ε3.x. The wave amplitude A is chosen so that A2 is a measure of the wave action flux in the x-direction. Equations of the type (1.1) were first derived by Ostrovsky and Pelinovsky (1970) for the case of a surface gravity wave travelling over variable depth (see also Johnson (1973) and Shuto (1974)); in other physical contexts, equations of the type (1.1) arise in plasma physics (Nlshikawa and Kaw (1975)) and for internal gravity waves (Grimshaw (1981a)).

In Section 2 we present a generalization of (1.1) which allows for more general kinds of inhomogeneity, includes transverse variations and dissipative effects, and also replaces the dispersive term in (1.1) (i.e. the term whose coefficient is ƛ) with a general linear operator. It will be shown how the linear long wave dispersion relation based on the linear long wave phase speed c0 allows the introduction of a set of rays, which in turn provide the natural coordinates for the evolution equation.