2 - The Korteweg-de Vries Equation and Related Problems in Water Wave Theory

Summary

INTRODUCTION.

The last decade and more has produced an altogether unlooked-for impetus in the study of certain partial differential equations by use of the inverse scattering transform. Two of the (now) standard equations which are susceptible to this technique arise quite naturally in the study of water waves: the Korteweg-de Vries (KdV) equation and the nonlinear Schrodinger (NLS) equation. This suggests the possibility that there exist other equations of a similar character which are also relevant in water wave theory. The similarity may merely be that the equations are generalisations (more terms, variable coefficients, etc.) which convert the problem into a non-integrable one. On the other hand a conceivable result is that we generate other integrable equations which are extensions of the classical equations to different - possibly higher dimensional - co-ordinate systems. The overall picture is that of a number of diverse equations which describe various aspects of the same underlying problem. This has the virtue that we can more readily compare and contrast the equations, and in some cases specifically relate one to another.

In this paper we shall collect together many of the varied forms of KdV (and to a lesser extent NLS) equations which arise in water wave theory. To emphasise the connecting themes the same variables and parameters will be used throughout. We shall start from an appropriate set of basic equations and thence develop both KdV and NLS equations for one spatial dimension. Since both equations describe alternative aspects of the same problem (by employing different limits in parameter space), it should be possible to match these two equations: this is readily demonstrated. We then turn to the two-dimensional problems which correspond to both the KdV and NLS equations. Some properties of the relevant similarity solutions in one and two dimensions are mentioned, together with matching to the near-field, i.e. to initial data. Finally, we briefly comment on a few other more involved equations of KdV-type which describe the rôle of other physical processes (such as viscous dissipation).