In Chapter 2 we presented some classical ideas in the theory of water waves. One particular concept that we introduced was the phenomenon of a balance between nonlinearity and dispersion, leading to the existence of the solitary wave, for example. Further, under suitable assumptions, this wave can be approximated by the sech2 function, which is an exact solution of the Korteweg–de Vries (KdV) equation; see Section 2.9.1. We shall now use this result as the starting point for a discussion of the equations, and of the properties of corresponding solutions, that arise when we invoke the assumptions of small amplitude and long wave-length. In the modern theories of nonlinear wave propagation – and certainly not restricted only to water waves – this has proved to be an exceptionally fruitful area of study.

The results that have been obtained, and the mathematical techniques that have been developed, have led to altogether novel, important and deep concepts in the theory of wave propagation. Starting from the general method of solution for the initial value problem for the KdV equation, a vast arena of equations, solutions and mathematical ideas has evolved. At the heart of this panoply is the soliton, which has caused much excitement in the mathematical and physical communities over the last 30 years or so. It is our intention to describe some of these results, and their relevance to the theory of water waves, where, indeed, they first arose.