Solitary waves propagating over variable topography

Solitary water waves are long nonlinear waves that can propagate steadily over long distances. They were first observed by Russell in 1837 in a now famous report [26] on his observations of a solitary wave propagating along a Scottish canal, and on his subsequent experiments. Some forty years later theoretical work by Boussinesq [8] and Rayleigh [25] established an analytical model. Then in 1895 Korteweg and de Vries [21] derived the well-known equation which now bears their names. Significant further developments had to wait until the second half of the twentieth century, when there were two parallel developments. On the one hand it became realised that the Korteweg-de Vries equation was a valid model for solitary waves in a wide variety of physical contexts. On the other hand came the discovery of the soliton by Kruskal and Zabusky [27], with the subsequent rapid development of the modern theory of solitons and integrable systems. In this chapter, we are mainly concerned with the behaviour of solitary waves as they propagate through a variable medium, with a particular emphasis on water waves over variable topography. But, first, we consider the well-known situation when the background medium is uniform. Solitary waves owe their existence to a balance between nonlinearity and wave dispersion. When both these effects are weak, the leading order model evolution equation is the Korteweg-de Vries (KdV) equation,