The theory of solitons is attractive; it is wide and deep, and it is intrinsically beautiful. It is related to even more areas of mathematics and has even more applications to the physical sciences than the many which are indicated in this book. It has an interesting history and a promising future. Indeed, the work of Kruskal and his associates which gave us the inverse scattering transform is a major achievement of twentieth-century mathematics. Their work was stimulated by a physical problem and is also a classic example of how computational results may lead to the development of new mathematics, just as observational and experimental results have done since the time of Archimedes.

This book originated from lectures given to classes of mathematics honours students at the University of Bristol in their final year. The aim was to make the essence of the method of inverse scattering understandable as easily as possible, rather than to expound the analysis rigorously or to describe the applications in detail. The present version of my lecture notes has a similar aim. It is intended for senior students and for graduate students, phyicists, chemists and engineers as well as mathematicians. The book will also help specialists in these and other subjects who wish to become acquainted with the theory of solitons, but does not go as far as the rapidly advancing frontier of research. The fundamentals are introduced from the point of view of a course of advanced calculus or the mathematical methods of physics.