An author makes his excuses in a preface, so here are mine. For a number of years at Bristol University we tried to find a suitable text to introduce fluid dynamics to second year mathematics students, and failed. The modern texts with the ‘right’ attitude to the subject were too hard for a first course, the older texts were dominated by potential theory and unrealistic examples. This text has been tried in draft form for several years on our students, and has been judged ‘hard, but interesting’. New work in mathematics is always hard, but I believe that the level chosen here is a suitable one.

I apologise to my colleagues for the gross over-simplification of their work and their subject which is committed in this book. And also for the errors and misapprehensions – students, beware! all texts have mistakes in them. I thank my colleagues for helpful comments and discussions over many years; I also thank a succession of seminar speakers for maintaining my awareness of the full range of fluid dynamics.

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.

The discovery of solitary waves

This book is an introduction to the theory of solitons and to the applications of the theory. Solitons are a special kind of localized wave, an essentially nonlinear kind. We shall define them at the end of this chapter, describing their discovery by Zabusky & Kruskal (1965). A solitary wave is the first and most celebrated example of a soliton to have been discovered, although more than 150 years elapsed after the discovery before a solitary wave was recognized as an example of a soliton. To lead to the definition of a soliton, it is helpful to study solitary waves on shallow water. We shall describe briefly in this section the properties of these waves, and then revise the elements of the theory of linear and nonlinear waves in order to build a foundation of the theory of solitons. Let us begin at the beginning, and relate a little history.

The solitary wave, or great wave of translation, was first observed on the Edinburgh to Glasgow canal in 1834 by J. Scott Russell. Russell reported his discovery to the British Association in 1844 as follows:

Epilogue

The essence of this book is the description of the method of inverse scattering. The book is too short to do more than outline the chief properties of solitons and indicate some lesser properties by a few remarks and problems. For that reason the chief properties stand out more clearly.

Research into the physical, earth and life sciences has led to the study of hundreds of nonlinear equations. We have mentioned only a little of this. Indeed, it is too wide to be described in any single volume. Each reader of this introduction to solitons may, however, go on to study the derivation of the equations used in his own specialist field. On having obtained an appropriate nonlinear system, it is natural to seek the waves of permanent form and to test whether they are stable and whether they are solitons. The existence of soliton interactions is the exception rather than the rule. Yet scores of nonlinear systems are already known to have soliton solutions. Even now it is not known how to ascertain definitely whether any given nonlinear system has solitons which may preserve their identities after interacting with one another. The preceding chapters, nonetheless, outline many ideas which may determine, or at least suggest, that a given nonlinear system has, or has not, soliton solutions and which indicate how the properties of the solitons may be found. The ideas are summarized in the following points.