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The study of the soliton as a stable particle-like state of nonlinear systems has so caught the imagination of physicists and mathematicians of all descriptions that it can genuinely claim to be one of the few interGisciplinary subjects of modern day mathematical physics. It may be that the topic of solitons will pass through this wide spectrum of subjects leaving their content unchanged, with only a shift in emphasis! Only time will tell.
The uninitiated reader who wishes to learn about the basic ideas in this subject may have a problem knowing where to start. As the bibliography at the back of the book shows, there is now a variety of books to choose from, each tackling the subject from a different angle. This book has been written for the reader who has no prior knowledge of solitons and nonlinear wave equations and who may find texts of up-to-date research papers too difficult a starting point. One of the unique features of this book is an attempt to combine the ideas of soliton theory with a study of the physical origins of the nonlinear equations concerned. We have also treated the inverse scattering (spectral) transform with a degree of rigour not usually attempted in other texts, although a full grasp of this technical material is not essential to an undérstanding of the applications of the theory. We have concentrated mainly on the theory and applications of classical solitons with only a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. Our treatment should, we hope, appeal to both the research worker and graduate student in applied mathematics, engineering, and theoretical physics.
Our choice of topics is based on two main aims. The first is the study of solutions of certain equations by the inverse scattering (spectral) transform, and the second is the study of general classes of physical systems which give rise to these equations. These two aims are mutually compatible, even though the number of equations solvable by the inverse scattering transform is small. It is precisely the integrable evolution equations such as the KdV, MKdV, sine-Gordon and NLS equations which are the most natural equations to arise on various time and space scales when studying weakly nonlinear dispersive systems of many types. The ubiquitousness: of these eouations surprises many people who tend to think of their occurrence in terms of ‘magic’. One of our aims is to show that this is not so: dispersive nonlinear systems with small or no damping should behave in the same way whether they occur in plasmas, classical fluids, lasers or nonlinear lattices. The evolution of long waves on the one hand, and harmonic wave envelopes in off and on resonant systems on the other hand, along with other nonlinear interactions, are classic phenomena which occur in many areas of physics and applied mathematics. These should now be regarded as standard effects and automatically invoked in any study. A further aim of this bookvi Solitons and Nonlinear Wave Equations is to encourage the interdisciplinary aspect in those students just beginning in the area. The study of solitons is a good example of an area of research which has brought together workers in different areas ranging from experimental physics to pure mathematics.
The beginnings of this subject is also an example of the value of careful numerical experiment allied with analytical methods. Now that problems in more than one space dimension are being tackled, numerical studies are becoming popular once more. For these reasons, we have thought it worthwhile to include a concise chapter on numerical methods and results.
With these comments in mind, we have tried to choose our chapters carefully. Most students of applied or engineering mathematics in the USA or UK will have studied little quantum mechanics. The basics of this subject, such as the concepts of reflection and transmission coefficients, square integrable wave functions and potential scattering are absolutely essential in studying the inverse scattering transform. For this reason, we have included a chapter on the elementary ideas of quantum mechanics and classical and quantum inverse problems. On the other hand, most students of theoretical physics will have studied very little classical applied mathematics including fluid mechanics or geophysical fluid dynamics. These students will not be so familiar with ideas such as multiple scaling or the method of stretched co-ordinates. We have attempted to work out some of the examples in detail from first principles because we feel that producing equations out of a hat is not a good idea. However, in certain cases, space has precluded a longer derivation. Obvious limitations on the length of the book have prevented us from dealing with other important aspects such as discrete systems, Lie algebraic methods, higher dimensions, perturbation theories and other integrable systems which require more complicated isospectral operators. It would have been nice to have included all these and we hope that colleagues who specialise in these areas are not offended that these aspects of the subject are excluded. We certainly hope we have assigned credit where it is due, although inevitably we will have omitted a number of key contributions through ignorance or oversight. A list of texts and books of edited research articles in which many of these aspects can be found is included at the beginning of the bibliography.
Because this book has been designed as a textbook of basic methods and ideas and not as an up-to-date book of the latest research results, we have in some chapters decided to relegate most of the referencing to chapter notes. Our introduction in Chapter 1, while partially historical, is nevertheless selective and the topics have been chosen in order to lead the reader gently up to the idea of quantum inverse scattering. We could make a list of at least a dozen names who have made major contributions to the subject but we believe that most would agree that M D Kruskal and V E Zakharov deserve the greatest credit for the conception and development of the inverse scattering method.
This book has been produced on the SCRIBE word processing system (Reid and Walker, 1979), including the bibliography and index. Camera ready copy (excluding figures and mathematical symbols) was generated on the Science and Engineering Research Council's Interactive Computing Facility DEC-10 machine at the Edinburgh Regional Computing Centre, and output on a DIABLO daisywheel printer. We would especially like to thank Jeff Phillips of ERCC for his valuable assistance with all problems arising during the use of the SERCPreface vii Ler faciiities’. Mathematical symbols and figures were added at the final stage, using a standard typewriter. Many of the figures were generated on the FR80 microfilm recorder at the €ERC Rutherford Appleton Laboratory using the ICF network. We are most grateful to Shona McVicar, who undertook the arduous job of inputting the bulk of the text from handwritten or typed copy. Tina Richardson at Imperial College and Mrs M Gardiner at Heriot-Watt University performed the skillful task of typing in the mathematical characters into the text. Thanks are also due to Miss A J Marsland, Mrs J Stewart, and Mrs I Sansonne for useful contributions to proof reading and typing duties. We would also like to thank Anne Trevillion and Anthony Watkinson of Academic Press for their almost infinite patience whilst several deadlines came and went. Several other readers made many useful suggestions, for which we are grateful, although we must accept final responsibility for remaining errors and omissions. We hope the reader will feel encouraged to bring to our notice corrections and suggestions for improvement of the text