x Preface because among their solutions are some very interesting ones that we call “solitons”. The original interest in solitons was just because they behave a lot more like particles than we would have imagined. But shortly after that, it became clear that there was something about these soliton equations that made them not only interesting, but also ridiculously easy as compared with most other wave equations. As we will see, in some ways it is like a magic trick. When you are impressed to see a magician pull a rabbit out of a hat or saw an assistant in half it is because you imagine these things to be impossible. You may later learn that these apparent miracles were really the result of the use of mirrors or a jacket with hidden pockets. In soliton theory, the role of the “mirrors” and “hidden pockets” is played by a surprising combination of algebra and geometry. Just like the magician’s secrets, these things are not obvious to a casual observer, and so we can understand why it might have taken math- ematicians so long to realize that they were hiding behind some of these wave equations. Now that the tricks have been revealed to us, however, we can do amazing things with soliton equations. In par- ticular, we can find and work with their solutions much more easily than we can for your average differential equation. Just as solitons have revealed to us secrets about the nature of waves that we did not know before (and have therefore benefited sci- ence and engineering), the study of these “tricks” of soliton theory has revealed hidden connections between different branches of math- ematics that also were hidden before. All of these things fall under the category of “soliton theory”, but it is the connections between analysis, algebra and geometry (more than the physical significance of solitons) that will be the primary focus of this book. Speaking personally, I find the interaction of these seemingly different mathe- matical disciplines as the underlying structure of soliton theory to be unbelievably beautiful. I know that some people prefer to work with the more general – and more diﬃcult – problems of analysis associ- ated with more general wave phenomena, but I hope that you will be able to appreciate the very specialized structure which is unique to the mathematics of solitons. About This Book Because it is such an active area of research, because it has deep con- nections to science and engineering, and because it combines many

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