Preface By covering a carefully selected subset of topics, offering detailed explanations and examples, and with the occasional assistance of technology, this book aims to introduce undergraduate students to a subject normally only encountered by graduate students and researchers. Because of its interdisciplinary nature (bringing together different branches of mathematics as well as having connections to science and engineering), it is hoped that this book would be ideal for a one semester special topics class, “capstone” or reading course. About Soliton Theory There are many different phenomena in the real world which we de- scribe as “waves”. For example, consider not only water waves but also electromagnetic waves and sound waves. Because of tsunamis, microwave ovens, lasers, musical instruments, acoustic considerations in auditoriums, ship design, the collapse of bridges due to vibration, solar energy, etc., this is clearly an important subject to study and understand. Generally, studying waves involves deriving and solv- ing some differential equations. Since these involve derivatives of functions, they are a part of the branch of mathematics known to professors as analysis and to students as calculus. But, in general, the differential equations involved are so difficult to work with that one needs advanced techniques to even get approximate information about their solutions. It was therefore a big surprise in the late 20th century when it was realized for the first time that some of these equations are much easier than they first appeared. These equations that are not as difficult as people might have thought are called “soliton equations” ix
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