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”

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