Glimpses of Soliton Theory (Second Edition)

    Alex Kasman / College of Charleston

    Exciting News: The second edition of "Glimpses of Soliton Theory" is now available. Like the first edition, this one is published by the American Mathematical Society in their Student Mathematical Library. The second edition features:

    • Improved exposition and illustrations throughout the book,
    • a new result which allows for detailed analysis and prediction of arbitrary KdV soliton solutions,
    • additional homework exercises and projects,
    • and examples of non-commutative integrable systems.
    • Plus, the eBook will be in color and feature active hyperlinks.

    Like the first edition of Glimpses of Soliton Theory (American Mathematical Society, 2010), this follow-up aims to introduce the algebro-geometric structure of soliton equations to undergraduate math majors.

    Solitons are solutions to certain very special differential equations that have applications in science and engineering. Aside from these practical applications, however, soliton theory is also amazing in the way that it ties together seemingly unrelated branches of mathematics. Unlike most nonlinear differential equations, soliton equations can be solved explicitly using algebraic methods and the set of all of the solutions has a rich geometric structure. This textbook allows undergraduate students to appreciate this "gem" of mathematics with only courses in calculus and linear algebra as prerequisites. (In particular, this book does not require prior experience with physics or differential equations.)

    Because of its interdisciplinary nature - combining aspects of algebra, geometry, analysis, and applied mathematics - this book would make an ideal textbook for a "capstone class" in mathematics. Moreover, carefully constructed examples and carefully selected topics also make it ideal for a reading course for a student wishing to learn this material independently.

    • Mathematica Notebook: One of the ways the book is able to address topics generally inaccessible to undergraduates is through the use of mathematical software. For instance, rather than introducing elliptic functions abstractly (which would require advanced experience with complex analysis), the computer manipulates these functions for the reader much as students first learning about trigonometric functions benefit from being able to graph and compute the values of the sine function on their calculator. In addition, although the reader will learn how to multiply differential operators "by hand", some homework exercises require the reader to compute products involving Lax operators that would be tedious to do without computer assistance.

      For the reader's convenience, I am making the following Mathematica Notebook available for download. It contains all of the commands used in the textbook (except those which the student is expected to write as part of a homework exercise), organized by chapter.

      Click Here to Download Mathematica Notebook

      (Note: If your browser opens the file as a text file in the browser window rather than saving it to your disk, simply use the "file" menu on your computer, select "save" and save it to the location of your choice with the suffix ".nb" in the filename.)

    • Errata: I very much regret that there are errors in the book which need correcting. However, such things seem to be unavoidable. The known errors (thus far) are:

      • Page 200: The coefficients in the operator K depend on the variable t as well as the variable x. To make this clear I should probably have written "βi(x,t)" instead of just "βi(x)".
      • Page 315: Bradley Yount at Eastern Washington University pointed out that the function given in Question 4 of Project III is not a solution to NLS. It should say:

        u(x,t)=a sech(2-1/2a(x+ct)) exp(-i/4(2cx+(c2-2a2)t)).

      • Page 326: Question 8 in Project X should refer to the KP I Equation rather than the KP II Equation.

      If you notice any other mistakes, please let me know.

    • Official AMS Websites: The American Mathematical Society maintains a webpage for this book at www.ams.org/bookpages/stml-100.

      In addition, the book is available for purchase from the AMS Bookstore at www.ams.org/bookstore-getitem/item=STML-100.