Glimpses of Soliton Theory is a textbook (published by the American Mathematical Society in 2010) that 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.)
(If you notice any other errors I should list here, please let me know.)
- Page 39: The summation notation for the Navier-Stokes equations on page 39 is humorously incorrect. Obviously, the sum is supposed to be from j=1 to j=4; anyone who has used TeX can guess what went wrong. (Apparently, this was noticed by Andrew Hone, though it was Anton Dzhamay (University of Northern Colorado) who actually told me about it.)
- Page 43: The equation in question 7b should say
Thanks to Bob Riehemann (Thomas More College) for catching the error and bringing it to my attention.
- Page 96: Hideshi Yamane (Kwansei Gakuin University) notes that the n appearing as a limit on the summation in the denominator of the first displayed equation at the top of the page is not the same as the n appearing elsewhere in Definition 5.1. So, it would be better if both occurrences of n in this paragraph were replaced with something else, like m'.
- Page 97: Anton Dzhamay also pointed out that the τ-function in Example 5.2 was mistypeset, although you can easily figure out that it was supposed to say
- Page 114: Example 6.1 should define P(f) as L(M(f)) instead of M(L(f)). (Thanks again, Anton.)
- Page 129: Iain Findlay points out that the 3rd line of last paragraph, should say "kernel of L" rather than "kernel of V".
- Page 217: Thanks to Paul Rigge (University of Michigan) for pointing out that Question 7 mistakenly refers to "the τ-function in part (c)" when it should say "the τ-function in part (b)".
- Page 236: The second line should begin with Φk not Φ2.
(The first subscript on the second line should be a k not a 2.)
- Page 269: Iain Findlay also noted that the middle parenthetical term and the final parenthetical term in Equation 2.1 should be (bc + ad).
- Page 282: Paul Rigge also noticed a problem in Project VII which refers to "Equation (7.48)" in the paper by Ascher and McLachlan. The relevant equation appears at the top of page 86 in that paper, but is not given an explicit number or name. (A partial explanation for this mysterious error involves noting that the same equation appears as Equation (7.48) in the textbook by Drazin and Johnson.)
- Official AMS Website:
Each book in the Student Mathematical Library series gets a webpage on the American Mathematical Society's server. The page for this book is www.ams.org/bookpages/stml-54.
In addition, the book can be purchased from the AMS Bookstore at www.ams.org/bookstore-getitem/item=STML-54.