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Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs

Alex Kasman
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 54
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[Reviewed by
William J. Satzer
, on

Solitons are explicit solutions to nonlinear partial differential equations. They are waves that behave in many respects like particles. The founding story of soliton theory, repeated so often it is now almost indistinguishable from myth, tells of John Scott Russell and his observation in 1834 of a peculiar solitary wave in a canal near Edinburgh. He followed this wave on horseback as it kept its speed and shape for a mile or two until he lost it. The response to this amazing discovery was … not much, mostly scoffing, because everyone thought that such a wave would disperse and distort and could not propagate. In 1895 Korteweg and de Vries modeled water waves in a canal, derived the KdV equation named after them and found a number of wave-like solutions that travel and maintain their shape. But not even they seemed particularly interested in what they found. It wasn’t until the twentieth century and computational work by Fermi-Pasta-Ulam and later Kruskal-Zabusky that the soliton got a name and some respect.

It’s an odd thing. We can write down many explicit exact solutions of the nonlinear KdV equation. Why is this possible when we usually can’t find even one explicit solution for most nonlinear partial differential equations? Moreover, an n-soliton solution to the KdV equation (with n peaks) bears an unusually close relationship to n individual one-soliton solutions: it looks almost — but not quite — like a linear combination of the others. Is there a geometric structure analogous to the vector spaces we see with solutions of ordinary differential equations?

This book explores the ramifications of these questions for advanced undergraduates who have had basic calculus and linear algebra. It’s very challenging material for undergraduates, but it presents an exciting opportunity too. As a capstone course, or for independent study, soliton theory ties together several important applications to science and engineering with an extraordinary range of mathematical topics from PDEs to elliptic curves, differential algebras and Grassmanians. The author doesn’t expect to bring students to the research frontiers with his book; his aim is rather to provide a “glimpse” that intrigues and engages. Partial differential equations and algebraic geometry meet in a most remarkable and unexpected way.

After an introductory review of differential equations that emphasizes the differences between linear and nonlinear equations, the author tells the story of solitons. He begins with James Scott Russell and continues to twentieth century developments and applications. These include examples in telecommunications (where solitons travel down optical fibers) and biology (where solitons play a role in DNA transcription and energy transfer).

The book’s real work begins with an examination of Korteweg and de Vries’ solution to the KdV equation. We see that the general solution they find can be written in terms of a Weierstrass -function. The connection to elliptic curves and algebraic geometry begins here.

The author makes extensive use of Mathematica throughout the book; in particular, that program is used to introduce the Weierstrass -function without requiring the background in complex analysis that would otherwise be necessary. (It is also used throughout for a variety of straightforward, if messy, calculations and for animations of wave dynamics.) This innovative use of Mathematica works well here where the object is to offer glimpses of a broad and subtle theory. It does, however, tie the book to the software — it would be unsatisfying to read the book without having access to Mathematica, and difficult to derive full benefit from it.

After dipping into algebraic geometry, the author goes on to discuss the n-soliton version of the KdV equation and its solutions that look asymptotically like linear combinations of solutions to one-soliton equations. The next few chapters try to explain the special nature of the KdV equations, and along the way discuss the algebra of differential operators, isospectral matrices, and the Lax form for KdV and other soliton equations. It’s only with an additional spatial dimension and analysis of the corresponding generalization of the KdV equation (the KP equation) that the picture gets a little clearer. We then finally get a glimmer of the geometry of the solution space, and a way to describe it using the Grassman cone.

This book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar.
