The soliton, "the great solitary wave", was first described by John Scott Russell in 1834 when he observed a single wave in a shallow, narrow channel that preserved its shape and speed as it propagated for a long distance. Russell had followed the wave on horseback until he could no longer keep up. He recorded his discovery and subsequent investigations in reports notable for their clarity and insight. In 1965 the physicists Kruskal and Zambusky coined the term *soliton* to indicate the stable particle-like properties of a solitary-wave solution of a nonlinear wave equation. In the intervening years there was very little investigation of the solitary wave and fewer than a dozen papers about it were published. Since 1965 thousands of papers on the soliton have been published with applications in dozens of fields ranging from astrophysics to neurophysiology to solid state physics.

*The Versatile Soliton* is an expository account of the subject for the general reader, requiring a very modest acquaintance with physics and very little mathematics. The author, Alexandre Filippov is a theoretical physicist at the Bogoliubov Laboratory of Theoretical Physics. An early version of this book was written in Russian and first published it in the former Soviet Union. This current edition in English is a revised and updated version. A primary goal of the book is to use the history of the soliton to show how a significant scientific idea develops with deep connections between apparently disparate ideas.

Beginning with a discussion of waves and oscillation, the author develops the history of the study of linear and nonlinear waves. He describes the state of understanding of water waves in the nineteenth century and tells the story of Russell's discovery and investigations. Physicists of Russell's time — particularly Airy and Stokes — dismissed his ideas as impossible according to their understanding of wave behavior. In 1895 Korteweg and de Vries found a relatively simple equation describing waves in shallow water (the KdV equation) and showed how solitary waves could be generated.

The middle section of the book discusses nonlinear oscillations of the pendulum, wave dispersion, and the phase and group velocity of waves. The approach follows Newton's view of waves as particles within a medium oscillating like tiny pendulums coupled to neighboring pendulums. These middle chapters use a bit more mathematics and include more equations than the first section.

The final section looks at solitons as they are seen by working scientists today. We see solitons as dislocations in crystals, in the Josephson junctions of superconductors, in tsunamis, in vortices of superfluids (such as liquid helium) and in the theory of elementary particles. It should be noted that for the author (and physicists in general) a soliton refers to the solution of any energy-conserved system that exhibits particle-like behavior. Mathematicians reserve the word for solitary wave solutions of partial differential equations (like the KdV equation) for which an inverse scattering transform can be formulated.

This is a book that reads very well locally, but does not quite fit together globally. If the reader samples selectively, this can be a very nice introduction to solitons and their applications. As an integrated presentation, however, it doesn't work so well. Continuity of the presentation is a problem; it is difficult in places to see where the author is going and why a given topic fits in. Some of the digressions (such as "What is theoretical physics?" and "Can men be on friendly terms with the computer?") don't seem to fit at all. In addition, it isn't really clear who the intended readers are. For mathematicians, physicists and other scientists, once you get past the very elementary sections there simply isn't enough detail. On the other hand, many of the topics are probably too specialized for the general reader who is likely to be either confused or intimidated by the equations that do appear.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.