BÄCKLUND AND DARBOUX TRANSFORMATIONS (GEOMETRY AND MODERN APPLICATIONS IN SOLITON THEORY),
C. Rogers, W. K. Schief, Cambridge University Press, 2002, pp: xv+403, (ISBN 0-521-81331-X hc) (ISBN 0-521-01288-0 pbk;); (Price: $35.00 pbk; $95.00 hc)

Have you ever watched a movie that seemed to have everything going for it: a great plot, good acting, big names, good character development, plenty of action, some drama, plus a touch of comedy and yet when the movie was over, you felt disconnected somewhat? Why was that? Perhaps, this solid movie deserved a high rating of say 4 or 5 stars but something keeps you from rating the movie great.. Perhaps, you had some trouble identifying with the story situation and characters as they live in a different world. Or, perhaps the movie did not have a strong ending. And, you have trouble imagining how it might all come to a satisfying conclusion.

This book is sort of like that kind of movie. If the reader is lucky enough to have a solid grounding in classical differential geometry of surfaces and also a fairly strong knowledge of non-linear wave equations and solitons, then this book probably flows well. On the other hand, if its a decade or so since the reader looked at a differential geometry text, the reader you may find it a little hard to get connected to the material. This situation doesn't imply that the material is boring; on the contrary, the reader may find himself/herself being drawn again and again to the puzzle. Interesting chapter introductions and wonderful illustrations of various surfaces are part of the compelling attraction. A plethora of maze of unusual names and terminology (Dini surfaces, AKNS representations, Calpso Equations, kinks, breathers, Tzizeica surfaces, isothermic surfaces, cyclides) forms another part of the attraction. Fortunately, the authors have provided has a fairly comprehensive index; derivations are, for the most part clearly done and explanations are, for the most part, straightforward.

Overall, the authors have done a thorough job of illustrating a special connection between classical differential geometry of surfaces and modern soliton theory. Chapter after chapter, with different wave equations, methods of generating soliton solutions from special surfaces are illustrated. As I approached the end of the book, I felt that surely a great theorem of some sort was lurking to tell us something like "for every non-linear wave equation with sufficiently nice properties, there exists a one-to-one correspondences between a classical differential geometry surface and soliton solutions with a unique superposition principle" But none was forthcoming.

Obviously, for the novice to the field, a short but solid refresher chapter in differential geometry along with some discussion of wave equations, and solitons would have been welcome introduction to the text. Some sort of synthesis of all the different wave equations and solution type solutions, giving a big picture view at the end of the book would also been welcome.

Although I found myself at times tiring of an seemingly endless stream of derivations of yet more soliton solutions, the reviewer also found that the book is a vast treasure house of interesting information on soliton theory, and, for the asking price, the book deserves a space on the shelf of any mathematician with even a passing interest in solitons.

Colin Carbno
Saskatchewan