Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory

by C. Rogers, W. K. Schief

Overview

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Bäcklund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory.

Product Details

ISBN-13:	9780521813310
Publisher:	Cambridge University Press
Publication date:	06/24/2002
Series:	Cambridge Texts in Applied Mathematics , #30
Pages:	432
Product dimensions:	5.98(w) x 9.02(h) x 1.14(d)

Table of Contents

Preface; Acknowledgements; General introduction and outline; 1. Pseudospherical surfaces and the classical Bäcklund transformation: the Bianchi system; 2. The motion of curves and surfaces. soliton connections; 3. Tzitzeica surfaces: conjugate nets and the Toda Lattice scheme; 4. Hasimoto Surfaces and the Nonlinear Schrödinger Equation: Geometry and associated soliton equations; 5. Isothermic surfaces: the Calapso and Zoomeron equations; 6. General aspects of soliton surfaces: role of gauge and reciprocal transfomations; 7. Bäcklund transformation and Darboux matrix connections; 8. Bianchi and Ernst systems: Bäcklund transformations and permutability theorems; 9. Projective-minimal and isothermal-asymptotic surfaces; A. The su(2)-so(3) isomorphism; B. CC-ideals; C. Biographies; Bibliography.

Editorial Reviews

"The book can serve as a very good introduction not only for students and young researchers but also for qualified scientists who would like to study nonlinear problems in connection with geometry of submanifolds. Work done in the last few years has proved that interactions between soliton theory and differential geometry are very profitable to both fields." Mathematical Reviews

From the Publisher

A textbook for a graduate or upper-level undergraduate course on the geometry of soliton theory, for students of applied mathematics or mathematical physics. Through the two transforms, it describes the many remarkable connections between results of classical differential geometry of the 19th and early 20th centuries, and recent soliton theory. Annotation c. Book News, Inc., Portland, OR (booknews.com)