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This title in other editionsOther titles in the Cambridge Texts in Applied Mathematics series:
Cambridge Texts in Applied Mathematics #30: Backlund and Darboux Transformations: Geometry and Modern Applications in Soliton Theoryby C. Rogers and Schief W. K.
Synopses & ReviewsPublisher Comments: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 BäcklundDarboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory.
Book News Annotation:A textbook for a graduate or upperlevel 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) Synopsis:Explores deep and fascinating connections between a ubiquitous class of physically important waves known as solitons.
Synopsis:This book explores the deep and fascinating connections that exist between a ubiquitous class of physically important waves known as solitons and the theory of transformations of a privileged class of surfaces as they were studied by eminent geometers of the nineteenth century. Thus, nonlinear equations governing soliton propagation and also mathematical descriptions of their remarkable interaction properties are shown to arise naturally out of the classical differential geometry of surfaces and what are termed BäcklundDarboux transformations.This text, the first of its kind, is written in a straightforward manner and is punctuated by exercises to test the understanding of the reader. It is suitable for use in higher undergraduate or graduate level courses directed at applied mathematicians or mathematical physics.
Table of ContentsPreface; 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. Projectiveminimal and isothermalasymptotic surfaces; A. The su(2)so(3) isomorphism; B. CCideals; C. Biographies; Bibliography.
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