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Klappentext 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! Bä cklund! and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Bä cklund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. Zusammenfassung This book explores the profound connections between a ubiquitous class of physically important waves known as solitons and the theory of transformations of a privileged class of surfaces. Punctuated with exercises! it is suitable for use in higher undergraduate or graduate level courses directed at applied mathematicians or mathematical physics. Inhaltsverzeichnis 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.
