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Full Description
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. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gauß-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Bäcklund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.
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.
