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Bäcklund and Darboux Transformations

Bäcklund and Darboux Transformations
Geometry and Modern Applications in Soliton Theory


Part of Cambridge Texts in Applied Mathematics

  • Authors:
  • C. Rogers, University of New South Wales, Sydney
  • W. K. Schief, University of New South Wales, Sydney
  • Date Published: July 2002
  • availability: Available
  • format: Paperback
  • isbn: 9780521012881

£ 44.99

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About the Authors
  • 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.

    • One of the first books to treat the geometry of soliton theory at graduate textbook level
    • No prior knowledge of soliton theory required
    • Straightforward account punctuated by exercises to test the understanding of the reader
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    Reviews & endorsements

    'It is an excellent book for graduate students and young researchers … very useful for scientists in this field.' Nieuw Archief voor Wiskunde

    'The book certainly is a recommendable book for everyone who is interested in these transformations as well as in the related geometry and modern applications.' Bulletin of the Belgian Mathematical Society

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    Product details

    • Date Published: July 2002
    • format: Paperback
    • isbn: 9780521012881
    • length: 432 pages
    • dimensions: 228 x 153 x 23 mm
    • weight: 0.596kg
    • contains: 48 b/w illus. 76 exercises
    • availability: Available
  • Table of Contents

    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

  • Authors

    C. Rogers, University of New South Wales, Sydney

    W. K. Schief, University of New South Wales, Sydney

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