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Lecture Notes in Mathematics

A Spectral Theory for Simply Periodic Solutions of the Sinh-Gordon Equation

Authors: Klein, Sebastian

  • Features a self-contained elaboration of the spectral theory for a specific integrable system
  • Provides detailed proofs of all necessary asymptotic estimates
  • Includes a complete treatment of singular spectral curves
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eBook $39.99
price for USA in USD (gross)
  • ISBN 978-3-030-01276-2
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.99
price for USA in USD
  • ISBN 978-3-030-01275-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

This book develops a spectral theory for the integrable system of 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation.  Such solutions (if real-valued) correspond to certain constant mean curvature surfaces in Euclidean 3-space.  Spectral data for such solutions are defined (following ideas of Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data.  Finally, a Jacobi variety and Abel map for the spectral curve are constructed and used to describe the change of the spectral data under translation of the solution u.  The book's primary audience will be research mathematicians interested in the theory of infinite-dimensional integrable systems, or in the geometry of constant mean curvature surfaces. 

 

Table of contents (21 chapters)

  • Introduction

    Klein, Sebastian

    Pages 3-20

  • Minimal Immersions into the 3-Sphere and the Sinh-Gordon Equation

    Klein, Sebastian

    Pages 21-24

  • Spectral Data for Simply Periodic Solutions of the Sinh-Gordon Equation

    Klein, Sebastian

    Pages 25-37

  • The Vacuum Solution

    Klein, Sebastian

    Pages 41-45

  • The Basic Asymptotic of the Monodromy

    Klein, Sebastian

    Pages 47-69

Buy this book

eBook $39.99
price for USA in USD (gross)
  • ISBN 978-3-030-01276-2
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.99
price for USA in USD
  • ISBN 978-3-030-01275-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
A Spectral Theory for Simply Periodic Solutions of the Sinh-Gordon Equation
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2229
Copyright
2018
Publisher
Springer International Publishing
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-030-01276-2
DOI
10.1007/978-3-030-01276-2
Softcover ISBN
978-3-030-01275-5
Series ISSN
0075-8434
Edition Number
1
Number of Pages
VIII, 334
Number of Illustrations
7 b/w illustrations
Topics