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

  • Sebastian¬†Klein

Part of the Lecture Notes in Mathematics book series (LNM, volume 2229)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Spectral Data

    1. Front Matter
      Pages 1-1
    2. Sebastian Klein
      Pages 3-20
  3. The Asymptotic Behavior of the Spectral Data

    1. Front Matter
      Pages 39-39
    2. Sebastian Klein
      Pages 41-45
    3. Sebastian Klein
      Pages 47-69
    4. Sebastian Klein
      Pages 71-84
    5. Sebastian Klein
      Pages 85-99
  4. The Inverse Problem for the Monodromy

    1. Front Matter
      Pages 111-111
    2. Sebastian Klein
      Pages 113-118
    3. Sebastian Klein
      Pages 119-145
  5. The Inverse Problem for Periodic Potentials (Cauchy Data)

    1. Front Matter
      Pages 173-173
    2. Sebastian Klein
      Pages 175-187
    3. Sebastian Klein
      Pages 189-207
  6. The Jacobi Variety of the Spectral Curve

    1. Front Matter
      Pages 219-219
    2. Sebastian Klein
      Pages 221-238
    3. Sebastian Klein
      Pages 307-308
  7. Back Matter
    Pages 309-334

About this book

Introduction

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. 

 

Keywords

37K10,53A10,58E12,35P05,35R30,37K25,40A05,42B05,46C05,46E15 Asymptotic Estimates Constant Mean Curvature Surfaces Integrable Systems Sinh-Gordon Equation Spectral Theory

Authors and affiliations

  • Sebastian¬†Klein
    • 1
  1. 1.School of Business Informatics & MathematicsUniversity of MannheimMannheimGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-01276-2
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-01275-5
  • Online ISBN 978-3-030-01276-2
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site