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

Springer, 2018. — 326 p. — (Lecture Notes in Mathematics 2229). — ISBN: 978-3-030-01275-5.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.Minimal Immersions into the 3-Sphere and the Sinh-Gordon Equation
Spectral Data for Simply Periodic Solutions of the Sinh-Gordon Equation
The Vacuum Solution
The Basic Asymptotic of the Monodromy
Basic Behavior of the Spectral Data
The Fourier Asymptotic of the Monodromy
The Consequences of the Fourier Asymptotic for the Spectral Data
Asymptotic Spaces of Holomorphic Functions
Interpolating Holomorphic Functions
Final Description of the Asymptotic of the Monodromy
Non-special Divisors and the Inverse Problem for the Monodromy
Divisors of Finite Type
Darboux Coordinates for the Space of Potentials
The Inverse Problem for Cauchy Data Along the Real Line
Estimate of Certain Integrals
Asymptotic Behavior of 1-Forms on the Spectral Curve
Construction of the Jacobi Variety for the Spectral Curve
The Jacobi Variety and Translations of the Potential
Asymptotics of Spectral Data for Potentials on a Horizontal Strip
Perspectives