# Spectral data for simply-periodic solutions of the sinh-Gordon equation

## TL;DR

This paper develops a spectral theory for simply periodic solutions of the sinh-Gordon equation, including spectral data characterization, inverse problem solution, and analysis of asymptotic behavior of solutions.

## Contribution

It introduces a spectral framework for complex-valued, simply periodic solutions of the sinh-Gordon equation, including spectral data definition, inverse problem solution, and Jacobi variety construction.

## Key findings

- Spectral data characterized for periodic Cauchy data.
- Inverse problem for spectral data solved.
- Asymptotic behavior of solutions analyzed.

## Abstract

This note summarizes results that were obtained by the author in his habilitation thesis (arXiv:1607.08792) concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions of the sinh-Gordon equation. Spectral data for such solutions are defined for periodic Cauchy data on a line (following 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 of such Cauchy data is answered. Finally a Jacobi variety for the spectral curve is constructed, and this is used to study the asymptotic behavior of the spectral data corresponding to actual simply periodic solutions of the sinh-Gordon equation on strips of positive height.