A spectral theory 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, defining spectral data, solving the inverse problem, and describing how spectral data change under translation.

Contribution

It introduces a spectral framework for complex solutions of the sinh-Gordon equation, including spectral data characterization and the construction of a Jacobi variety and Abel map.

Findings

Spectral data are asymptotically characterized for solutions.

The inverse problem is solved along a line, reconstructing solutions from spectral data.

A Jacobi variety and Abel map are constructed to describe spectral data changes.

Abstract

In this work a spectral theory for 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation is developed. Spectral data for such solutions are defined (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 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 is constructed; they are used to describe the change of the spectral data under translation of the solution u.