Topological Charge of the real periodic finite-gap Sine-Gordon solutions

TL;DR

This paper derives an explicit formula for the topological charge of real periodic finite-gap solutions to the sine-Gordon equation, overcoming previous difficulties in extracting this from Theta-functional expressions.

Contribution

The authors develop a new method to explicitly compute the topological charge for sine-Gordon solutions, including an analog of Fourier-Laurent transform on Riemann surfaces.

Findings

Explicit formula for the topological charge is obtained and proved.

A new method based on Riemann surface transforms is introduced.

An analog of Fourier-Laurent transform on Riemann surfaces is defined.

Abstract

An effective description of the inverse spectral data corresponding to the real periodic and quasiperiodic solutions for the sine-gordon equation is obtained. In particular, the explicit formula for the so-called topological charge of the solutions is found and proved. As it was understood already 20 years ago, it is very hard to extract any formula for this quantity from the Theta-functional expressions. A new method was developed by the authors for this goal. In the appendix 3 an analog of the Fourier-Laurent integral transform on the Riemann surfaces is defined, based on the constructions of the Soliton Theory. It is a natural continuous analog of the discrete Krichever-Novikov bases developed in the late 80s for the needs of the Operator Quantization of String Theory.