Topological Phenomena in the Real Periodic Sine-Gordon Theory

TL;DR

This paper investigates the topological properties of real finite-gap solutions in the periodic Sine-Gordon theory, revealing how their connected components influence conserved quantities and topological charge calculations.

Contribution

It provides an explicit description of the solution components, enabling the calculation of topological charge and demonstrating invariance of conservation law averages across components.

Findings

Connected components of solutions are explicitly characterized.

Topological charge can be computed using the explicit description.

Averaging of conservation laws is consistent across components.

Abstract

The set of real finite-gap Sine-Gordon solutions corresponding to a fixed spectral curve consists of several connected components. A simple explicit description of these components obtained by the authors recently is used to study the consequences of this property. In particular this description allows to calculate the topological charge of solutions (the averaging of the $x$-derivative of the potential) and to show that the averaging of other standard conservation laws is the same for all components.