Multiscale limit for finite-gap Sine-Gordon Solutions and Calculation of Topological Charge using Theta-functional Formulae

TL;DR

This paper develops a multiscale limit method for spectral curves to compute the topological charge density of real finite-gap Sine-Gordon solutions directly from theta-functional formulas.

Contribution

It introduces a multiscale limit approach for spectral curves, enabling direct calculation of topological charge density from theta-functional formulas for finite-gap solutions.

Findings

Multiscale limit method for spectral curves established

Direct computation of topological charge density achieved

Enhanced understanding of finite-gap Sine-Gordon solutions

Abstract

In this paper, we introduce the so-called multiscale limit for spectral curves, associated with real finite-gap Sine-Gordon solutions. This technique allows to solve the old problem of calculating the density of topological charge for real finite-gap Sine-Gordon solutions directly from the $\theta$-functional formulas.