Calculation of Topological Charge of Real Finite-Gap sine-Gordon solutions using Theta-functional formulae

TL;DR

This paper develops a method to compute the topological charge density of real finite-gap sine-Gordon solutions directly from theta-functional formulas by deforming the spectral curve to a simpler singular case.

Contribution

It introduces a novel multiscale limit approach that simplifies the calculation of topological charges from theta-functional solutions of the sine-Gordon equation.

Findings

Topological charge density can be computed via spectral curve deformation.

The method reduces complex calculations to two simpler cases.

The approach provides explicit formulas for real finite-gap solutions.

Abstract

The most basic characteristic of x-quasiperiodic solutions u(x,t) of the sine-Gordon equation u_{tt}-u_{xx}+\sin u=0 is the topological charge density denoted $\bar n$. The real finite-gap solutions u(x,t) are expressed in terms of the Riemann theta-functions of a non-singular hyperelliptic curve $\Gamma$ and a positive generic divisor D of degree g on $\Gamma$, where the spectral data $(\Gamma, D)$ must satisfy some reality conditions. The problem addressed in note is: to calculate $\bar n$ directly from the theta-functional expressions for the solution u(x,t). The problem is solved here by introducing what we call the multiscale or elliptic limit of real finite-gap sine-Gordon solutions. We deform the spectral curve to a singular curve, for which the calculation of topological charges reduces to two special easier cases.