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

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.