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.
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-fu
nctions 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.
