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

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