##### 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. 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 this dissertation is: to calculate the topological charge density from the theta-functional expressions for the solution u(x, t). This problem has remained unsolved since it was first raised by S.P. Novikov in 1982. The problem is solved here by introducing a new limit of real finite-gap sine-Gordon solutions, which we call the multiscale or elliptic limit. We deform the spectral curve to a singular nodal curve, having elliptic curves as components, for which the calculation of topological charges reduces to two special easier cases.