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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.
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
We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial genera
The Klein-Gordon equation is solved approximately for the Hulth{e}n potential for any angular momentum quantum number $ell$ with the position-dependent mass. Solutions are obtained reducing the Klein-Gordon equation into a Schr{o}dinger-like differen
Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of the nonlinear Klein-Gordon-Maxwell system and nonlinear Schroedinger-Maxwell system with subcritical nonlinearity. We prove that the number of one
We consider the dynamics $tmapstotau_t$ of an infinite quantum lattice system that is generated by a local interaction. If the interaction decomposes into a finite number of terms that are themselves local interactions, we show that $tau_t$ can be ef