Calculation of Topological Charge of Real Finite-Gap sine-Gordon solutions using Theta-functional formulae
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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2008
- DOI:
- 10.48550/arXiv.0812.2494
- arXiv:
- arXiv:0812.2494
- Bibcode:
- 2008arXiv0812.2494G
- Keywords:
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- Mathematical Physics