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Topological charge of the real periodic finite‐gap Sine‐Gordon solutions

An effective description of the inverse spectral data corresponding to the real periodic and quasi‐periodic finite‐gap (algebro‐geometrical) solutions for the sine‐Gordon equation is obtained. In particular, the explicit formula for the so‐called topological charge of the solutions is found and proven. As was already understood 20 years ago, it is very hard to extract any formula for this quantity from the Θ‐functional expressions. A new method was developed by the authors for this goal. In Appendix C an analogue of the Fourier‐Laurent integral transform on the Riemann surface is defined, based on the constructions of soliton theory. It is a natural continuous analogue of the discrete Krichever‐Novikov bases developed in the late 1980s for the needs of the operator quantization of string theory. © 2003 Wiley Periodicals, Inc.