The Inverse Spectral Problem Method for the Periodic Differential–Difference Sine–Gordon Equation

This paper is devoted to the investigation of the periodic discrete sine–Gordon equation by means of the inverse spectral problem method. The associated discrete Lax pair is studied, and the spectral theory of the corresponding periodic linear difference problem is established. Explicit representations of fundamental solutions are obtained in terms of polynomial structures, and the spectral data are characterized via a hyperelliptic curve. Using these results, trace formulas are derived that provide an explicit reconstruction of the solution in terms of the spectral parameters. An algorithm for solving the inverse spectral problem is formulated based on the evolution of the spectral data. Finally, we derive a system of evolution equations for the spectral parameters, which may be regarded as a discrete analogue of the Dubrovin equations.

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