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r-matrix and algebraic-geometric solution for integrable symplectic map

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Chinese Science Bulletin

Abstract

A new Lax matrix is introduced for the integrable symplectic map (ISM), and the non-dynamical (i.e. constant)r-matrix of ISM is obtained. Moreover, an effective approach is systematically presented to construct the explicit solution (here, the explicit solution means algebraic-geometric solution expressed by the Riemann-Theta function) of a soliton system or nonlinear evolution equation from Lax matrix,r-matrix, and the theory of nonlinearization through taking the Toda lattice as an example. The given algebraic-geometric solution of the Toda lattice is almost-periodic and includes the periodic and finite-band solution.

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Authors and Affiliations

  1. Institute of Mathematics and School of Mathematical Sciences, Peking University, 100871, Beijing, China

    Zhijun Qiao

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  1. Zhijun Qiao
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Qiao, Z. r-matrix and algebraic-geometric solution for integrable symplectic map. Chin. Sci. Bull. 44, 114–118 (1999). https://doi.org/10.1007/BF02884730

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  • DOI: https://doi.org/10.1007/BF02884730

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