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Algebro-Geometric Quasi-Periodic Solutions to the Bogoyavlensky Lattice 2(3) Equations

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Abstract

The theory of tetragonal curves is established and first applied to the study of algebro-geometric quasi-periodic solutions of discrete soliton equations. Using the zero-curvature equation and the discrete Lenard equation, we derive the hierarchy of Bogoyavlensky lattice 2(3) equations associated with a discrete \(4\times 4\) matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix of this hierarchy, we introduce a tetragonal curve and associated Riemann theta functions and explore the algebro-geometric properties of Baker–Akhiezer functions and a class of meromorphic functions on the tetragonal curve. The straightening out of various flows is precisely given through the Abel map and Abelian differentials. We finally obtain algebro-geometric quasi-periodic solutions of the entire hierarchy of Bogoyavlensky lattice 2(3) equations.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon request.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11931017, 12271490).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, 450001, Henan, People’s Republic of China

    Minxin Jia, Xianguo Geng & Jiao Wei

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  1. Minxin Jia
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  2. Xianguo Geng
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Correspondence to Xianguo Geng.

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Jia, M., Geng, X. & Wei, J. Algebro-Geometric Quasi-Periodic Solutions to the Bogoyavlensky Lattice 2(3) Equations. J Nonlinear Sci 32, 98 (2022). https://doi.org/10.1007/s00332-022-09858-x

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