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Transactions of the American Mathematical Society

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The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices


Authors: Jiao Wei, Xianguo Geng and Xin Zeng
Journal: Trans. Amer. Math. Soc. 371 (2019), 1483-1507
MSC (2010): Primary 37K10, 37K20, 14H42, 37K40
DOI: https://doi.org/10.1090/tran/7349
Published electronically: September 10, 2018
MathSciNet review: 3885186
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Abstract: Starting with a discrete $ 3\times 3$ matrix spectral problem, the hierarchy of Bogoyavlensky lattices which are pure differential-difference equations are derived with the aid of the Lenard recursion equations and the stationary discrete zero-curvature equation. By using the characteristic polynomial of Lax matrix for the hierarchy of stationary Bogoyavlensky lattices, we introduce a trigonal curve $ \mathcal {K}_{m-1}$ of arithmetic genus $ m-1$ and a basis of holomorphic differentials on it, from which we construct the Riemann theta function of the trigonal curve, the related Baker-Akhiezer function, and an algebraic function carrying the data of the divisor. Based on the theory of trigonal curves, the Riemann theta function representations of the Baker-Akhiezer function, the meromorphic function, and in particular, that of solutions of the hierarchy of Bogoyavlensky lattices are obtained.


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Additional Information

Jiao Wei
Affiliation: School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan 450001, People’s Republic of China
Email: weijiaozzu@sohu.com

Xianguo Geng
Affiliation: School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan 450001, People’s Republic of China
Email: xggeng@zzu.edu.cn

Xin Zeng
Affiliation: School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan 450001, People’s Republic of China
Email: xzeng@zzu.edu.cn

DOI: https://doi.org/10.1090/tran/7349
Keywords: Bogoyavlensky lattices, trigonal curve, Baker--Akhiezer function, Riemann theta function solutions.
Received by editor(s): May 4, 2016
Received by editor(s) in revised form: November 24, 2016, and June 5, 2017
Published electronically: September 10, 2018
Additional Notes: This work was supported by National Natural Science Foundation of China (Grant nos. 11331008 and 11871440
The second author (Xianguo Geng) is the corresponding author.
Article copyright: © Copyright 2018 American Mathematical Society