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Finite Genus Solutions to a Hierarchy of Integrable Semi-Discrete Equations

Finite Genus Solutions to a Hierarchy of Integrable Semi-Discrete Equations

Year:    2025

Author:    Yaru Xu, Minxin Jia, Xianguo Geng, Yunyun Zhai

East Asian Journal on Applied Mathematics, Vol. 15 (2025), Iss. 1 : pp. 80–112

Abstract

Resorting to the discrete zero-curvature equation and the Lenard recursion equations, a hierarchy of integrable semi-discrete nonlinear evolution equations is derived from a $3 \times 3$ matrix spectral problem with three potentials. Based on the characteristic polynomial of the Lax matrix for the hierarchy, a trigonal curve is introduced, and the properties of the corresponding three-sheeted Riemann surface are studied, including the genus, three kinds of Abelian differentials, Riemann theta functions. The asymptotic properties of the Baker-Akhiezer function and fundamental meromorphic functions defined on the trigonal curve are analyzed with the established theory of trigonal curves. As a result, finite genus solutions of the whole integrable semi-discrete nonlinear evolution hierarchy are obtained.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/eajam.2023-195.251023

East Asian Journal on Applied Mathematics, Vol. 15 (2025), Iss. 1 : pp. 80–112

Published online:    2025-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    33

Keywords:    Integrable semi-discrete nonlinear evolution equation trigonal curve Baker-Akhiezer function finite genus solution.

Author Details

Yaru Xu

Minxin Jia

Xianguo Geng

Yunyun Zhai