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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 5258375, 9 pages
https://doi.org/10.1155/2017/5258375
Research Article

Algebro-Geometric Solutions for a Discrete Integrable Equation

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Huanhe Dong; moc.621@gnodshtam

Received 25 April 2017; Revised 4 August 2017; Accepted 24 October 2017; Published 14 November 2017

Academic Editor: Chris Goodrich

Copyright © 2017 Mengshuang Tao and Huanhe Dong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.