TY - JOUR
T1 - Finite Genus Solutions to a Hierarchy of Integrable Semi-Discrete Equations
AU - Xu , Yaru
AU - Jia , Minxin
AU - Geng , Xianguo
AU - Zhai , Yunyun
JO - East Asian Journal on Applied Mathematics
VL - 1
SP - 80
EP - 112
PY - 2025
DA - 2025/01
SN - 15
DO - http://doi.org/10.4208/eajam.2023-195.251023
UR - https://global-sci.org/intro/article_detail/eajam/23742.html
KW - Integrable semi-discrete nonlinear evolution equation, trigonal curve, Baker-Akhiezer function, finite genus solution.
AB - <p style="text-align: justify;">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.</p>