Algebro-geometric Constructions of a Hierarchy of Integrable Semi-discrete Equations

Abstract A hierarchy of integrable semi-discrete equations is deduced in terms the discrete zero curvature equation as well its bi-Hamiltonian structure gotten through trace identity. The above separated into soluble ordinary differential according to relationship between elliptic variables and potentials, from which continuous flow straightened out via Abel–Jacobi coordinates resorting algebraic curves theory. Eventually, meromorphic function Baker–Akhiezer are introduced successively on hyperelliptic curve algebro-geometric solutions expressed Riemann theta can be obtained two functions mentioned above.