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.
Highlights
As we all know, the generation of integrable system, determination of exact solution, and the properties of the conservation laws are becoming more and more rich [1–5]; in particular, the discrete integrable systems have many applications in statistical physics, quantum physics, and mathematical physics [6–11]
According to the loop algebras, we introduce the following discrete spectral problems
In order to acquire the algebro-geometric solutions of systems (16), we first introduce the Riemann surface Γ of the hyperelliptic curve with genus N: 2N
Summary
The generation of integrable system, determination of exact solution, and the properties of the conservation laws are becoming more and more rich [1–5]; in particular, the discrete integrable systems have many applications in statistical physics, quantum physics, and mathematical physics [6–11]. It is worth discussing the properties of discrete integrable systems, such as Darboux transformations [12, 13], Hamiltonian structures [14–16], exact solutions [17], and the transformed rational function method [18].
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