本项目将对与 离散矩阵谱问题相联系的离散可积模型进行研究。运用代数几何方法构造离散可积孤子方程族的拟周期解。在此种方法下，借助离散的零曲率方程，将推导出一整族离散可积孤子方程。然后依据代数曲线理论以及广义的朗格朗日插值函数，在Abel-Jacobi 坐标下，试图给出相应孤子方程的离散流和连续流的代数几何构造。最后通过在超椭圆曲线上引入亚纯函数以及Baker-Akhiezer向量，并分析它们的代数和特征以及在无穷远处的渐近性质，最终将给出由精确黎曼theta 函数形式表示的整族离散可积孤子方程的拟周期解

The discrete integrable models associated with the discrete matrix spectral problems will be detailedly researched in this programme. The algebro-geometric methods will be used to investigate the discrete integrable soliton hierarchies. By using this method, the discrete integrable soliton hierachies will be obtained with the aid of the discrete zero-curvature equation. On the basis of the theory of algbraic curves and generalized Lagrange interpolation, the algebro-geometric constructons of the various flows of the derived discrete integrable soliton hierarchies are obtained under the Abel-Jacobi coordinates. The meromorphic function and the Baker-Akhiezer vectors are introduced on the hyperelliptic curve,with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the Baker-Akhiezer functions, Algebro-geometric solutions of the corrsponding discrete intgrable soliton hierarchies areconstructed, which can be explicitly given by theta functions on the Riemann surface.