本项目基于(1+1)维连续孤子方程的Darboux-Bäcklund变换（DBT），探讨连续和离散可积系统之间的内在结构，构造新的DBT和可积离散系统，研究离散可积系统的可积性质和求解，以及DBT在可积离散几何中的应用。具体研究内容包括：1)从(1+1)维连续孤立子方程的DBT变换出发，构造新的离散可积系统，建立连续和离散可积系统可积性质之间的对应关系；2）在公共的有限维可积Hamilton系统平台上，统一构造连续和离散可积系统的代数几何解；3）构造超（对称）可积系统的Darboux变换和新的Grassmann离散可积系统；4）构造具有复杂对称性可积系统的Darboux变换；5）探索给定孤立子方程最大允许的DBT问题;以及6) DBT在可积离散几何中的应用。研究成果将进一步发展和丰富可积系统的数学理论，深化对可积系统的理解。

The project is to explore the intrinsic relationship among continuous, semi-discrete and fully-discrete integrable systems with the aid of the Darboux-Bäcklund transformations (DBTs) of continuous ones. The aim is to construct novel discrete system integrable both differential-difference equations and integrable difference equations, investigate integrable properties and algebra-geometric solutions of the resulting discrete integrable systems. Especially we will focus on the following topics: 1) constructing novel integrable discrete systems from (1+1) dimensional continuous soliton equations via DBTs, and establishing relationship among continuous and discrete integrable systems; 2) synchronously constructing algebra-geometric solutions for the related continuous and discrete integrable systems on the platform of finite-dimensional Liouville integrable Hamiltonian systems; 3) Darboux transformations of supersymmetric integrable systems and discrete integrable systems on Grassmann algebras; 4) Darboux transformations for the soliton equations which have complex symmetries; 5) how many DBTs for a given integrable system are allowed; and 6) the applications of DBTs to integrable discrete geometry. The study will enrich the mathematic theory of integrable systems and deepen the understandings to various integrable systems.