本项目研究与非线性Schrodinger 方程相联系的若干连续和离散可积系统的可积性。主要研究内容包括: 离散NLS 方程和一些重要的离散可积方程的联系； 完全各向异性的修正Landau-Lifshitz方程的规范等价结构和可积离散； 连续和离散矩阵Painleve 方程簇的构造和性质；在非线性光学和激光物理中有重要作用的短脉冲方程和复的短脉冲方程的Hamilton 理论；立方非线性项包含非线性相邻格点的离散不可积NLS 方程的动力学和解，如空间周期解，对应的离散系统的不动点和稳定性，不动点附近轨道的性质，离散孤子解。本项目所涉及的与非线性Schrodinger 方程相联系的系统有广泛的物理应用。研究这些方程的可积性、动力学特征和孤子解不仅在可积系统理论上而且对于揭示所描述的物理现象的本质都具有重要意义，将给与非线性Schrodinger 方程相联系的若干系统带来新发现和新进展。

In this project, we will study the integrability for several continuous and discrete integrable equations related to the nonlinear Schrodinger (NLS) equation. We will focus on the following topics: the relation between the discrete NLS equation and several important discrete integrable equations; gauge-equivalent structure and integrable discretization for a modified Landau-Lifshitz equation with anisotropy; the construction of continuous and discrete Painleve equation hierarchy ; the Hamilton theory for the short pules equation and the complex short pules equation which have important applications in nonlinear optic; the dynamics and solution for the discrete non-integrable NLS equation with nonlinear hopping, such as exact spatial period solutions, fixed point and its stability, the property of orbits near fixed points, and discrete soliton. These equations discussed in this project have widely physical applications. Thus, the study on their integrability, dynamic, and soliton is very important not only for the theory of integrable system but also for the discovery of the physical phenomenon. The investigation of this project will give us new finds and new progression of these equations related to NLS equation.