空间离散可积系统理论中若干重要问题的研究

本项目研究空间离散可积系统理论中若干重要和前沿的问题。主要研究内容包括：(1)空间离散可积方程Cauchy问题的逆散射与对应的可积方程Cauchy问题的逆散射之间的联系，重点讨论空间离散KdV方程的Cauchy问题和KdV方程的Cauchy问题之间的联系；(2)几个重要的有物理背景的空间离散可积方程可积性，如离散多分量短脉冲方程和离散多分量复短脉冲方程Cauchy问题的精确和数值求解；(3)非线性非局部可积方程的研究，包括非局部Hirota 方程，非局部Sasa-Satusma方程，非局部Heisenberg铁磁链方程，非局部完全各向异性Landau-Lifshitz方程的可积离散和可积性；(4)离散的矩阵Painlevé方程簇的构造和求解。. 本项目研究的科学问题引领性创新性强。通过本项目的研究，我们力求对空间离散可积系统理论中这些重要问题带来有价值的新发现新进展。

This project explores some important and frontier problems in the theory of spatially discrete integrable systems. We will focus on the following topics:.(1) Relation between Cauchy problems for spatially discrete integrable equations and Cauchy problems for the corresponding integrable equations via inverse scattering transform. Especially, the connection between Cauchy problem for the spatially discrete KdV equation and Cauchy problem for the KdV equation; .(2)Integrability of several vital spatially discrete integrable equations in fields of physics. For example, the exact solutions and numerical simulation of the Cauchy problem for the discrete multi-component short-pulse equation and the discrete multi-component complex short-pulse equation; .(3) On the nonlinear nonlocal integrable equation. We will focus on integrable discretization and integrability of several nonlinear nonlocal integrable equations, including nonlocal Hirota equation, nonlocal Sasa-Satusma equation, nonlocal Heisenberg ferromagnet equation and nonlocal anisotropic Landau-Lifshitz equation..(4)Construction and exact solutions of hierarchy of the discrete Matrix Painleve equation. .Through the study of this project, we hope to gain valuable new discovery and development for those important topics in the theory of spatially discrete integrable system.