可积微分差分方程初边值问题研究的Fokas方法

可积系统的初边值问题在可积系统的研究中一直占据重要地位。Fokas方法是研究可积系统初边值问题的有效手段，近年来已被广泛应用于研究连续可积系统的初边值问题。本项目拟拓展Fokas方法至研究离散可积系统的初边值问题。主要研究内容包括：（1）推广Fokas方法至研究可积微分差分方程在正整数集上的初边值问题；（2）研究可积微分差分方程在有限整数集上的初边值问题，其中着重研究关于离散变量是周期的边值问题；（3）沟通可积微分差分方程的线性化边值条件与R-矩阵理论、Bäcklund变换、对称等可积性质的内在联系；（4）探究2+1维连续与离散可积系统初边值问题研究中的一些公开问题。研究成果有助于深化人们对可积微分差分方程结构的理解，丰富离散可积系统的数学理论内涵。

It is very important and essential to study initial-boundary value problems associated with integrable nonlinear evolution systems. In recent years, the so-called Fokas method has been extensively used to analyse initial-boundary value problems for integrable continuous nonlinear evolution equations. In this project, we will develop the Fokas method to study initial-boundary value problems for integrable discrete nonlinear evolution equations. The studies mainly focus on: (1) expanding the Fokas method to analyse initial-boundary value problems for integrable differential-difference equations on positive integers; (2) initial-boundary value problems for integrable differential-difference equations on finite sets of integers, especially in the case of n-periodic boundary value problems; (3) relationship between linearizable boundary conditions and integrable properties such as the R-matrix theory, Bäcklund transform and symmetries; (4) open problems associated with initial-boundary value problems for (2+1)-dimensional integrable systems. Our research will deepen the understanding to integrable differential-difference equations and enrich the mathematical theory of integrable discrete systems.