可积系统、特殊函数与正交多项式相关问题研究

The project focuses on the study of related problems among integrable systems, special functions, orthogonal polynomials and combinatorics. Discrete integrable systems are one of important topics in the field of integrable systems, in the same time, discrete integrable systems are closely tied up with other branches of mathematics. We will use bilinear method to construct new ways of integrable discretization. We will also construct the connection between integrable systems and orthogonal polynomials and study integrable systems from the aspect of orthogonal polynomials. Painleve equations are basic ones in the integrable systems， its theory is called the theory of nonlinear special functions. Heun equations are Fuchsian type differential equaitons with four singularities. There exists correspondence between Heun and Painleve equations. We will study the Painleve equations from the aspect of Heun equations and give new explanation to rational and special function solutions of Painlevé equations. Integrable systems are also closely related to combinatorics. Integrable combinatorics has become a new title. We will start from Hankel type determinant solutions of integrable systems to study related combinatorics to give new knowledge of integrable systems and look for new relation between integrable systems and combinatorics.

本项目的研究目标定位在对可积系统与特殊函数、正交多项式、组合数等相关问题的研究。离散可积系统是可积系统研究的热点之一，同时,离散可积系统与其他数学分支存在紧密联系。我们将利用双线性方法寻找孤子方程可积离散化的新途径；建立离散可积系统与正交多项式之间的联系，以正交多项式为工具研究可积系统。Painleve方程是最基本的可积系统，关于它的理论也被称为非线性特殊函数理论。Huen方程是具有四个奇点的Fuchsian型方程，它与Painleve方程之间存在对应。我们将从Heun方程的角度研究Painleve方程，给出Painleve方程有理解和特殊函数解的奇特性质的新解释；可积系统与组合数学之间存在密切联系，可积组合学成为一个新的研究方向。我们将从可积系统的Hankel型行列式解出发寻找相关的组合数，给予可积系统内在结构的新认识，深入研究可积系统与组合数学之间的联系。

孤立子理论也被认为是特殊函数的理论，许多孤子方程存在包括Airy函数、Hermite函数等特殊函数表示的解。正交多项式满足三项递推公式，与离散可积系统存在紧密联系。该项目在可积离散化与数值模拟，正交多项式与离散可积系统、Heun方程及其相关特殊函数、离散可积系统相关的组合数学、连续可积系统的复化及其动力学性质方面取得了进展。项目按照计划，顺利执行，完成了预期的研究目标。