离散可积系统与椭圆函数

Discrete systems and their integrability have undergone a remarkable development during the last two decades, and particularly, they pushed the emergence of discrete complex analysis and discrete differential geometry, which are regarded as new developing mathematical tools. The study of integrability of discrete systems forms at the present time the most promising route towards a general theory of difference equations and discrete systems. We would mainly be working on the following 3 topics: 1) The application of elliptic functions in the discrete systems. For those multi-dimensionally consistent systems we will try to derive their elliptic counterparts (here we mean the new lattice equations containing the terms defined by the points of some elliptic curves). We are also interested in constructing solutions in terms of elliptic functions for discrete systems. This will need to use properties of elliptic functions. 2) Integrable characteristics of discrete systems. We would develop approaches to symmetries and conservation laws from Lax pairs. The possible role that a pseudo-difference operator plays in discrete systems would be also investigated. 3) The derivation and study of what we call "non-potential" lattices. We will try to find relations of multi-dimensionally consistent systems and their possible non-potential forms. Particularly we will investigate the non-potential forms of Kadomtsev-Petviashvili type, the integrability of which we expect would exhibit new mathematical features. The research around all the above 3 topics is hoped to deeply understand discrete systems as well as develop new mathematical tools for discrete systems.

离散系统及其可积性的研究在过去20年中取得很大进展，并推动了离散复分析、离散微分几何等新的数学工具的发展，为目前研究差分方程和离散系统一般理论提供了有效途径。本项目将主要关注以下3个方面。1）椭圆函数在离散系统中的应用。研究离散系统的椭圆化途径，利用椭圆函数性质构造离散可积系统的精确解。2）离散系统的可积特征。从Lax对及拟差分算子出发研究离散系统的可积特征，探索可积特征之间的内在联系。3）具有非势形式(non-potential form)的离散可积系统。寻找多维相容系统的非势形式，发展相关研究方法，重点研究非势形式的离散Kadomtsev-Petviashvili方程。藉此研究实现认识离散可积系统，发展离散系统的数学方法的目标。

近年来离散可积系统和特殊函数理论的进展带来差分方程理论的革命，形成了一个集复分析、代数几何、表示论、伽罗华理论、谱分析、特殊函数理论、图论、差分几何等诸多数学物理分支于一体的新领域。本项目主要研究椭圆函数在离散系统中的应用、离散系统的可积特征、具有非势形式的离散可积系统等，目的在于进一步认识离散可积系统，发展椭圆函数理论和离散系统的数学方法，同时继续关注多维相容系统的构造以及多维相容性在可积性研究中的应用。具体研究成果包括：1）利用椭圆曲线定义离散色散关系，并建立了直接线性化方法的椭圆格式，2）探讨了Sylvester矩阵方程与连续可积系统的联系，3）建立了基于Sylvester方程的可积系统的椭圆化途径，4）提出了“相容三重组”的概念并应用于构建离散可积系统有理解和周期解，5）发展了基于谱问题和散射数据构造多维相容系统无穷守恒量的方法，6）研究了在直接离散中保持可积性的途径，7）给出了半离散AKNS系统非平凡守恒律的组合学公式并建立了严格的连续极限，8）建立了半离散KP系统与半离散AKNS系统的平方本征函数对称约束联系，9）构造了一族多维相容的二元离散可积系统，10）对离散Boussinesq系统的研究，11）完成了基于Wronskian的修正KdV方程的解的完整分类，12）提出了求解非局部方程的双线性-约化方法。这些研究使我们对离散和连续可积系统的数学结构有了更加深刻的认识。所获成果都具有原创性，或建立了新的框架，或发展了新的方法，或揭示了新的联系。项目执行期间，共有4名硕士研究生、6名博士研究生毕业，一位博士后出站；共发表SCI收录论文18篇。我们满意地完成了该项目。