离散可积系统中的变换及其几何

The development of new mathematical tools such as discrete complex analysis and discrete differential geometry, has provided a new effective approach for the research on the theory of discrete integrable systems. We would mainly be working on the following 3 topics: 1) Bäcklund transformation of discrete systems. For discrete integrable systems we will try to derive a routine for deriving BT, its classification and further applications including exact solutions and Lax pairs. 2) Discrete differential geometry related to discrete systems. Starting from surface theory and the motion of curves on different geometries we construct, we investigate the related discrete systems and its properties, to find the correspondences with continuous cases. 3) Bi-differential graded algebras for discrete integrable systems. We are interested in finding a general approach for constructing properties of integrable systems in the frame of bi-differential calculus. And then we systematically study the case of discrete integrable systems. The research around all the above 3 topics is hoped to deeply understand discrete systems with the aid of different new mathematical tools.

离散复分析、离散微分几何等新数学工具的发展，为离散可积系统理论的研究提供了新的有效途径。本项目将主要关注以下3个方面。1）离散系统中的Bäcklund变换。研究离散系统中Bäcklund变换的构造分类，以及其应用。2）离散系统中的离散微分几何。从曲面论或者曲线运动出发研究离散可积系统，探索其与连续系统的对应。3）离散可积系统中的双微分分次代数。在双微分分次代数下探寻对于系统性质的一般化构造方法，并用该方法研究离散可积系统。从三方面借助不同的数学工具更好地认识离散可积系统。

Bäcklund变换是研究离散可积系统的重要工具，双微分分次代数和离散几何是研究离散可积系统的新颖视角。本项目从离散可积系统Bäcklund变换、多分量离散可积系统、双微分分次代数、离散几何等方面展开研究：提出了多个Bäcklund变换构造性方法，构造了新的多维相容性立方体，导出了扭曲型ABS方程、H3*方程等，构建了Bäcklund变换与周期函数加法公式的联系，得到了Bäcklund变换的双线性形式，构造了一系列非自治方程的Casorati形式有理解；提出了一种构造多维相容的N分量离散可积系统的系统性方法，并导出了高阶单分量新的可积方程、离散Toda型方程、King-Schief方程、多分量离散Painleve方程、非局域ABS方程等系统；双微分分次代数框架下构造了方程Lax对的直接方式，并得到了H1方程等一系列可积系统；建立了曲线运动与H3方程、离散sine-Gordon方程、Q1(0)方程、H3*方程、A1(0)方程等系统的联系。