复域差分和Painlevé差分方程的研究

Using the Nevanlinna theory of the value distribution of meromorphic functions and theory of complex differential equation, we investigate some important problems of the difference equations containing some difference Painlevé equation. Firstly, we seek the expression of the coefficients of the difference equation containing a difference Painlevé I equation or a difference III equation, which has an non-rational meromorphic solution of finite order; Secondly, we discuss whether the difference equations containing some difference Painlevé equation has an asymptotic solution in a certain domain; Thirdly, we establish the difference counterpart of Malmquist type theorem of generalized differential equations; Last but not least, we establish the difference analogue of the logarithmic derivative lemma in some angular domain or annular domain. Difference equations have been the focus on the content of applied mathematics and physics, and have a wide range of applications, such as quantum physics, control theory, signal processing and so on. It is difficult to solve the main extant problem if we only use the Nevanlinna theory of the value distribution of meromorphic functions. In this project, we completely solve two or three open questions of difference equations by using a combination of the value distribution of meromorphic functions and theory of complex differential equation and complex difference equations and some innovative research methods, which can be positive in the developments and connections in different branch of mathematics.

综合应用复域微分方程理论和亚纯函数值分布理论，深入研究涉及Painlevé型复差分方程的几个重要问题。第一、若包含Painlevé I、III 型差分方程存在一个有穷级超越亚纯解，给出系数的具体表达式；第二、研究包含Painlevé型差分方程在特定区域内渐进解的存在性；第三、研究一般代数微分方程的Malmquist定理的差分模拟；第四、建立环域或角域中差分形式的对数导数引理。复差分方程一直是应用数学与物理学的研究热点，其研究成果与方法在量子力学、信号处理、控制理论等方面都有广泛应用。现存复差分方程的主要问题如果仅用亚纯函数值分布理论已很难解决。本项目旨在将复域微分方程、Nevanlinna 值分布理论和复差分方程问题结合起来研究，创新研究方法，完全解决两到三个现存的复差分方程的开问题。这对复分析的发展、促进不同数学分支间的交叉均有重要意义。

通过三年的项目执行期，项目组成员不仅按照申请书的计划，具体研究涉及Painlevé型复差分方程的几个重要问题。同时，以本项目书为依托，我们对相应的微分方程对应同样的进行了研究。除此之外，我们还讨论了某类差分方程，得到一系列结果。与本项目有关的研究结果，以论文的形式在国内外学术期刊发表7篇。其中SCI收录期刊7篇，2篇被 SCI 期刊接受。3篇审稿。