Nevanlinna理论在几类复差分方程中的应用

The project intends to use Nevanlinna theory of meromorphic functions to investigate properties of solutions of the difference equations related to some types such as the Painlevé type and the Fermat type equations. 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. Based on Nevanlinna theory, we will seek some new crossing points between value distribution theory，complex differential equations and complex difference equations. Furthermore, we will try to look for a general way to solve the problems in difference counterparts of Nevanlinna theory. The study of this project is the cross-over study between Nevanlinna theory and complex difference equations, and is an interesting subject of complex analysis recently. This project is devoted to Nevanlinna theory's connections and applications in difference theory, and to improving some recent results within the field of complex difference equations and difference analogues of Nevanlinna theory. In this project, we mainly continue to study the related problems by some innovative research methods, which can be positive in the developments and connections in different branch of mathematics.

申请者拟利用Nevanlinna理论研究涉及潘勒韦型、费尔马型等复差分方程亚纯解的存在性、解的增长级等相关问题。差分方程一直是应用数学与物理学的研究热点，如在控制理论、量子力学、信号处理等都具有广泛的应用。本项目将在Nevanlinna理论的基础上，寻求不同理论之间的新交叉点，通过对亚纯函数值分布论、复微分方程、差分方程等理论的深入探究，找到解决相关差分问题的一般途径和方法。本项目的研究是Nevanlinna理论与复差分方程的交叉研究，是近几年单复变函数论中比较活跃的研究领域。对此项目的研究，目的在于丰富差分Nevanlinna理论的内涵与应用，创新研究方法，精确差分上的Nevanlinna理论以及复差分方程解的性质等相关结果，这对复分析的发展和促进不同数学分支间的交叉应用，都具有重要的科学意义。

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，具体研究了涉及潘勒韦型、费尔马型等复差分方程解的相关性质。同时，以本项目为依托，我们对上述方程的q阶差分对应同样进行了研究。除此之外，我们还讨论了亚纯函数及其函数位移，差分（q阶差分）算子，差分多项式的分担公共值问题，得到了一系列结果。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11篇。其中 SCI 收录期刊10篇，核心期刊1篇。