复域上的差分Painleve方程

If the solutions of second order differential equations satisfy Painleve property, then all the equations can be expressed by six differential equations, which are called Painleve equations (I)-(VI). Difference Painleve equations can be deduced from Painleve equations by continuous limit. In this proposal, we plan to investigate the following three interesting problems: (1) If a finite order transcendental meromorphic function is a solutions of difference Painleve equations I, is this solution of order 5/2? (2) If we have a solutions of difference Painleve equations II, how should we get another new solutions according to coefficients of the equations. (3) How should we obtain difference Painleve equations IV from equations family. In order to study these three problems, we proceed to use “singularity confinement” and difference Nevanlinna theory in this proposal. The problems in this proposal are interesting, it will be useful for differential Painleve equations and other subjects.

如果二阶微分方程的解满足Painleve性质，那么二阶微分方程可以退化成六类方程，这六类方程我们称之Painleve方程。差分Painleve方程是Painleve方程的离散形式，通过取解的连续极限，差分Painleve方程可以由对应的六类Painleve方程得到。本项目针对差分Painleve方程主要研究以下三个问题：（1）如果一个超越亚纯函数是差分Painleve方程I的解，那么它的增长级是否是5/2。 (2) 已知差分Painleve方程II的一个亚纯解，能否根据方程的系数和已知解构造出一个新的方程的解。(３)如何由一个方程族得到差分Painleve方程IV的形式。本项目将会采用极点限制方法结合差分Nevenlinna理论来解决以上三个问题。对差分Painleve方程的研究有利于了解微分Painleve方程的性质，即有理论价值，又有物理应用意义。

复域上的差分Painleve方程是一类特殊的非线性差分方程，对复域上的差分Painleve方程的研究是近十几年内比较关注的课题之一，对复域上的差分Painleve方程的研究不仅具有重大的理论价值，还具有实际意义。目前，比较关注的问题是差分Painleve方程的分类、差分Painleve方程之间的相互转化以及差分Painleve方程解的一些性质。对差分Painleve方程相关问题的研究，促进了差分Nevanlinna理论的形成，以及推动了差分形式的值分布的发展。本项目主要研究了差分Painleve方程的分类问题，以及差分Painleve方程解的性质，围绕这个问题，本项目持续3年，一共发表5篇文章，举办2次国际会议，参加国内外会议数次。