微分方程在数学（如特殊函数的构造）和物理（如天体的运动）等学科中起着非常重要的作用。在应用方面，很多问题的关键之处就是构造微分方程的精确解。不过，一般来讲，构造精确解相对比较困难，所以对方程的解的性质的研究也非常重要，比如解的增长性。本项目针对代数微分方程、差分方程的亚纯解和偏微分方程的rogue wave解研究以下四个问题：.（1）构造一个具有物理背景的三阶代数微分方程的所有亚纯解。.（2）对一类差分方程的所有亚纯解进行分类，即将Eremenko对某些代数微分方程亚纯解的分类推广到差分方程。.（3）关于亚纯解的增长性，验证可分解的二阶代数微分方程满足Hayman猜想。.（4）构造耦合AB系统的rogue wave解。

Differential equations play important roles in mathematics (e.g. defining special functions) and physics (e.g. celestial motion). For many applications, exact solutions of these differential equations may provide crucial information or explanations to them. However, constructing exact solutions in general is quite difficult, and so it is also very important to study properties of solutions, e.g. the growth of them. In this research proposal, we focus on the following four questions on meromorphic solutions of differential or difference equations and rogue wave solutions of some partial differential equations: .(1) Construct all meromorphic solutions of a third order algebraic differential equation with physical background. .(2) Classify all meromorphic solutions of certain difference equations and thus generalize Eremenko's result on the classification of meromorphic solutions of certain differential equations to difference equations. .(3) On the growth of meromorphic solutions, verify that second order factorizable algebraic differential equations satisfy Hayman's conjecture. .(4) Construct rogue wave solutions of a coupled "AB" system.