Abstract

We show that all local holomorphic solutions of all equations constituting the hierarchies of the first and second Painlevé equations can be analytically continued to meromorphic functions on the whole complex plane. We also present a new conceptual proof of the fact that all local holomorphic solutions of the first, second, and fourth Painlevé equations are globally meromorphic.

中文翻译：

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Painlevé方程解及其层级的整体亚纯

摘要

我们表明，构成第一个和第二个Painlevé方程的层级的所有方程的所有局部全纯解都可以解析地延续为整个复杂平面上的亚纯函数。我们还提供了一个新的概念证明，证明了第一，第二和第四个Painlevé方程的所有局部全纯解都是全局亚纯的。