Painleve VI方程的Picard和Chazy解
Picard and Chazy solutions to the Painleve' VI equation
摘要：
我研究了一个特殊的家庭的Painlevé VI方程的参数$β=γ=0, δ=1/2$和$2α=(2μ-1)^2$的解决方案，为$2μ\in\interi$。我表明，半整数$μ$的情况下是可积的，解决方案有两种类型：所谓的皮卡德解决方案和所谓的Chazy解决方案。给出了它们的显式公式，并完全确定了它们在奇点$0,1,\infty$附近的渐近行为和它们的非线性单值性。本文研究PVI $μ$方程解的解析延拓的结构，对于任何$μ$使得$2μ\in\interi$。作为一个应用，我对所有的代数解进行了分类。对于$μ$半整数，证明了它们与平面上的正多边形或星形多边形一一对应。对于$μ$整数，证明了所有的代数解都属于一个单参数有理解族。
英文摘要：
I study the solutions of a particular family of Painlevé VI equations with the parameters $β=γ=0, δ=1/2$ and $2α=(2μ-1)^2$, for $2μ\in\interi$. I show that the case of half-integer $μ$ is integrable and that the solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. I give explicit formulae for them and completely determine their asymptotic behaviour near the singular points $0,1,\infty$ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVI$μ$ equation for any $μ$ such that $2μ\in\interi$. As an application, I classify all the algebraic solutions. For $μ$ half-integer, I show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For $μ$ integer, I show that all algebraic solutions belong to a one-parameter family of rational solutions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
