摘要翻译：
本文研究了Chazy III、IX和X方程。对于Chazy III方程，通过二次变换，将其转化为一个三阶有理型常微分方程。对于这个方程，我们得到了它的亚纯解，其自由参数本质上是两个。我们还证明了与此方程相关的系统允许用$tanh(t)$解的新的特解。对于Chazy IX方程，我们通过二次变换将Chazy IX方程转化为一阶常微分方程组。对于该系统，我们给出了两个新的双分形B{a}cklund变换，并给出了该系统的全纯条件。利用这个全纯条件，我们得到了一个包含Chazy IX方程的二元偏微分系统，该系统满足相容条件，并且允许行波解。对于Chazy X方程，我们通过双分形变换将Chazy X方程转化为一阶常微分方程组。对于该系统，我们给出了两个双分形B{a}cklund变换。其中一个是新的。我们还给出了这个系统的全纯条件。多亏了这个全纯条件，我们才能恢复这个系统。
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英文标题：
《Studies on the Chazy equations》
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作者：
Yusuke Sasano
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Classical Analysis and ODEs        经典分析与颂歌
分类描述：Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics
特殊函数、正交多项式、调和分析、Ode、微分关系、变分法、逼近、展开、渐近
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一级分类：Mathematics        数学
二级分类：Dynamical Systems        动力系统
分类描述：Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations
微分方程和流动的动力学，力学，经典的少体问题，迭代，复杂动力学，延迟微分方程
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英文摘要：
  In this paper, we study the Chazy III,IX and X equations. For the Chazy III equation, by making the birational transformations the Chazy III equation is transformed into a third-order ordinary differential equation of rational type. For this equation, we find its meromorphic solutions, whose free parameters are essentially two. We also show that the system associated with this equation admits new special solutions solved by $tanh(t)$. For the Chazy IX equation, we transform the Chazy IX equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two new birational B{\"a}cklund transformations. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we obtain a new partial differential system in two variables involving the Chazy IX equation, This system satisfies the compatibility condition, and admits a travelling wave solution. For the Chazy X equation, we transform the Chazy X equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two birational B{\"a}cklund transformations. One of them is new. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we can recover this system.
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PDF链接：
https://arxiv.org/pdf/0708.3537