Schwarzian导数、Painlevé XXV-Ermakov方程与Bäcklund变换
Schwarzian derivative, Painlevé XXV-Ermakov equation and Bäcklund transformations
AI总结：
本文重新审视了Schwarzian导数在非线性常微分方程中的作用，通过变量替换证明其是构造解的关键工具，并获得了联系线性与非线性方程的两族Bäcklund变换。
摘要：
重新审视了Schwarzian导数在研究非线性常微分方程中的作用。Painlevé XXV-Ermakov方程、Ermakov方程和正规形式的三阶线性方程所容许的解与不变性，被证明是基于Schwarzian方程的解。从Riccati方程和Riccati链的二阶元素作为可线性化方程的最简单例子出发，通过引入适当的变量替换，说明了Schwarzian导数如何在解的构造中代表关键工具。获得了联系所研究的线性与非线性方程的两族Bäcklund变换。给出并讨论了一些具有相关应用的例子。
英文摘要：
The role of Schwarzian derivative in the study of nonlinear ordinary differential equations is revisited. Solutions and invariances admitted by Painlevé XXV-Ermakov equation, Ermakov equation and third order linear equation in a normal form are shown to be based on solutions of the Schwarzian equation. Starting from the Riccati equation and the second order element of the Riccati chian as the simplest examples of linearizable equations, by introducing a suitable change of variables, it is shown how the Schwarzian derivative represents a key tool in the construction of solutions. Two families of Bäcklund transformations which link the linear and nonlinear equations under investigation are obtained. Some examples with relevant applications are given and discussed.
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