Abstract

We obtain second-order nonlinear differential equations (and the associated Bäcklund transformations) with an arbitrary analytic function of the independent variable. These equations (which are not of Painlevé type in general) under certain constraints imposed on an arbitrary analytic function can be reduced, in particular, to the second, third or fourth Painlevé equation. We consider the properties of the Bäcklund transformations for the second-order nonlinear differential equations generated by two systems of two first-order nonlinear differential equations with quadratic nonlinearities in derivatives of the unknown functions.

中文翻译：

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一些二阶非线性微分方程的 Bäcklund 变换

摘要

我们获得具有自变量的任意解析函数的二阶非线性微分方程（以及相关的贝克伦德变换）。这些方程（通常不是 Painlevé 类型）在施加于任意解析函数的某些约束下可以简化为第二、第三或第四 Painlevé 方程。我们考虑由两个一阶非线性微分方程的两个系统生成的二阶非线性微分方程的 Bäcklund 变换的性质，未知函数的导数具有二次非线性。