微分方程的自洽化 I：Painlevé 方程 I-IV 的 Sundman 变换
On the autonomisation of differential equations I: Sundman transformations for Painlevé equations I-IV
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中文总结 AI 辅助
本文通过构造等价自治系统并应用 Sundman 变换，将前四个 Painlevé 方程自洽化，给出解映射及 Airy 和 Weber-Hermite 函数系数的特例。
中文摘要 AI 辅助
我们考虑将 Sundman 变换应用于前四个 Painlevé 方程 $P_I$-$P_{IV}$，而 $P_V$ 和 $P_{VI}$ 留待未来工作。由于 Painlevé 方程是非自治的，这首先需要构造一个等价的自治系统，我们详细讨论这一过程，并将其描述为利用 Lie 对称性对自治系统进行降阶的逆过程。其次，我们将 Sundman 变换应用于该自治系统，并将所得方程与相应的非自治方程进行识别。我们给出了该非自治方程的解与原始 Painlevé 方程的解之间的显式映射。通过 Sundman 变换发现的非自治方程的有趣特例包括：对于 $P_{II}$，一个系数通过 Airy 函数表达的方程；对于 $P_{IV}$，一个系数通过 Weber-Hermite 函数表达的方程。我们还讨论了一种替代的自洽化过程。最后，我们指出了未来工作的多种可能方向以及有趣的开放问题。
英文摘要
We consider the application of Sundman transformations to the first four Painlevé equations $P_I$-$P_{IV}$, with $P_V$ and $P_{VI}$ being left for future work. Since the Painlevé equations are nonautonomous, this requires first of all the construction of an equivalent autonomous system, a process which we discuss in detail, and which we characterize as being inverse to the use of Lie symmetries to obtain a reduction of order of an autonomous system. Secondly, we apply a Sundman transformation to this autonomous system, and identify the resulting equation with a corresponding nonautonomous equation. We give an explicit mapping between solutions of this nonautonomous equation and the original Painlevé equation. Interesting special cases of the nonautonomous equation found by Sundman transformation include, for $P_{II}$, an equation with coefficients expressed via Airy functions, and for $P_{IV}$, an equation with coefficients expressed via Weber-Hermite functions. We also discuss an alternative autonomisation process. Finally, a variety of possible directions for future work are identified, along with interesting open problems.
