Resurgent-transseries solutions to Painlevé equations may be recursively constructed out of these nonlinear differential-equations—but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connection-formulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear resurgent Stokes data lack, however, a first-principle computational approach, hence are hard to determine generically. In the Painlevé I and Painlevé II contexts, nonlinear Stokes data get further hindered as these equations are resonant, with non-trivial consequences for the interconnections between transseries sectors, bridge equations, and associated Stokes coefficients. In parallel to this, the Painlevé I and Painlevé II equations are string-equations for two-dimensional quantum (super) gravity and minimal string theories, where Stokes data have natural ZZ-brane interpretations. This work conjectures for the first time the complete, analytical, resurgent Stokes data for the first two Painlevé equations, alongside their quantum gravity or minimal string incarnations. The method developed herein, dubbed “closed-form asymptotics”, makes sole use of resurgent large-order asymptotics of transseries solutions—alongside a careful analysis of the role resonance plays. Given its generality, it may be applicable to other distinct (nonlinear, resonant) problems. Results for analytical Stokes coefficients have natural structures, which are described, and extensive high-precision numerical tests corroborate all analytical predictions. Connection-formulae are explicitly constructed, with rather simple and compact final results encoding the full Stokes data, and further allowing for exact monodromy checks—hence for an analytical proof of our Painlevé I results.

中文翻译：

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Painlevé 方程和二维量子（超）引力的 Resurgent Stokes 数据

Painlevé 方程的 Resurgent-transseries 解可以递归地从这些非线性微分方程中构造出来——但需要在复平面上全局定义 Stokes 数据。Stokes 数据明确地构造了连接公式，它描述了与这些解决方案相关的非线性 Stokes 现象，通过 Stokes transitions 作用于 transseries 的实现。然而，非线性复兴 Stokes 数据缺乏第一性原理计算方法，因此很难进行一般性确定。在 Painlevé I 和 Painlevé II 的背景下，非线性 Stokes 数据受到进一步阻碍，因为这些方程是共振的，对跨系列扇区、桥方程和相关的 Stokes 系数之间的互连具有重要影响。与此同时，Painlevé I 和 Painlevé II 方程是二维量子（超）引力和最小弦理论的弦方程，其中 Stokes 数据具有自然的 ZZ 膜解释。这项工作首次推测了前两个 Painlevé 方程的完整的、分析的、复活的 Stokes 数据，以及它们的量子引力或最小弦化身。此处开发的方法被称为“闭式渐近法”，它仅使用跨序列解的重生大阶渐近法——以及对共振作用的仔细分析。鉴于其普遍性，它可能适用于其他不同的（非线性、共振）问题。分析 Stokes 系数的结果具有描述的自然结构，广泛的高精度数值测试证实了所有分析预测。