Classification of the real Painlev\'{e} I transcendents by zeros and connection problem: an asymptotic study

TL;DR

This paper analyzes the asymptotic behavior and connection problem of Painlevé I solutions by examining zeros and Stokes multipliers, classifying solutions, and constructing a phase diagram with numerical validation.

Contribution

It introduces a detailed asymptotic analysis of Stokes multipliers and zero parameters, providing a comprehensive classification and connection framework for real Painlevé I solutions.

Findings

Classified solutions into oscillatory, separatrix, and singular types.

Derived full asymptotic expansions of Stokes multipliers.

Constructed a phase diagram in the (r,b)-plane with numerical validation.

Abstract

In this paper, we study the asymptotic behavior and connection problem of Painlev\'e I (PI) equation through a detailed analysis of the Stokes multipliers associated with its solutions. Focusing on the regime where the derivative at the real zeros of the solution becomes large, we apply the complex WKB method to derive full asymptotic expansions of the Stokes multipliers. These expansions allow us to classify real solutions of PI according to their behavior at the zeros, distinguishing between oscillatory, separatrix, and singular types solutions on the negative real axis. Furthermore, we resolve the connection problem between the large negative asymptotics and the location of positive zeros by establishing full asymptotic expansions of the zero parameters. Our approach enables the construction of a precise phase diagram in the $(r,b)$-plane, where $r$ is the location of a zero and $b$ is the derivative at that point. Numerical simulations are provided to validate the theoretical results. This work extends prior studies on monodromy asymptotics and contributes a comprehensive framework for understanding the global structure of real PI solutions through their local zero data.