Classification of the Real Painlevé I Transcendents by Zeros and Connection Problems: An Asymptotic Study

ABSTRACT In this paper, we study the asymptotic behavior and connection problem of the first Painlevé (PI) equation through a detailed analysis of the Stokes multipliers associated with its real 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 solutions on the negative real axis. Our approach enables the construction of an asymptotic description of the phase diagram in the ‐plane, where is the location of a zero and is the derivative at that point. 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. 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.