On the asymptotics of real solutions for the Painlev\'{e} I equation

TL;DR

This paper refines the asymptotic formulas for real solutions of the Painlevé I equation as the variable approaches negative infinity, improving error estimates and correcting previous typos using the Riemann-Hilbert method.

Contribution

It provides more precise asymptotic error estimates for Painlevé I solutions and their Hamiltonians, and corrects existing literature errors using advanced analytical techniques.

Findings

Improved error bounds for oscillatory asymptotics.

Established precise singular asymptotics for Hamiltonians.

Corrected typos in previous asymptotic descriptions.

Abstract

In this paper, we revisit the asymptotic formulas of real Painlev\'e I transcendents as the independent variable tends to negative infinity, which were initially derived by Kapaev with the complex WKB method. Using the Riemann-Hilbert method, we improve the error estimates of the oscillatory type asymptotics and provide precise error estimates of the singular type asymptotics. We also establish the corresponding asymptotics for the associated Hamiltonians of real Painlev\'e I transcendents. In addition, two typos in the mentioned asymptotic behaviors in literature are corrected.