On the asymptotics of real solutions for the Painlev\'{e} I equation
Classical Analysis and ODEs
2024-09-06 v1
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.
Cite
@article{arxiv.2409.03313,
title = {On the asymptotics of real solutions for the Painlev\'{e} I equation},
author = {Wen-Gao Long and Jun Xia},
journal= {arXiv preprint arXiv:2409.03313},
year = {2024}
}