Connection problem of the first Painlevé transcendent between poles and negative infinity

Wen-Gao Long, Yu-Tian Li, Qing-hai Wang

Published 2021-12-17, updated 2023-01-19Version 2

We consider a connection problem of the first Painlev\'{e} equation ($\mathrm{P_I}$), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable $t$ tends to negative infinity for real $\mathrm{P_I}$ functions. We get a classification of the real $\mathrm{P_I}$ functions in terms of $(p,H)$ so that they behave differently at the negative infinity, where $p$ is the location of a pole and $H$ is the free parameter in the Laurent series. Some limiting-form connection formulas of $\mathrm{P_I}$ functions are obtained for large $H$. Specifically, for the real tritronqu\'{e}e solution, the large-$n$ asymptotic formulas of $p_n$ and $H_n$ are obtained, where $p_n$ is the $n$-th pole on the real line in the ascending order and $H_n$ is the associated free parameter. Our approach is based on the complex WKB method (also known as the method of uniform asymptotics) introduced by Bassom, Clarkson, Law and McLeod in their study on the connection problem of the second Painlev\'{e} transcendent [Arch. Rational Mech. Anal., 1998, pp. 241-271]. Several numerical simulations are carried out to verify our main results. Meanwhile, we obtain the phase diagram of \PI~solutions in the $(p,H)$ plane, which somewhat resembles the Brillouin zones in solid-state physics. The asymptotic and numerical results obtained in this paper partially answer Clarkson's open question on the connection problem of the first Painlev\'{e} transcendent.