Connection problem of the first Painlev\'{e} transcendent between poles and negative infinity

TL;DR

This paper investigates the connection problem of the first Painlevé equation, classifying real solutions based on pole location and Laurent series parameters, and deriving asymptotic formulas using complex WKB methods.

Contribution

It provides a new classification of real Painlevé I solutions in terms of pole location and Laurent series parameters, with asymptotic formulas and phase diagrams.

Findings

Classification of solutions in the (p,H) parameter space

Asymptotic formulas for large H and pole index n

Phase diagram of solutions resembling Brillouin zones

Abstract

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.