Global Asymptotics of the Second Painlev\'e Equation in Okamoto's Space

TL;DR

This paper investigates the asymptotic behavior of solutions to the second Painlevé equation within Okamoto's space, revealing the nature of solutions at infinity, their pole structure, and the dynamics of exceptional lines.

Contribution

It provides a detailed analysis of the solutions' asymptotics, pole distribution, and the dynamical role of exceptional lines in Okamoto's space for the second Painlevé equation.

Findings

Solutions that do not vanish at infinity have infinitely many poles.

The union of exceptional lines acts as a repellor in the dynamical system.

The limit set of solutions is shown to be compact and connected.

Abstract

We study the solutions of the second Painlev\'e equation in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painlev\'e equation. By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repellor for the dynamics in Okamoto's space. Moreover, we show that the limit set of the solutions exists and is compact and connected.