Okamoto's space for the first Painlev\'e equation in Boutroux coordinates

TL;DR

This paper analyzes the asymptotic behavior of solutions to the first Painlevé equation using Okamoto's space, revealing properties of the solution set and employing complex geometric techniques.

Contribution

It provides a detailed geometric analysis of the solution space for the first Painlevé equation using explicit blow-up constructions of Okamoto's space.

Findings

The complex limit set of solutions is non-empty, compact, and invariant.

Infinity set acts as a repellor for the dynamics.

New proofs for solutions near equilibrium points.

Abstract

We study the completeness and connectedness of asymptotic behaviours of solutions of the first Painlev\'e equation $\op{d}^2y/\op{d}x^2=6\, y^2+ x$, in the limit $x\to\infty$, $x\in\C$. This problem arises in various physical contexts including the critical behaviour near gradient catastrophe for the focusing nonlinear Schr\"odinger equation. We prove that the complex limit set of solutions is non-empty, compact and invariant under the flow of the limiting autonomous Hamiltonian system, that the infinity set of the vector field is a repellor for the dynamics and obtain new proofs for solutions near the equilibrium points of the autonomous flow. The results rely on a realization of Okamoto's space, i.e., the space of initial values compactified and regularized by embedding in $\C\Proj 2$ through an explicit construction of nine blow-ups.