Okamoto's space for the first Painlevé equation in Boutroux's coordinates

J. J. Duistermaat and N. Joshi

Abstract

We study the completeness and connectedness of asymptotic behaviours of solutions of the first Painlevé equation \(d^2y/dx^2 = 6 y^2 + x\), in the limit \(x\to\infty\), \(x\in{\mathbb C}\). This problem arises in various physical contexts including the critical behaviour near gradient catastrophe for the focusing nonlinear Schrödinger 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 \({\mathbb{CP}}2\) through an explicit construction of nine blow-ups.

Keywords: Asymptotics, initial-value space, the first Painlevé equation.

AMS Subject Classification: Primary 34M55;; secondary 58F30; 34C40.