Archive for Rational Mechanics and Analysis

, Volume 202, Issue 3, pp 707–785 | Cite as

Okamoto’s Space for the First Painlevé Equation in Boutroux Coordinates

Article

Abstract

We study the completeness and connectedness of asymptotic behaviours of solutions of the first Painlevé equation d2 y/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, that is, the space of initial values compactified and regularized by embedding in \({{\mathbb C}{\mathbb P} 2}\) through an explicit construction of nine blowups.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Mathematics and Statistics F07The University of SydneySydneyAustralia

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