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

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

2010-10-27

arxiv.org/abs/1010.5563v2

Mathematics

Classical Analysis and ODEs

68 pages, 7 figures; added affiliation and further information about first author, information about how the collaborative res

Scientific paper

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.

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