|Presenter||Nalini Joshi (The University of Sydney)|
|Session||Analysis and Partial Differential Equations|
I will report on results obtained with Duistermaat, Howes and Radnovic for the completeness and connectedness of asymptotic behaviours of solutions of the first, second and fourth Painlev\'e equations in the limit $x\to\infty$, $x\in\mathbb C$. 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 $\mathbb P^2$ through an explicit construction of nine blow-ups.
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