# Dubrovin Conjecture and Singularities of Painleve-I Tritronquee solution

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AR2W02 - Mathematics of beyond all-orders phenomena

Abstract:  Painleve Equations arise in many context. In particular Grava, Klein & Dubrovin proposed universality in blow up of focussing NLS equation that required absence of singularity of Tritronquee solutions of P-I in a sector of width 8 \pi/5. This conjecture, usually known as Dubrovin Conjecture, remained an important open problem for a while until it was proved by the authors in 2014. We present the analysis of this proof that require among other results precise asymptotics includuing a two scale expansion in a regime where erstwhile exponentially small terms become $O(1)$ in a manner described for generic ODEs by Costin \& Costin (2001). There has been more recent work that extends the singularity results in different directions.    *Joint work with Ovidiu Costin & Min Huang

This talk is part of the Isaac Newton Institute Seminar Series series.

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