Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of $P_I$

Abstract

We show that the tritronquée solution of the Painlevé equation $\P 1$, $y"=6y^2+z$ which is analytic for large $z$ with $\arg z\in (-\frac{3\pi }{5},\pi )$ is pole-free in a region containing the full sector $z\ne 0,\arg z\in [-\frac{3\pi }{5},\pi ]$ and the disk $z:|z|<37/20$. This proves in particular the Dubrovin conjecture, an open problem in the theory of Painlevé transcendents. The method, building on a technique developed in Costin, Huang, Schlag (2012), is general and constructive. As a byproduct, we obtain the value of the tritronquée and its derivative at zero within less than 1/100 rigorous error bounds.

References in corpus (2)

• Poles of Integrale Tritronquee and Anharmonic Oscillators. Asymptotic localization from WKB analysis
• On Borel summation and Stokes phenomena of nonlinear differential systems

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