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arXiv · 1209.1009

Proof of the Dubrovin conjecture and analysis of the tritronqu\'ee solutions of $P_I$

O. Costin·M. Huang·S. Tanveer
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Abstract

We show that the tritronqu\'ee solution of the Painlev\'e equation $\P1$, $ 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 Painlev\'e 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 tritronqu\'ee and its derivative at zero within less than 1/100 rigorous error bounds.

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BibTeXRIS

O. Costin, M. Huang, S. Tanveer. 2012-09-05. Proof of the Dubrovin conjecture and analysis of the tritronqu\'ee solutions of $P_I$. https://doi.org/10.1215/00127094-2429589

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