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

TL;DR

This paper proves the Dubrovin conjecture by analyzing the pole-free regions of the tritronquée solution to Painlevé I, providing new insights and precise values at zero with rigorous bounds.

Contribution

It establishes the pole-free region for the tritronquée solution of Painlevé I and confirms the Dubrovin conjecture using a general, constructive method.

Findings

Proves the Dubrovin conjecture for Painlevé I

Identifies pole-free regions of the tritronquée solution

Provides precise values of the solution and its derivative at zero

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.