Rigorous analytical approximation of tritronquee solution to Painleve-1 and the first singularity

TL;DR

This paper presents a rigorous analytical method to approximate the tritronque9e solution of Painleve-1, accurately locating its nearest singularity with tight bounds, confirming previous numerical results.

Contribution

The authors develop a new rigorous analytical approach to approximate Painleve-1's tritronque9e solution and precisely determine its closest singularity.

Findings

Confirmed the singularity location at x = -2.3841687675 with high accuracy

Provided rigorous bounds for the approximation in domain D

Validated previous numerical estimates of the singularity

Abstract

We use a recently developed method to determine approximate expression for tritronqu\'{e}e solution for P-1: $y^{\prime \prime} + 6 y^2 - x=0$ in a domain $D$ with rigorous bounds. In particular we rigorously confirm the location of the closest singularity from the origin to be at $x= - \frac{770766}{323285} = -2.3841687675\cdots$ to within $5 \times 10^{-6}$ accuracy, in agreement with previous numerical calculation.