On the Riemann-Hilbert approach to asymptotics of tronqu\'ee solutions of Painlev\'e I

TL;DR

This paper uses the Riemann-Hilbert method to analyze the asymptotic behavior of special solutions to Painlevé I, providing detailed non-perturbative insights through explicit local parametrices.

Contribution

It introduces an explicit local parametrix involving error functions to improve understanding of non-perturbative effects in Painlevé I solutions.

Findings

Explicit local parametrix constructed around stationary points.

Enhanced understanding of non-perturbative contributions.

Detailed asymptotic expansions for tronquée solutions.

Abstract

In this paper, we revisit large variable asymptotic expansions of tronqu\'ee solutions of the Painlev\'e I equation, obtained via the Riemann-Hilbert approach and the method of steepest descent. The explicit construction of an extra local parametrix around the recessive stationary point of the phase function, in terms of complementary error functions, makes it possible to give detailed information about non-perturbative contributions beyond standard Poincar\'e expansions for tronqu\'ee and tritronqu\'ee solutions.