Exponentially-improved asymptotics and numerics for the (un)perturbed first Painlev\'e equation

TL;DR

This paper develops advanced asymptotic and numerical methods for the first Painlevé equation, accounting for exponential small terms and the Stokes phenomenon, providing highly accurate solutions and singularity locations.

Contribution

It introduces exponentially-improved asymptotics for the perturbed first Painlevé equation, including detailed analysis of the Stokes phenomenon and numerical validation.

Findings

Asymptotic formulas for singularity locations are derived.

Numerical examples demonstrate high accuracy of the asymptotic approximations.

Precise computation of zeros and poles for the unperturbed solution.

Abstract

The solutions of the perturbed first Painlev\'e equation $y"=6y^2-x^\mu$, $\mu>-4$, are uniquely determined by the free constant $C$ multiplying the exponentially small terms in the complete large $x$ asymptotic expansions. Full details are given, including the nonlinear Stokes phenomenon, and the computation of the relevant Stokes multipliers. We derive asymptotic approximations, depending on $C$, for the locations of the singularities that appear on the boundary of the sectors of validity of these exponentially-improved asymptotic expansions. Several numerical examples illustrate the power of the approximations. For the tri-tronqu\'ee solution of the unperturbed first Painlev\'e equation we give highly accurate numerics for the values at the origin and the locations of the zeros and poles.