Numerical Approach to Painlev\'e Transcendents on Unbounded Domains

TL;DR

This paper introduces a multidomain spectral method for accurately computing Painlevé transcendents on unbounded domains, especially effective for solutions defined by asymptotic series near infinity, demonstrated on the Painlevé I tritronquée solution.

Contribution

The paper presents a novel spectral approach that approximates Painlevé transcendents on unbounded domains without truncating asymptotic series, improving computational accuracy.

Findings

Method accurately computes Painlevé I tritronquée solution.

Spectral approach works on straight lines within the asymptotic sector.

No need for series truncation at finite points.

Abstract

A multidomain spectral approach for Painlev\'e transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and approximates the solution on straight lines lying entirely within said sector without the need of evaluating truncations of the series at any finite point. The accuracy of the method is illustrated for the example of the tritronqu\'ee solution to the Painlev\'e I equation.