Numerical Approach to Painlevé Transcendents on Unbounded Domains

Abstract

A multidomain spectral approach for Painlevé 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 to evaluate truncations of the series at any finite point. The accuracy of the method is illustrated for the example of the tritronquée solution to the Painlevé I equation.