### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 068, 10 pages      arXiv:1807.04442      https://doi.org/10.3842/SIGMA.2018.068
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

### Numerical Approach to Painlevé Transcendents on Unbounded Domains

Christian Klein and Nikola Stoilov
Institut de Mathématiques de Bourgogne, UMR 5584, Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21078 Dijon Cedex, France

Received April 18, 2018, in final form July 02, 2018; Published online July 12, 2018

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 of evaluating 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.

Key words: Painlevé equations; spectral methods.

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