On the autonomisation of differential equations I: Sundman transformations for Painlevé equations I-IV
Abstract
We consider the application of Sundman transformations to the first four Painlevé equations PI-PIV, with PV and PVI being left for future work. Since the Painlevé equations are nonautonomous, this requires first of all the construction of an equivalent autonomous system, a process which we discuss in detail, and which we characterize as being inverse to the use of Lie symmetries to obtain a reduction of order of an autonomous system. Secondly, we apply a Sundman transformation to this autonomous system, and identify the resulting equation with a corresponding nonautonomous equation. We give an explicit mapping between solutions of this nonautonomous equation and the original Painlevé equation. Interesting special cases of the nonautonomous equation found by Sundman transformation include, for PII, an equation with coefficients expressed via Airy functions, and for PIV, an equation with coefficients expressed via Weber-Hermite functions. We also discuss an alternative autonomisation process. Finally, a variety of possible directions for future work are identified, along with interesting open problems.
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BibTeX
@misc{gordoa2026autonomisation,
title = {{On the autonomisation of differential equations I: Sundman transformations for Painlevé equations I-IV}},
author = {Pilar R. Gordoa and Alexandru Iosif and Andrew Pickering},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12636},
eprint = {2609.12636},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12636},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the autonomisation of differential equations I: Sundman transformations for Painlevé equations I-IV
AU - Pilar R. Gordoa
AU - Alexandru Iosif
AU - Andrew Pickering
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12636
UR - https://arxiv.org/abs/2609.12636
AB - We consider the application of Sundman transformations to the first four Painlevé equations P_I-P_IV, with P_V and P_VI being left for future work. Since the Painlevé equations are nonautonomous, this requires first of all the construction of an equivalent autonomous system, a process which we discuss in detail, and which we characterize as being inverse to the use of Lie symmetries to obtain a reduction of order of an autonomous system. Secondly, we apply a Sundman transformation to this autonomous system, and identify the resulting equation with a corresponding nonautonomous equation. We give an explicit mapping between solutions of this nonautonomous equation and the original Painlevé equation. Interesting special cases of the nonautonomous equation found by Sundman transformation include, for P_II, an equation with coefficients expressed via Airy functions, and for P_IV, an equation with coefficients expressed via Weber-Hermite functions. We also discuss an alternative autonomisation process. Finally, a variety of possible directions for future work are identified, along with interesting open problems.
ER -