Journal of Nonlinear Mathematical Physics

Volume 25, Issue 4, July 2018, Pages 604 - 617

Lie point symmetries and ODEs passing the Painlevé test

Authors
D. Levi
Dipartimento di Matematica e Fisica, Roma Tre University and INFN, Sezione Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy, levi@roma3.infn.it
D. Sekera
Institute of Physics, Laboratory for Particle Physics and Cosmology, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland, david.sekera@epfl.ch
P. Winternitz
Département de mathématiques et de statistique and Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal (QC) H3C 3J7, Canada, wintern@crm.umontreal.ca
Received 29 April 2018, Accepted 10 May 2018, Available Online 6 January 2021.
DOI
https://doi.org/10.1080/14029251.2018.1503435How to use a DOI?
Abstract

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order n = 2, . . . , 5. Among the 6 ODEs identifying the Painlevé transcendents only PIII, PV and PVI have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlevé transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlevé test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.

Copyright
© 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
25 - 4
Pages
604 - 617
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2018.1503435How to use a DOI?
Copyright
© 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - D. Levi
AU  - D. Sekera
AU  - P. Winternitz
PY  - 2021
DA  - 2021/01
TI  - Lie point symmetries and ODEs passing the Painlevé test
JO  - Journal of Nonlinear Mathematical Physics
SP  - 604
EP  - 617
VL  - 25
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1503435
DO  - https://doi.org/10.1080/14029251.2018.1503435
ID  - Levi2021
ER  -