Share This Article:

Equivalence Problem of the Painlevé Equations

Abstract Full-Text HTML Download Download as PDF (Size:200KB) PP. 297-303
DOI: 10.4236/apm.2013.32042    2,249 Downloads   4,529 Views  
Author(s)    Leave a comment

ABSTRACT

The manuscript is devoted to the equivalence problem of the Painlevé equations. Conditions which are necessary and sufficient for second-order ordinary differential equations y=F (x ,y, y) to be equivalent to the first and second Painlevé equation under a general point transformation are obtained. A procedure to check these conditions is found.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

S. Khamrod, "Equivalence Problem of the Painlevé Equations," Advances in Pure Mathematics, Vol. 3 No. 2, 2013, pp. 297-303. doi: 10.4236/apm.2013.32042.

References

[1] S. Lie, “Klassifikation und Integration von Gewonlichen Differentialgleichungen Zwischen x,y, Die Eine Gruppe von Transformationen Gestaten. III,” Archiv for Matematik og Naturvidenskab, Vol. 8, No. 4, 1883, pp. 371-427.
[2] R. Liouville, “Sur les Invariants de Certaines Equations Differentielles et sur Leurs Applications,” Journal de l’école Polytechnique, Vol. 59, 1889, pp. 7-76.
[3] A. M. Tresse, “Détermination des Invariants Ponctuels de l'équation Différentielle Ordinaire du Second Ordre y''=ω(x,y,y'),” Preisschriften der Furstlichen Jablonowski’schen Gesellschaft XXXII, Leipzig, 1896.
[4] E. Cartan, “Sur les Variétés à Connexion Projective,” Bulletin de la Société Mathématique de France, Vol. 52, 1924, pp. 205-241.
[5] M. V. Babich and L. A. Bordag, “Projective Differential Geometrical Structure of the Painleve Equations,” Journal of Differential Equations, Vol. 157, No. 2, 1999, pp. 452-485.
[6] V. V. Kartak, “Explicit Solution of the Problem of Equivalence for Some Painleve Equations,” CUfa Math Journal, Vol. 1, No. 3, 2009, pp. 1-11.
[7] N. H. Ibragimov, “Invariants of a Remarkable Family of Nonlinear Equations,” Nonlinear Dynamics, Vol. 30, No. 2, 2002, pp. 155-166. doi:10.1023/A:1020406015011
[8] N. H. Ibragimov and S. V. Meleshko, “Linearization of Third-Order Ordinary Differential Equations by Point Transformations,” Archives of ALGA, Vol. 1, 2004, pp. 71-93.
[9] N. H. Ibragimov and S. V. Meleshko, “Linearization of Third-Order Ordinary Differential Equations by Point Transformations,” Journal of Mathematical Analysis and Applications, Vol. 308, No. 1, 2005, pp. 266-289. doi:10.1016/j.jmaa.2005.01.025

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.