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Equivalence Problem of the Painlevé Equations

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DOI: 10.4236/apm.2013.32042    2,249 Downloads   4,529 Views  
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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.

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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.


[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

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