Equivalence Problem of the Painlevé Equations
Author(s)
Sopita Khamrod*
Affiliation(s)
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, Thailand.
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, Thailand.
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.
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.
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
[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