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 TMF, 2015, Volume 182, Number 2, Pages 256–276 (Mi tmf8657)

Equivalence of second-order ordinary differential equations to Painlevé equations

Yu. Yu. Bagderina

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa, Russia

Abstract: All Painlevé equations except the first belong to one type of equations. In terms of invariants of these equations, we obtain criteria for the equivalence to the second Painlevé equation and to equation XXXIV in the list of $50$ equations without movable critical points. We find new necessary conditions of equivalence for the third and fourth and also special cases of the fifth and sixth Painlevé equations. We compare the invariants we use with invariants previously introduced by other authors and compare the obtained results.

Keywords: Painlevé equation, equivalence, invariant

 Funding Agency Grant Number Russian Science Foundation 14-11-00078

DOI: https://doi.org/10.4213/tmf8657

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English version:
Theoretical and Mathematical Physics, 2015, 182:2, 211–230

Bibliographic databases:

Revised: 11.08.2014

Citation: Yu. Yu. Bagderina, “Equivalence of second-order ordinary differential equations to Painlevé equations”, TMF, 182:2 (2015), 256–276; Theoret. and Math. Phys., 182:2 (2015), 211–230

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf8657
• https://doi.org/10.4213/tmf8657
• http://mi.mathnet.ru/eng/tmf/v182/i2/p256

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. Yu. Bagderina, “Equivalence of second-order ODEs to equations of first Painlevé equation type”, Ufa Math. J., 7:1 (2015), 19–30
2. Yu. Yu. Bagderina, N. N. Tarkhanov, “Solution of the equivalence problem for the third Painlevé equation”, J. Math. Phys., 56:1 (2015), 013507
3. Yu. Yu. Bagderina, “Invariants of a family of scalar second-order ordinary differential equations for Lie symmetries and first integrals”, J. Phys. A-Math. Theor., 49:15 (2016), 155202
4. P. V. Bibikov, “On Lie's problem and differential invariants of ODEs $y-F(x,y)$”, Funct. Anal. Appl., 51:4 (2017), 255–262
5. P. Bibikov, A. Malakhov, “On Lie problem and differential invariants for the subgroup of the plane Cremona group”, J. Geom. Phys., 121 (2017), 72–82
6. P. V. Bibikov, “Generalized Lie problem and differential invariants for the third order ODEs”, Lobachevskii J. Math., 38:4, SI (2017), 622–629
7. P. Bibikov, A. Malakhov, “On classification problems in the theory of differential equations: algebra plus geometry”, Publ. Inst. Math.-Beograd, 103:117 (2018), 33–52
8. I. Kossovskiy, D. Zaitsev, “Normal form for second order differential equations”, J. Dyn. Control Syst., 24:4 (2018), 541–562
9. Yu. Yu. Bagderina, “Necessary conditions of point equivalence of second-order ODEs to the sixth Painlevé equation”, J. Math. Sci. (N. Y.), 242:5 (2019), 595–607
10. Peter A. Clarkson, “Open Problems for Painlevé Equations”, SIGMA, 15 (2019), 006, 20 pp.
11. Bibikov P.V., “Classification of Second Order Linear Ordinary Differential Equations With Rational Coefficients”, Lobachevskii J. Math., 40:1, SI (2019), 14–23
12. Yu. Yu. Bagderina, “Point equivalence of second-order ordinary differential equations to the fifth Painlevé equation with one and two nonzero parameters”, Theoret. and Math. Phys., 202:3 (2020), 295–308
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