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

This article is cited in 12 scientific papers (total in 12 papers)

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

Full text: PDF file (537 kB)
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English version:
Theoretical and Mathematical Physics, 2015, 182:2, 211–230

Bibliographic databases:

Received: 17.02.2014
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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    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  mathnet  crossref  mathscinet  isi  elib
    2. Yu. Yu. Bagderina, N. N. Tarkhanov, “Solution of the equivalence problem for the third Painlevé equation”, J. Math. Phys., 56:1 (2015), 013507  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    6. P. V. Bibikov, “Generalized Lie problem and differential invariants for the third order ODEs”, Lobachevskii J. Math., 38:4, SI (2017), 622–629  crossref  mathscinet  zmath  isi  scopus
    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  crossref  isi  scopus
    8. I. Kossovskiy, D. Zaitsev, “Normal form for second order differential equations”, J. Dyn. Control Syst., 24:4 (2018), 541–562  crossref  mathscinet  isi  scopus
    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  mathnet  crossref
    10. Peter A. Clarkson, “Open Problems for Painlevé Equations”, SIGMA, 15 (2019), 006, 20 pp.  mathnet  crossref
    11. Bibikov P.V., “Classification of Second Order Linear Ordinary Differential Equations With Rational Coefficients”, Lobachevskii J. Math., 40:1, SI (2019), 14–23  crossref  mathscinet  isi  scopus
    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  mathnet  crossref  crossref  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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