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Necessary Conditions for Point Equivalence of Second-Order Odes to the Sixth Painlevé Equation

Journal of Mathematical Sciences volume 242, pages 595–607 (2019)Cite this article

We consider the equivalence problem for projective-type scalar second-order ordinary differential equations with respect to invertible point changes of variables. Invariants of the equivalence transformation group of this family of equations are used to find necessary conditions for equivalence to the sixth Painlevé equation.

References

  1. R. Liouville, “Sur les invariants de certaines équations différentielles et sur leurs applications,” J. École Polytechnique, 59, 7–76 (1889).

    MATH  Google Scholar 

  2. A. Tresse, “Sur les invariants différentiels des groupes continus de transformations,” Acta Math., 18, 1–88 (1894).

    MathSciNet  Article  Google Scholar 

  3. E. Cartan, “Sur les variétés à connexion projective,” Bull. Soc. Math. France, 52, 205–241 (1924).

    MathSciNet  Article  Google Scholar 

  4. S. Lie, “Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten. III,” Arch. Mat. Naturvidensk., 8, 371–458 (1883).

    MATH  Google Scholar 

  5. E. L. Ince, Ordinary Differential Equations, Longmans, Green & Co, London (1926).

  6. N. Kamran, K. G. Lamb, and W. F. Shadwick, “The local equivalence problem for y′′ = F(x, y, y′) and the Painlevé transcendents,” J. Differential Geom., 22, 139–150 (1985).

    MathSciNet  Article  Google Scholar 

  7. A. V. Bocharov, V. V. Sokolov, and S. I. Svinolupov, “On some equivalence problems for differential equations,” Preprint ESI-54, International Erwin Schrödinger Institute for Mathematical Physics, Vienna (1993).

  8. J. Hietarinta and V. Dryuma, “Is my ODE a Painlevé equation in disguise?” J. Nonlinear Math. Phys., 9, Suppl. 1, 67–74 (2002).

    MathSciNet  Article  Google Scholar 

  9. R. Dridi, “On the geometry of the first and second Painlevé equations,” J. Phys. A, 42, 125201 (2009).

    MathSciNet  Article  Google Scholar 

  10. V. V. Kartak, “Point classification of second order ODEs and its application to Painlevé equations,” J. Nonlinear Math. Phys., 20, Suppl. 1, 110–129 (2013).

    MathSciNet  Article  Google Scholar 

  11. Yu. Yu. Bagderina, “Invariants of a family of scalar second-order ordinary differential equations,” J. Phys. A, 46, 295201 (2013).

    MathSciNet  Article  Google Scholar 

  12. V. V. Kartak, “‘Painlevé 34’ equation: equivalence test,” Commun. Nonlinear. Sci. Numer. Simul., 19, No. 9, 2993–3000 (2014).

    MathSciNet  Article  Google Scholar 

  13. Yu. Yu. Bagderina, “Equivalence of second-order ODEs to equations of first Painlevé equation type,” Ufa Math. J., 7, 19–30 (2015).

    MathSciNet  Article  Google Scholar 

  14. Yu. Yu. Bagderina and N. N. Tarkhanov, “Solution of the equivalence problem for the third Painlevé equation,” J. Math. Phys., 56, 013507 (2015).

    MathSciNet  Article  Google Scholar 

  15. Yu. Yu. Bagderina, “Equivalence of second-order ordinary differential equations to Painlevé equations,” Theoret. Math. Phys., 182, No. 2, 211–230 (2015).

    MathSciNet  Article  Google Scholar 

  16. R. Fuchs, “Sur quelques équations différentielles linéaires du second ordre,” C. R. Acad. Sci. Paris, 141, 555–558 (1905).

    MATH  Google Scholar 

  17. B. Gambier, “Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes,” Acta Math., 33, 1–55 (1910).

    MathSciNet  Article  Google Scholar 

  18. M. Jimbo and T. Miwa, “Studies of holonomic quantum fields. XVII,” Proc. Japan Acad. Ser. A, 56, 405–410 (1980).

    MathSciNet  Article  Google Scholar 

  19. P. J. Forrester and N. S. Witte, “Application of the τ -function theory of Painlevé equations to random matrices: PVI, the JUE, CyUE, cJUE and scaled limits,” Nagoya Math. J., 174, 29–114 (2014).

    Article  Google Scholar 

  20. H. Sakai, “Rational surfaces associated with affine root systems and geometry of the Painlevé equations,” Comm. Math. Phys., 220, 165–220 (2001).

    MathSciNet  Article  Google Scholar 

  21. A. S. Fokas, R. A. Leo, L. Martina, and G. Soliani, “The scaling reduction of the three-wave resonant system and the Painlevé VI equation,” Phys. Lett. A, 115, 329–332 (1986).

    MathSciNet  Article  Google Scholar 

  22. A. V. Kitaev, “On similarity reductions of the three-wave resonant system to the Painlevé equations,” J. Phys. A, 23, 3543–3553 (1990).

    MathSciNet  Article  Google Scholar 

  23. L. J. Mason and N. M. Woodhouse, “Self-duality and the Painlevé transcendents,” Nonlinearity, 6, 569–581 (1993).

    MathSciNet  Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ufa Federal Research Center, Ufa, Russia

    Yu. Yu. Bagderina

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  1. Yu. Yu. Bagderina
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Correspondence to Yu. Yu. Bagderina.

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Dedicated to M. A. Semenov–Tian-Shansky on the occasion of his 70th birthday

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 473, 2018, pp. 17–33.

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Bagderina, Y.Y. Necessary Conditions for Point Equivalence of Second-Order Odes to the Sixth Painlevé Equation. J Math Sci 242, 595–607 (2019). https://doi.org/10.1007/s10958-019-04499-0

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  • DOI: https://doi.org/10.1007/s10958-019-04499-0