Necessary conditions of point equivalence of second-order ODEs to the sixth Painlevé equation

Yu. Yu. Bagderina

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 25, Tome 473 (2018), pp. 17-33

Résumé

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

Bibliographie

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

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

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

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

[5] E. L. Ains, Obyknovennye differentsialnye uravneniya, ONTI, Kharkov, 1939

[6] N. Kamran, K. G. Lamb, W. F. Shadwick, “The local equivalence problem for $y''=F(x,y,y')$ and the Painlevé transcendents”, J. Diff. Geom., 22 (1985), 139–150 | DOI | MR | Zbl

[7] A. V. Bocharov, V. V. Sokolov, S. I. Svinolupov, On some equivalence problems for differential equations, Preprint ESI-54, International Erwin Schrödinger Institute for Mathematical Physics, Vienna, 1993, 12 pp. | Zbl

[8] J. Hietarinta, V. Dryuma, Is my ODE a Painlevé equation in disguise?, J. Nonlin. Math. Phys., 9, Suppl. 1 (2002), 67–74 | DOI | MR | Zbl

[9] R. Dridi, “On the geometry of the first and second Painlevé equations”, J. Phys. A: Math. Theor., 42 (2009), 125201 | DOI | MR | Zbl

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

[11] Yu. Yu. Bagderina, “Invariants of a family of scalar second-order ordinary differential equations”, J. Phys. A: Math. Theor., 46 (2013), 295201 | DOI | MR | Zbl

[12] V. V. Kartak, “"Painlevé 34" equation: equivalence test”, Commun. Nonlin. Sci. Numer. Simul., 19:9 (2014), 2993–3000 | DOI | MR

[13] Yu. Yu. Bagderina, “Ekvivalentnost ODU vtorogo poryadka uravneniyam tipa pervogo uravneniya Penleve”, Ufimskii matematicheskii zhurnal, 7:1 (2015), 19–30 | Zbl

[14] Yu. Yu. Bagderina, N. N. Tarkhanov, “Solution of the equivalence problem for the third Painlevé equation”, J. Math. Phys., 56 (2015), 013507 | DOI | MR | Zbl

[15] Yu. Yu. Bagderina, “Ekvivalentnost obyknovennykh differentsialnykh uravnenii vtorogo poryadka uravneniyam Penleve”, Teor. matem. fizika, 182:2 (2015), 256–276 | DOI | Zbl

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

[17] B. Gambier, “Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes”, Acta Mathematica, 33 (1910), 1–55 | DOI | MR

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

[19] P. J. Forrester, N. S. Witte, “Application of the $\tau$-function theory of Painlevé equations to random matrices: PVI, the JUE, CyUE, cJUE and scaled limits”, Nagoya Math. J., 174 (20014), 29–114 | DOI | MR

[20] H. Sakai, “Rational surfaces associated with affine root systems and geometry of the Painlevé equations”, Commun. Math. Phys., 220 (2001), 165–220 | DOI | MR | Zbl

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

[22] A. V. Kitaev, “On similarity reductions of the three-wave resonant system to the Painlevé equations”, J. Phys. A: Math. Gen., 23 (1990), 3543–3553 | DOI | MR | Zbl

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

[24] K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painlevé, a Modern Theory of Special Functions, Vieweg, Braunschweig, 1991 | MR

[25] M. V. Babich, L. A. Bordag, “Projective differential geometrical structure of the Painlevé equations”, J. Differ. Equations, 157 (1999), 452–485 | DOI | MR | Zbl

[26] Yu. I. Manin, Sixt Painlevé Equation, Universal Elliptic Curve, and Mirror of $\mathbb{P}^2$, Preprint MPI 96-114, Max–Planck–Institut für Mathematik, Bonn, 1996, 20 pp. | MR

[27] 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 (2016), 155202 | DOI | MR | Zbl