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.
This publication is cited in the following 16 articles:
Dmitry I. Sinelshchikov, “Linearizabiliy and Lax representations for cubic autonomous and non-autonomous nonlinear oscillators”, Physica D: Nonlinear Phenomena, 448 (2023), 133721
Dmitry I. Sinelshchikov, Partha Guha, A. Ghose Choudhury, “Lax representation and a quadratic rational first integral for second-order differential equations with cubic nonlinearity”, Communications in Nonlinear Science and Numerical Simulation, 112 (2022), 106553
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
Peter A. Clarkson, “Open Problems for Painlevé Equations”, SIGMA, 15 (2019), 006, 20 pp.
P. V. Bibikov, “Classification of second order linear ordinary differential equations with rational coefficients”, Lobachevskii J. Math., 40:1, SI (2019), 14–23
D. I. Sinelshchikov, “On first integrals for some non-autonomous lienard-type equations”, International Conference on Numerical Analysis and Applied Mathematics (Icnaam-2018), AIP Conf. Proc., 2116, eds. T. Simos, C. Tsitouras, Amer. Inst. Phys., 2019, 270009
Yu. Yu. Bagderina, “Necessary conditions of point equivalence of second-order odes to the fifth Painleve equation”, Vii International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 1205, IOP Publishing Ltd, 2019, 012004
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
I. Kossovskiy, D. Zaitsev, “Normal form for second order differential equations”, J. Dyn. Control Syst., 24:4 (2018), 541–562
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
P. V. Bibikov, “On Lie's problem and differential invariants of ODEs y″=F(x,y)”, Funct. Anal. Appl., 51:4 (2017), 255–262
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
P. V. Bibikov, “Generalized Lie problem and differential invariants for the third order ODEs”, Lobachevskii J. Math., 38:4, SI (2017), 622–629
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
Yu. Yu. Bagderina, “Equivalence of second-order ODEs to equations of first Painlevé equation type”, Ufa Math. J., 7:1 (2015), 19–30
Yu. Yu. Bagderina, N. N. Tarkhanov, “Solution of the equivalence problem for the third Painlevé equation”, J. Math. Phys., 56:1 (2015), 013507
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