Abstract
We solve the equivalence problem for the Painlevé IV equation, formulating the necessary and sufficient conditions in terms of the invariants of point transformations for an arbitrary second-order differential equation to be equivalent to the Painlevé IV equation. We separately consider three pairwise nonequivalent cases: both equation parameters are zero, a = b = 0; only one parameter is zero, b = 0; and the parameter b ≠ 0. In all cases, we give an explicit point substitution transforming an equation satisfying the described test into the Painlevé IV equation and also give expressions for the equation parameters in terms of invariants.
Similar content being viewed by others
References
V. I. Gromak and N. A. Lukashevich, Analytic Properties of Solutions of Painleve Equations [in Russian], Izdatel’stvo Universitetskoe, Minsk (1990).
N. A. Kudryashov, Analytic Theory of Nonlinear Differential Equations [in Russian], IKI, Moscow (2004).
A. R. Its and V. Yu. Novokshenov, The Isomonodromic Deformation Method in the Theory of Painleve Equations (Lect. Notes Math., Vol. 1191), Springer, Berlin (1986).
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM Stud. Appl. Math., Vol. 4), SIAM, Philadelphia (1981).
N. Kamran, K. G. Lamb, and W. F. Shadwick, J. Differ. Geom., 22, 139–150 (1985).
J. Hietarinta and V. Dryuma, J. Nonlinear Math. Phys. (Suppl. 1), 9, 67–74 (2002).
M. V. Babich and L. A. Bordag, J. Differ. Equations, 157, 452–485 (1999).
A. V. Bocharov, V. V. Sokolov, and S. I. Svinolupov, “On some equivalence problem for differential equations,” Preprint ESI 54, Intl. Erwin Schrödinger Inst. Math. Phys., Vienna (1993).
R. Dridi, J. Phys. A, 42, 125201 (2009).
V. V. Kartak, Ufim. Mat. Zh., 1, No. 3, 46–56 (2009); ar**v:0909.1987v1 [math.CA] (2009).
V. V. Kartak, J. Nonlinear Math. Phys., 18, 613–640 (2011); ar**v:1106.6124v1 [nlin.SI] (2011).
A. Tresse, Acta Math., 18, 1–88 (1894).
A. Tresse, Détermination des invariants ponctuels de l’équation différentielle ordinaire de second ordre y″ = w(x, y, y′), S. Hirzel, Leipzig (1896).
R. Liouville, J. de L’ École Polytechnique, 59, 7–76 (1889).
E. Cartan, Bull. Soc. Math. France, 52, 205–241 (1924).
S. Lie, Theorie der Transformationsgruppen III, Teubner, Leipzig (1930).
G. Thomsen, Abhandlungen Hamburg, 7, 301–328 (1930).
C. Grissom, G. Thompson, and G. Wilkens, J. Differ. Equations, 77, 1–15 (1989).
L. A. Bordag and V. S. Dryuma, “Investigation of dynamical systems using tools of the theory of invariants and projective geometry,” NTZ-Preprint 24/95, Zentrum für Höhere Studien, Univ. Leipzig, Leipzig (1995); ar**v:solv-int/9705006v1 (1997).
V. V. Dmitrieva and R. A. Sharipov, “On the point transformations for the second order differential equations I,” ar**v:solv-int/9703003v1 (1997).
R. A. Sharipov, “On the point transformations for the equation y″ = P +3Qy′+3Ry′2+Sy′3,” ar**v:solv-int/9706003v1 (1997).
R. A. Sharipov, “Effective procedure of point classification for the equations y″ = P +3Qy′ +3Ry′2 + S y′3,” ar**v:math/9802027v1 (1998).
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry: Methods and Applications [in Russian], Fizmatlit, Moscow (1986); English transl. (Grad. Texts Math., Vols. 93, 104, and 124), Vols. 1–3, Springer, New York (1992, 1985, and 1990).
C. Bandle and L. A. Bordag, Nonlinear Anal. A, 50, 523–540 (2002).
D. Cox, J. Little, and D. O’shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, New York (1997).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 173, No. 2, pp. 245–267, November, 2012.
Rights and permissions
About this article
Cite this article
Kartak, V.V. Solution of the equivalence problem for the Painlevé IV equation. Theor Math Phys 173, 1541–1564 (2012). https://doi.org/10.1007/s11232-012-0132-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-012-0132-4