Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 029, 14 pages      arXiv:1612.03674      https://doi.org/10.3842/SIGMA.2017.029

Isomonodromy for the Degenerate Fifth Painlevé Equation

Primitivo B. Acosta-Humánez a, Marius van der Put b and Jaap Top b
a) Universidad Simón Bolívar, Barranquilla, Colombia
b) University of Groningen, Groningen, The Netherlands

Received December 12, 2016, in final form May 01, 2017; Published online May 09, 2017

Abstract
This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlevé equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann-Hilbert morphism is an isomorphism. As a consequence these equations have the Painlevé property and the Okamoto-Painlevé space is identified with a moduli space of connections. Using MAPLE computations, one obtains formulas for the degenerate fifth Painlevé equation, for the Bäcklund transformations.

Key words: moduli space for linear connections; irregular singularities; Stokes matrices; monodromy spaces; isomonodromic deformations; Painlevé equations.

pdf (385 kb)   tex (23 kb)

References

  1. Chekhov L., Mazzocco M., Rubtsov V., Painlevé monodromy manifolds, decorated character varieties and cluster algebras, Int. Math. Res. Not., to appear, arXiv:1511.03851.
  2. Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents. The Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006.
  3. Gromak V.I., On the theory of Painlevé's equations, Differential Equations 11 (1975), 285-287.
  4. Inaba M., Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence, J. Algebraic Geom. 22 (2013), 407-480, math.AG/0602004.
  5. Inaba M., Iwasaki K., Saito M.-H., Dynamics of the sixth Painlevé equation, in Théories asymptotiques et équations de Painlevé, Sémin. Congr., Vol. 14, Soc. Math. France, Paris, 2006, 103-167.
  6. Inaba M., Iwasaki K., Saito M.-H., Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I, Publ. Res. Inst. Math. Sci. 42 (2006), 987-1089, math.AG/0309342.
  7. Inaba M., Iwasaki K., Saito M.-H., Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II, in Moduli Spaces and Arithmetic Geometry, Adv. Stud. Pure Math., Vol. 45, Math. Soc. Japan, Tokyo, 2006, 387-432, math.AG/0605025.
  8. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau $-function, Phys. D 2 (1981), 306-352.
  9. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  10. Ohyama Y., Kawamuko H., Sakai H., Okamoto K., Studies on the Painlevé equations. V. Third Painlevé equations of special type ${\rm P}_{\rm III}(D_7)$ and ${\rm P}_{\rm III}(D_8)$, J. Math. Sci. Univ. Tokyo 13 (2006), 145-204.
  11. Ohyama Y., Okumura S., R. Fuchs' problem of the Painlevé equations from the first to the fifth, in Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Contemp. Math., Vol. 593, Amer. Math. Soc., Providence, RI, 2013, 163-178, math.CA/0512243.
  12. Okamoto K., Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé. Espaces des conditions initiales, Japan. J. Math. 5 (1979), 1-79.
  13. Okamoto K., Isomonodromic deformation and Painlevé equations, and the Garnier system, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986), 575-618.
  14. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, ${\rm P}_{{\rm II}}$ and ${\rm P}_{{\rm IV}}$, Math. Ann. 275 (1986), 221-255.
  15. Okamoto K., Studies on the Painlevé equations. IV. Third Painlevé equation ${\rm P}_{{\rm III}}$, Funkcial. Ekvac. 30 (1987), 305-332.
  16. Okamoto K., The Hamiltonians associated to the Painlevé equations, in The Painlevé property, CRM Ser. Math. Phys., Springer, New York, 1999, 735-787.
  17. van der Put M., Families of linear differential equations related to the second Painlevé equation, in Algebraic Methods in Dynamical Systems, Banach Center Publ., Vol. 94, Polish Acad. Sci. Inst. Math., Warsaw, 2011, 247-262.
  18. van der Put M., Families of linear differential equations and the Painlevé equations, in Geometric and Differential Galois Theories, Sémin. Congr., Vol. 27, Soc. Math. France, Paris, 2013, 207-224.
  19. van der Put M., Saito M.-H., Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble) 59 (2009), 2611-2667, arXiv:0902.1702.
  20. van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003.
  21. van der Put M., Top J., A Riemann-Hilbert approach to Painlevé IV, J. Nonlinear Math. Phys. 20 (2013), suppl. 1, 165-177, arXiv:1207.4335.
  22. van der Put M., Top J., Geometric aspects of the Painlevé equations ${\rm PIII}(\rm D_6)$ and ${\rm PIII}(\rm D_7)$, SIGMA 10 (2014), 050, 24 pages, arXiv:1207.4023.
  23. Saito M.-H., Takebe T., Classification of Okamoto-Painlevé pairs, Kobe J. Math. 19 (2002), 21-50, math.AG/0006028.
  24. Saito M.-H., Takebe T., Terajima H., Deformation of Okamoto-Painlevé pairs and Painlevé equations, J. Algebraic Geom. 11 (2002), 311-362, math.AG/0006026.
  25. Saito M.-H., Terajima H., Nodal curves and Riccati solutions of Painlevé equations, J. Math. Kyoto Univ. 44 (2004), 529-568, math.AG/0201225.
  26. Witte N.S., New transformations for Painlevé's third transcendent, Proc. Amer. Math. Soc. 132 (2004), 1649-1658, math.CA/0210019.

Previous article  Next article   Contents of Volume 13 (2017)