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A Review of the Sixth Painlevé Equation

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Abstract

For the Painlevé VI transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of poles close to a critical point.

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Notes

  1. This differs from the terminology of singularity theory, where a critical point is a zero of the first derivative of a function.

  2. Note that, in general, for a series like (37), one expects convergence for \(0<|x|<r\) and \(|x^{n+2im\nu }e^{im\phi }|<\epsilon _{nm}\), where \(r,~\epsilon _{nm}>0\) are sufficiently small. Thus

    $$\begin{aligned} \ln |x|-\mathfrak {I}\phi +\max \Biggl \{-\ln r,\sup _{m>0,n\ge 1}\left| {\ln \epsilon _{nm}\over m} \right| \Biggr \} < 2\nu \arg x < -\mathfrak {I}\phi +\min \Biggl \{\ln r,\inf _{m<0,n\ge 1}\left| {\ln \epsilon _{nm}\over m} \right| \Biggr \}. \end{aligned}$$

  3. \( \mathcal{A}^2={\bigl [(1-\sigma )^2-(\theta _\infty -1-\theta _1)^2\bigr ]\bigl [(1-\sigma )^2-(\theta _\infty -1+\theta _1)^2\bigr ] \over 4(1-\sigma )^2}, \)    \( \mathcal{B}={(\theta _\infty -1)^2-\theta _1^2+(1-\sigma )^2\over 2(1-\sigma )^2}\).

References

  1. Anosov, D.V., Bolibruch, A.A.: The Riemann–Hilbert Problem. Aspects of Mathematics, vol. E22. Friedr. Vieweg and Sohn, Braunschweig (1994)

  2. Andreev, A.V., Kitaev, A.V.: Connection formulas for asymptotics of the fifth Painlevé transcendent on the real axis. Nonlinearity 13, 1801–1840 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boalch, P.: From Klein to Painleve’, via Fourier, Laplace and Jimbo. Proc. Lond. Math. Soc. 90(1), 167–208 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brezhnev, Y.V.: \(\tau \)-function solution of the Sixth Painlevé Transcendent. Theor. Math. Phys. 161(3), 1616–1633 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. Nauka, Moskow, 1979; English transl., North-Holland, Amsterdam, (2000)

  6. Bruno, A.D.: Asymptotic behaviour and expansion of a solution of an ordinary differential equation. Russian Math. Surv. 59, 429–480 (2004)

    Article  Google Scholar 

  7. Bruno, A.D., Goryuchkina, I.V.: Asymptotic expansions of solutions of the sixth Painlevé equation. Moscow Math Soc, pp 1–104, (2010)

  8. Dubrovin, B.: Geometry of 2D Topological Field Theories. Lecture Notes in Math vol. 1620, pp. 120–348 (1996)

  9. Dubrovin, B.: The Painlevé property, one century later. In: Conte, R. (ed.) Painlevé Transcendents in Two-Dimensional Topological Field Theory. Springer, Berlin (1999)

  10. Dubrovin, B., Mazzocco, M.: Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141, 55–147 (2000)

  11. Dubrovin, B., Mazzocco, M.: Canonical structure and symmetries of the Schlesinger equations. Commun. Math. Phys. 271(2), 289–373 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fokas, A., Its, A., Kapaev, A., Novokshenov, V.: Painlevé Transcendents: The Riemann–Hilbert Approach. AMS, Washington, DC (2006)

    Book  Google Scholar 

  13. Fricke, R., Klein, F.: Vorlesungen uber die Theorie der automorphen Funktionen. I, Druck und Verlag von B.G.Teubner, Leipzig, (1897), p. 366

  14. Fuchs R.: Uber lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singularen Stellen. Mathematische Annalen LXIII, (1907), 301–321

  15. Gambier, B.: Sur des equations differentielles du second ordre et du Premier Degré dont l’Intégrale est à points critiques fixes. Acta Math. 33, 1–55 (1910)

    Article  MathSciNet  Google Scholar 

  16. Gamayun, O., Iorgov, N., Lisovyy, O.: Conformal field theory of Painlevé VI. J. High Energy Phys. 2012(10), 38 (2012). arXiv:1207.0787

  17. Guzzetti, D.: On the critical behaviour, the connection problem and the elliptic representation of a Painlevé 6 equation. Math. Phys. Anal. Geom. 4, 293–377 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Guzzetti, D.: The elliptic representation of the general Painlevé VI equation. Commun. Pure Appl. Math. LV, 1280–1363 (2002)

  19. Guzzetti, D.: Matching procedure for the sixth Painlevé equation. J. Phys. A: Math. Gen. 39, 1197–1231 (2006)

    Article  MathSciNet  Google Scholar 

  20. Guzzetti, D.: The logarithmic asymptotics of the sixth Painlevé equation. J. Phys. A: Math. Theor. 41 (2008), 205201(46p)

  21. Guzzetti, D.: Solving the sixth Painlevé equation: towards the classification of all the critical behaviours and the connection formulas. Int. Math. Res. Notices (2011), rnr071, 62 p. doi:10.1093/imrn/rnr071

  22. Guzzetti, D.: Poles distribution of PVI transcendents close to a critical point. Physica D (2012). doi:10.1016/j.physd.2012.02.015

    MathSciNet  Google Scholar 

  23. Guzzetti, D.: Tabulation of Painlevé 6 transcendents. Nonlinearity 25, 3235–3276 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Iorgov, N., Lisovyy, O., Shchechkin, A., Tykhyy, Y.: Painlevé functions and conformal blocks. Constr. Approx. 39, 255–272 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hitchin, N.: Twistor spaces, Einstein metrics and isometric deformations. J. Differ. Geom. 42, 30–112 (1995)

    MATH  MathSciNet  Google Scholar 

  26. Its, A.R., Novokshenov, V.Y.: The Isomonodromic Deformation Method in the Theory of Painleve Equations. Lecture Notes in Math, vol. 1191, (1986)

  27. Iwasaki, K.: An area-preserving action of the modular group on cubic surfaces of the Painlevé VI equation. Commun. Math. Phys. 242, 185–219 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painleve’. Aspects of Mathematics, vol. 16, (1991)

  29. Jimbo, M.: Monodromy Problem and the Boundary Condition for Some Painlevé Trascendents. Publ. RIMS, Kyoto University, vol. 18, pp. 1137–1161 (1982)

  30. Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients (I). Physica D 2, 306 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  31. Jimbo, M., Miwa, T.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients (II). Physica D 2, 407–448 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  32. Jimbo, M., Miwa, T.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients (III). Physica D 4, 26 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kaneko, K.: Painlevé transcendents which are meromorphic at a fixed singularity. Proc. Jpn. Acad. A 82, 71–76 (2006)

    Article  MATH  Google Scholar 

  34. Lisovyy, O., Tykhyy, Y.: Algebraic solutions of the sixth Painleve’ equation. http://arxiv.org/abs/0809.4873 (2008)

  35. Manin, Y.I.: Sixth Painlevé Equation, Universal Elliptic Curve, and Mirror of \({ P}^2\). Geometry of differential equations. Am. Math. Soc. Transl. Ser 2, 186, 131–151 (1998)

  36. Mazzocco, M.: Rational solutions of the Painlevé VI equation. J. Phys. A: Math. Gen. 34, 2281–2294 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mazzocco, M.: Irregular isomonodromic deformations for Garnier systems and Okamoto’s canonical transformations. J. Lond. Math. Soc. 70(2), 405–419 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  38. Okamoto, K.: Studies on the Painlevé equations I, the six Painlevé equation. Ann. Math. Pura Appl. 146, 337–381 (1987)

    Article  MATH  Google Scholar 

  39. Painlevé, P.: Sur les equations differentielles du second ordre et d’Ordre Supérieur, dont l’Intégrale Générale est uniforme. Acta Math. 25, 1–86 (1900)

    Article  Google Scholar 

  40. Painlevé, P.: Sur les équations différentielles du second ordre à points critiques fixes. CRAS 143, 1111–1117 (1906)

    Google Scholar 

  41. Picard, E.: Mémoire sur la Théorie des functions algébriques de deux variables. Journal de Liouville 5, 135–319 (1889)

    Google Scholar 

  42. Sato, M., Miwa, T., Jimbo, M.: Holonomic Quantum Fields. II—The Riemann–Hilbert Problem. Publ. RIMS. Kyoto. University, vol. 15, pp. 201–278 (1979)

  43. Schlesinger, L.: Uber eine Klasse von Differentsial System Beliebliger Ordnung mit Festen Kritischer Punkten. J. Math. 141, 96–145 (1912)

    MATH  Google Scholar 

  44. Shimomura, S.: A family of solutions of a nonlinear ordinary differential equation and its application to Painlevé equations (III), (V), (VI). J. Math. Soc. Jpn. 39, 649–662 (1987)

  45. Umemura, H.: Painlevé Birational automorphism groups and differential equations. Nagoya Math. J. 119, 1–80 (1990)

    MATH  MathSciNet  Google Scholar 

  46. Umemura, H.: On the Irreducibility of the First Differential Equation of Painlevé. Algebraic Geometry and Commutative Algebra in honour of Masayoshi NAGATA, Kinokuniya, Tokyo, pp. 771–789 (1987)

  47. Umemura, H.: Second proof of the irreducibility of the first differential equation of Painlevé. Nagoya Math. J. 117, 125–171 (1990)

    MATH  MathSciNet  Google Scholar 

  48. Watanabe, H.: Birational canonical transformations and classical solutions of the sixth Painlevé equation. Ann. Scuola Norm. Sup. Pisa Cl Sci. 27, 379–425 (1999)

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  1. SISSA, Intenational School of Advanced Studies, 34136 , Trieste, Italy

    Davide Guzzetti

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Communicated by Percy Deift and Alexander Its.

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Guzzetti, D. A Review of the Sixth Painlevé Equation. Constr Approx 41, 495–527 (2015). https://doi.org/10.1007/s00365-014-9250-6

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