Asymptotic expansions of solutions of the sixth Painlevé equation

Bruno, A.; Goryuchkina, I.

doi:10.1090/S0077-1554-2010-00186-0

Asymptotic expansions of solutions of the sixth Painlevé equation
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by A. D. Bruno and I. V. Goryuchkina
Translated by: O. Khleborodova

Trans. Moscow Math. Soc. 2010, 1-104

DOI: https://doi.org/10.1090/S0077-1554-2010-00186-0

Published electronically: December 28, 2010

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Abstract:

We obtain all asymptotic expansions of solutions of the sixth Painlevé equation near all three singular points $x=0, x=1$, and $x=\infty$ for all values of four complex parameters of this equation. The expansions are obtained for solutions of five types: power, power-logarithmic, complicated, semiexotic, and exotic. They form 117 families. These expansions may contain complex powers of the independent variable $x$. First we use methods of two-dimensional power algebraic geometry to obtain those asymptotic expansions of all five types near the singular point $x=0$ for which the order of the leading term is less than 1. These expansions are called basic expansions. They form 21 families. All other asymptotic equations near three singular points are obtained from basic ones using symmetries of the equation. The majority of these expansions are new. Also, we present examples and compare our results with previously known ones.

References

Mark J. Ablowitz and Harvey Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR 642018
Alexander D. Bruno, Local methods in nonlinear differential equations, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1989. Part I. The local method of nonlinear analysis of differential equations. Part II. The sets of analyticity of a normalizing transformation; Translated from the Russian by William Hovingh and Courtney S. Coleman; With an introduction by Stephen Wiggins. MR 993771, DOI 10.1007/978-3-642-61314-2
Alexander D. Bruno, Power geometry in algebraic and differential equations, North-Holland Mathematical Library, vol. 57, North-Holland Publishing Co., Amsterdam, 2000. Translated from the 1998 Russian original by V. P. Varin and revised by the author. MR 1773512
A. D. Bryuno, Power expansions of solutions of an algebraic or a differential equation, Dokl. Akad. Nauk 380 (2001), no. 2, 155–159 (Russian). MR 1873273
A. D. Bryuno, Power asymptotics of solutions of an ordinary differential equation, Dokl. Akad. Nauk 392 (2003), no. 3, 295–300 (Russian). MR 2088468
A. D. Bryuno, Power-logarithmic expansions of solutions of an ordinary differential equation, Dokl. Akad. Nauk 392 (2003), no. 4, 439–444 (Russian). MR 2081499
A. D. Bryuno, Nonpower asymptotics of solutions of an ordinary differential equation, Dokl. Akad. Nauk 392 (2003), no. 5, 586–591 (Russian). MR 2082848
A. D. Bryuno, Asymptotic behavior and expansions of solutions of an ordinary differential equation, Uspekhi Mat. Nauk 59 (2004), no. 3(357), 31–80 (Russian, with Russian summary); English transl., Russian Math. Surveys 59 (2004), no. 3, 429–480. MR 2116535, DOI 10.1070/RM2004v059n03ABEH000736
A. D. Bryuno, Compound expansions of solutions of an ordinary differential equation, Dokl. Akad. Nauk 406 (2006), no. 6, 730–733 (Russian). MR 2347360
A. D. Bryuno, Exotic expansions of solutions of an ordinary differential equation, Dokl. Akad. Nauk 416 (2007), no. 5, 583–587 (Russian); English transl., Dokl. Math. 76 (2007), no. 2, 729–733. MR 2458919, DOI 10.1134/S1064562407050237
—, On exotic expansions of solutions of an ordinary differential equation. Preprint, Keldysh Inst. Applied Math., 2010, no. 27, 27 pp. (Russian)
A. D. Bryuno and I. V. Goryuchkina, Expansions of solutions of the sixth Painlevé equation, Dokl. Akad. Nauk 395 (2004), no. 6, 733–737 (Russian). MR 2114829
—, Expansions of solutions of the sixth Painlevé equation near a nonsingular point. Preprint, Keldysh Inst. Applied Math., 2005, no. 4, 19 pp. (Russian)
A. D. Bryuno and I. V. Goryuchkina, Expansions of solutions of the sixth Painlevé equation in the cases $a=0$ and $b=0$, Dokl. Akad. Nauk 410 (2006), no. 3, 295–300 (Russian). MR 2450920
A. D. Bryuno and I. V. Goryuchkina, All the asymptotic expansions of solutions of the sixth Painlevé equation, Dokl. Akad. Nauk 417 (2007), no. 3, 298–302 (Russian); English transl., Dokl. Math. 76 (2007), no. 3, 851–855. MR 2455353, DOI 10.1134/S1064562407060129
—, A survey of all asymptotic expansions of solutions of the equation P6. Preprint, Keldysh Inst. Applied Math., 2007, no. 60, 16 pp. (Russian)
—, Methods of analysis of asymptotic expansions of solutions of the equation P6. Preprint, Keldysh Inst. Applied Math., 2007, no. 61, 30 pp. (Russian)
—, All basic asymptotic expansions of solutions of the equation P6 in the case $a\cdot b\neq 0$. Preprint, Keldysh Inst. Applied Math., 2007, no. 62, 30 pp. (Russian)
—, All asymptotic expansions of solutions of the equation P6 in the case $a\cdot b= 0$. Preprint, Keldysh Inst. Applied Math., 2007, no. 70, 30 pp. (Russian)
—, All asymptotic expansions of solutions of the equation P6 obtained from basic ones. Preprint, Keldysh Inst. Applied Math., 2007, no. 77, 28 pp. (Russian)
A. D. Bryuno and I. V. Goryuchkina, Boutroux asymptotics of solutions of Painlevé equations and power geometry, Dokl. Akad. Nauk 422 (2008), no. 2, 157–160 (Russian); English transl., Dokl. Math. 78 (2008), no. 2, 681–685. MR 2477348, DOI 10.1134/S1064562408050104
A. D. Bryuno and I. V. Goryuchkina, Asymptotic behavior of the solutions of the third Painlevé equation, Dokl. Akad. Nauk 422 (2008), no. 6, 729–732 (Russian); English transl., Dokl. Math. 78 (2008), no. 2, 765–768. MR 2477241, DOI 10.1134/S1064562408050335
A. D. Bryuno and I. V. Goryuchkina, Asymptotic behavior of the solutions of the fourth Painlevé equation, Dokl. Akad. Nauk 423 (2008), no. 4, 443–448 (Russian); English transl., Dokl. Math. 78 (2008), no. 3, 868–873. MR 2498533, DOI 10.1134/S1064562408060173
A. D. Bryuno and I. V. Goryuchkina, All expansions of solutions of the sixth Painlevé equation near its nonsingular point, Dokl. Akad. Nauk 426 (2009), no. 5, 586–591 (Russian); English transl., Dokl. Math. 79 (2009), no. 3, 397–402. MR 2573669, DOI 10.1134/S1064562409030260
A. D. Bryuno and I. V. Goryuchkina, Basic expansions of solutions of the sixth Painlevé equation in the generic case, Differ. Uravn. 45 (2009), no. 1, 19–33 (Russian, with Russian summary); English transl., Differ. Equ. 45 (2009), no. 1, 18–32. MR 2597090, DOI 10.1134/S0012266109010030
Bruno [Bryuno], A. D., Chukhareva, I. V., Power expansions of solutions of the sixth Painlevé equation. Preprint, Keldysh Inst. Applied Math., 2003, no. 49. (Russian)
A. D. Bryuno and T. V. Shadrina, An axisymmetric boundary layer on a needle, Tr. Mosk. Mat. Obs. 68 (2007), 224–287 (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (2007), 201–259. MR 2429271, DOI 10.1090/s0077-1554-07-00165-3
Goryuchkina, I. V., Expansions of solutions of the sixth Painlevé equation in power series in real power of $x$. Differentsialnye Uravneniya 41 (2004), no. 6, 854. (Russian)
—, On power and logarithmic expansions of solutions of the sixth Painlevé equation near singular points. Abstracts of talks, XXVI Conf. of Young Scientists, Math. Department, Moscow State Univ., Moscow, 2004, pp. 39–40. (Russian)
—, On power and logarithmic expansions of solutions of the sixth Painlevé equation near singular points. Proc. XXVI Conf. of Young Scientists, Math. Department Moscow State Univ. Iz-vo MGU, Moscow, 2004, pp. 63–68. (Russian)
—, Power and logarithmic expansions of solutions of the sixth Painlevé equation, in: Modern Methods of the Theory of Boundary Problems, VGU, Voronezh, 2004, pp. 63–64. (Russian)
—, Expansions of solutions of the sixth Painlevé equation near singularities, in: Intern. Conf. on Differential Equations and Dynamical Systems, Suzdal, July 10–15, 2006, Sobor, Vladimir, 2006, pp. 75–77. (Russian)
—, Exotic expansions of solutions of the sixth Painlevé equation. Preprint, Keldysh Inst. Applied Math., 2007, no. 3, 29 pp. (Russian)
Goursat, Édouard, Cours d’analyse mathématique, Tomes I–III. Éditions Jacques Gabay, Sceaux, 1992.
Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov, Painlevé transcendents, Mathematical Surveys and Monographs, vol. 128, American Mathematical Society, Providence, RI, 2006. The Riemann-Hilbert approach. MR 2264522, DOI 10.1090/surv/128
Sophie Kowalevski, Sur le probleme de la rotation d’un corps solide autour d’un point fixe, Acta Math. 12 (1889), no. 1, 177–232 (French). MR 1554772, DOI 10.1007/BF02391879
Kudryashov N. A., Analytic theory of nonlinear differential equations. Inst. Computer Research, Izhevsk, 2004.
A. N. Kuznetsov, On the existence of solutions, entering at a singular point, of an autonomous system that has a formal solution, Funktsional. Anal. i Prilozhen. 23 (1989), no. 4, 63–74, 96 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 4, 308–317 (1990). MR 1035375, DOI 10.1007/BF01078945
Rozov, M. Kh., Painlevé equation, Encyclopedia Math., Sovetskaya Entsiklopediya, Moscow, 1984, vol. 4, pp. 233–234. (Russian)
Tikhomirov, V. M., Fréchet derivative. Encyclopedia Math., Sovetskaya Entsiklopediya, Moscow, 1984, vol. 5, p. 666. (Russian)
Chukhareva, I. V., Singularities of solutions of the VI Painlevé equation. Intern. Youth Conf “Gagarinskie Chteniya XXX”, Abstracts of talks, Moscow, 2004, pp. 72–73. (Russian)
M. J. Ablowitz, A. Ramani, and H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of $P$-type. I, J. Math. Phys. 21 (1980), no. 4, 715–721. MR 565716, DOI 10.1063/1.524491
M. J. Ablowitz, A. Ramani, and H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of $P$-type. II, J. Math. Phys. 21 (1980), no. 5, 1006–1015. MR 574872, DOI 10.1063/1.524548
Alexander D. Bruno, Power geometry as a new calculus, Analysis and applications—ISAAC 2001 (Berlin), Int. Soc. Anal. Appl. Comput., vol. 10, Kluwer Acad. Publ., Dordrecht, 2003, pp. 51–71. MR 2022739, DOI 10.1007/978-1-4757-3741-7_{4}
Y. F. Chang, J. M. Greene, M. Tabor, and J. Weiss, The analytic structure of dynamical systems and self-similar natural boundaries, Phys. D 8 (1983), no. 1-2, 183–207. MR 724588, DOI 10.1016/0167-2789(83)90317-2
B. Dubrovin and M. Mazzocco, Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), no. 1, 55–147. MR 1767271, DOI 10.1007/PL00005790
Fuchs, L., $\ddot \textrm {U}$ber Differentialgleichungen deren integrale feste Verzweigungspunkte besitzen. Sitz. Acad. Wiss. Berlin, 1884, 669–720.
Fuchs, R., Sur quelques eq́uations différentielles linéaires du second ordres. C. R. Acad. Sci. Paris 141 (1905), 555–558.
B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes, Acta Math. 33 (1910), no. 1, 1–55 (French). MR 1555055, DOI 10.1007/BF02393211
R. Garnier, Sur des équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses points critiques fixes, Ann. Sci. École Norm. Sup. (3) 29 (1912), 1–126 (French). MR 1509146
—, Etude de l’intégrale générale de l’équations VI de M. Painlevé. Ann. Ec. Norm. (3) 34 (1917), 243–353.
Goryuchkina, I. V., About power-logarithmic expansions of solutions to the sixth Painlevé equation. Intern. Conf. on Differential Equations and Dynamical Systems, Suzdal, July 5–10, 2004, Sobor, Vladimir, 2004, pp. 259–260.
—, Asymptotic expansions of solutions to the sixth Painlevé equation. ACA 2006. 12th. International Conference on Applications of Computer Algebra. Abstracts of Presentations. Sofia, 2006. p. 50.
Valerii I. Gromak, Ilpo Laine, and Shun Shimomura, Painlevé differential equations in the complex plane, De Gruyter Studies in Mathematics, vol. 28, Walter de Gruyter & Co., Berlin, 2002. MR 1960811, DOI 10.1515/9783110198096
Davide Guzzetti, On the critical behavior, the connection problem and the elliptic representation of a Painlevé VI equation, Math. Phys. Anal. Geom. 4 (2001), no. 4, 293–377 (2002). MR 1888007, DOI 10.1023/A:1014265919008
Davide Guzzetti, The elliptic representation of the general Painlevé VI equation, Comm. Pure Appl. Math. 55 (2002), no. 10, 1280–1363. MR 1912098, DOI 10.1002/cpa.10045
Davide Guzzetti, The elliptic representation of the sixth Painlevé equation, Théories asymptotiques et équations de Painlevé, Sémin. Congr., vol. 14, Soc. Math. France, Paris, 2006, pp. 83–101 (English, with English and French summaries). MR 2353463
Davide Guzzetti, Matching procedure for the sixth Painlevé equation, J. Phys. A 39 (2006), no. 39, 11973–12031. MR 2266210, DOI 10.1088/0305-4470/39/39/S02
Davide Guzzetti, The logarithmic asymptotics of the sixth Painlevé equation, J. Phys. A 41 (2008), no. 20, 205201, 46. MR 2450511, DOI 10.1088/1751-8113/41/20/205201
Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, and Masaaki Yoshida, From Gauss to Painlevé, Aspects of Mathematics, E16, Friedr. Vieweg & Sohn, Braunschweig, 1991. A modern theory of special functions. MR 1118604, DOI 10.1007/978-3-322-90163-7
Michio Jimbo, Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982), no. 3, 1137–1161. MR 688949, DOI 10.2977/prims/1195183300
Hironobu Kimura, The construction of a general solution of a Hamiltonian system with regular type singularity and its application to Painlevé equations, Ann. Mat. Pura Appl. (4) 134 (1983), 363–392. MR 736747, DOI 10.1007/BF01773512
Marta Mazzocco, Picard and Chazy solutions to the Painlevé VI equation, Math. Ann. 321 (2001), no. 1, 157–195. MR 1857373, DOI 10.1007/PL00004500
Kazuo Okamoto, Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{\textrm {VI}}$, Ann. Mat. Pura Appl. (4) 146 (1987), 337–381. MR 916698, DOI 10.1007/BF01762370
Painlevé, P., Leçons sur la théorie analytique des équations différentielles, professées à Stokholm. Paris, 1897.
P. Painlevé, Mémoire sur les équations différentielles dont l’intégrale générale est uniforme, Bull. Soc. Math. France 28 (1900), 201–261 (French). MR 1504376
P. Painlevé, Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math. 25 (1902), no. 1, 1–85 (French). MR 1554937, DOI 10.1007/BF02419020
Emile Picard, Démonstration d’un théorème général sur les fonctions uniformes liées par une relation algébrique, Acta Math. 11 (1887), no. 1-4, 1–12 (French). Extrait d’une lettre adressée à M. Mittag-Leffler. MR 1554745, DOI 10.1007/BF02418039
—, Mémoire sur la théorie des fonctions algébriques de deux variables. Journal de Liouville 5 (1889), 135–319.
Poincaré, H., Sur les intégrales irrégulières des équations linéaires. C. R. Acad. Sci. Paris. 101 (1885), 939–941. Oeuvres. Vol. IV, 611–613.
—, Sur les intégrales irrégulières des équations linéaires. C. R. Acad. Sci. Paris 101 (1885), 990–991. Oeuvres. Vol. IV, 614–615.
—, Sur les intégrales irrégulières des équations linéaires, Acta Math. 8 (1886), 295–344. Oeuvres. Vol. I, 290–332.
Shun Shimomura, Painlevé transcendents in the neighbourhood of fixed singular points, Funkcial. Ekvac. 25 (1982), no. 2, 163–184. MR 694910
Shun Shimomura, Series expansions of Painlevé transcendents in the neighbourhood of a fixed singular point, Funkcial. Ekvac. 25 (1982), no. 2, 185–197. MR 694911
Shun Shimomura, Supplement to: “Series expansions of Painlevé transcendents in the neighbourhood of a fixed singular point”, Funkcial. Ekvac. 25 (1982), no. 3, 363–371. MR 707567
Shun Shimomura, A family of solutions of a nonlinear ordinary differential equation and its application to Painlevé equations $(\textrm {III})$, $(\textrm {V})$ and $(\textrm {VI})$, J. Math. Soc. Japan 39 (1987), no. 4, 649–662. MR 905630, DOI 10.2969/jmsj/03940649
Kyoichi Takano, Reduction for Painlevé equations at the fixed singular points of the first kind, Funkcial. Ekvac. 29 (1986), no. 1, 99–119. MR 865217
Humihiko Watanabe, Birational canonical transformations and classical solutions of the sixth Painlevé equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), no. 3-4, 379–425 (1999). MR 1678014