Abstract
The Painlevé equations are six families P I , P II , P III , P IV , P V , P VI of second order differential equations in \(U = \mathbb{C},\ \ \mathbb{C}^{{\ast}} = \mathbb{C} -\{ 0\},\text{ or }\mathbb{C} -\{ 0,1\}\) of the form \(f_{xx} = R(x,f,f_{x})\) where R is holomorphic for x ∈ U and rational in f and f x .
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References
Babbitt, D.G., Varadarajan, V.S.: Local moduli for meromorphic differential equations. Asterisque 169–170, 1–217 (1989)
Balser, W., Jurkat, W.B., Lutz, D.A.: Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations. J. Math. Anal. Appl. 71, 48–94 (1979)
Bǎnicǎ, C., Stǎnǎşilǎ, O.: Algebraic Methods in the Global Theory of Complex Spaces. Editura Academiei, Buchurest (1976)
Boalch, P.: Symplectic manifolds and isomonodromic deformations. Adv. Math. 163, 137–205 (2001)
Boalch, P.: Quasi-hamiltonian geometry of meromorphic connections. Duke Math. J. 139, 369–404 (2007)
Bobenko, A.I., Its, A.R.: The Painlevé III equation and the Iwasawa decomposition. Manuscripta Math. 87, 369–377 (1995)
Bolibruch, A., Its, A.R., Kapaev, A.A.: On the Riemann-Hilbert-Birkhoff inverse monodromy problem and the Painlevé equations. Algebra i Analiz 16, 121–162 (2004)
Cattani, E., Kaplan, A., Schmid, W.: Degeneration of Hodge structures. Ann. Math. (2) 123(3), 457–535 (1986)
Cecotti, S., Vafa, C.: Topological–anti-topological fusion. Nuclear Phys. B 367(2), 359–461 (1991)
Cecotti, S., Vafa, C.: On classification of N = 2 supersymmetric theories. Comm. Math. Phys. 158(3), 569–644 (1993)
Conte, R. (ed.): The Painlevé property: one century later. CRM Series in Mathematical Physics. Springer, New York (1999).
Dorfmeister, J.: Generalized Weierstraß representations of surfaces. In: Surveys on Geometry and Integrable Systems. Adv. Stud. Pure Math. 51, 55–111 (2008)
Dorfmeister, J., Guest, M.A., Rossman, W.: The tt ∗ structure of the quantum cohomology of \(\mathbb{C}P^{1}\) from the viewpoint of differential geometry. Asian J. Math. 14, 417–438 (2010)
Dorfmeister, J., Pedit, F., Wu, H.: Weierstrass type representations of harmonic maps into symmetric spaces. Commun. Anal. Geom. 6, 633–668 (1998)
Dubrovin, B.: Geometry and integrability of topological-antitopological fusion. Comm. Math. Phys. 152(3), 539–564 (1993)
Dubrovin, B.: Painlevé equations in 2D topological field theories. In: [Co99], pp. 287–412
Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori. Invent. Math. 146, 507–593 (2001)
Ferus, D., Pinkall, U., Timmreck, M.: Constant mean curvature planes with inner rotational symmetry in Euclidean 3-space. Math. Z. 215, 561–568 (1994)
Flaschka, H., Newell, A.C.: Monodromy- and spectrum-preserving deformations I. Comm. Math. Phys. 76, 65–116 (1980)
Fokas, A.S., Ablowitz, M.J.: On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23, 2033–2042 (1982)
Fokas, A.S., Mugau, U., Zhou, X.: On the solvability of Painlevé I, III, and IV. Inverse Prob. 8, 757–785 (1992)
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 (2006)
Gordoa, P.R., Joshi, N., Pickering, A.: Mappings preserving locations of movable poles: II. The third and fifth Painlevé equations. Nonlinearity 14, 567–582 (2001)
Gromak, V.I.: The solutions of Painlevé’s third equation. Differ. Equ. 9, 1599–1600 (1973)
Gromak, V.I.: Algebraic solutions of the third Painlevé equation. Dokl. Akad. Nauk BSSR 23, 499–502 (1979)
Gromak, V.I., Lukashevich, N.A.: Special classes of solutions of Painlevé equations. Differ. Equ. 18, 317–326 (1980)
Gromak, V.I., Laine, I., Shimomura, S.: Painlevé Differential Equations in the Complex Plane. Walter de Gruyter, Berlin (2002)
Guest, M.A.: From Quantum Cohomology to Integrable Systems. Oxford Univ. Press, Oxford (2008)
Guest, M.A., Lin, C.-S.: Nonlinear PDE aspects of the tt ∗ equations of Cecotti and Vafa. J. Reine Angew. Math. 689, 1–32 (2014)
Hertling, C.: tt ∗ geometry, Frobenius manifolds, their connections, and the construction for singularities. J. Reine Angew. Math. 555, 77–161 (2003)
Hertling, C., Sabbah, C.: Examples of non-commutative Hodge structures. J. Inst. Math. Jussieu 10(3), 635–674 (2011)
Hertling, C., Sevenheck, Ch.: Nilpotent orbits of a generalization of Hodge structures. J. Reine Angew. Math. 609, 23–80 (2007)
Hertling, C., Sevenheck, Ch.: Limits of families of Brieskorn lattices and compactified classifying spaces. Adv. Math. 223, 1155–1224 (2010)
Heu, V.: Stability of rank 2 vector bundles along isomonodromic deformations. Math. Ann. 344, 463–490 (2009)
Heu, V.: Universal isomonodromic deformations of meromorphic rank 2 connections on curves. Ann. Inst. Fourier 60, 515–549 (2010)
Hinkkanen, A., Laine, I.: Solutions of modified third Painlevé equation are meromorphic. J. Anal. Math. 85, 323–337 (2001)
Inaba, M., Saito, M.-H.: Moduli of unramified irregular singular parabolic connections on a smooth projective curve. Kyoto J. Math. 53, 433–482 (2013)
Its, A.R., Niles, D.: On the Riemann-Hilbert-Birkhoff inverse monodromy problem associated with the third Painlevé equation. Lett. Math. Phys. 96, 85–108 (2011)
Its, A.R., Novokshenov, V.Yu.: The isomonodromic deformation method in the theory of Painlevé equations. Lecture Notes in Mathematics, vol. 1191. Springer, Berlin (1986)
Its, A.R., Novokshenov, V.Yu.: On the effective sufficient conditions for solvability of the inverse monodromy problem for the systems of linear ordinary differential equations. Funct. Anal. i Prilozhen. 22(3), 25–36 (1988)
Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé. A modern theory of special functions. Aspects of Mathematics, vol. E 16. Vieweg, Braunschweig (1991)
Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica D 2, 407–448 (1981)
Kitaev, A.V.: The method of isomonodromy deformations and the asymptotics of solution of the “complete” third Painlevé equation. Math. USSR Sb. 62, 421–444 (1988)
Kobayashi, Sh.: Real forms of complex surfaces of constant mean curvature. Trans. AMS 363(4), 1765–1788 (2011)
Lukashevich, N.A.: On the theory of Painlevé’s third equation. Differ. Equ. 3, 994–999 (1967)
Malgrange, B.: La classification des connexions irrégulières à une variable. In: Séminaire de’l ENS, Mathématique et Physique, 1979–1982. Progress in Mathematics, vol. 37, pp. 353–379. Birkhäuser, Boston (1983)
Malgrange, B.: Sur les déformations isomonodromiques, I, II. In: Séminaire de’l ENS, Mathématique et Physique, 1979–1982. Progress in Mathematics, vol. 37, pp. 401–438. Birkhäuser, Boston (1983)
Mansfield, E.L., Webster, H.N.: On one-parameter families of solutions of Painlevé III. Stud. Appl. Math. 101, 321–341 (1998)
Matano, T., Matumiya, A., Takano, K.: On some Hamiltonian structures of Painlevé systems, II. J. Math. Soc. Jpn. 51, 843–866 (1999)
Matumiya, A.: On some Hamiltonian structure of Painlevé systems, III. Kumamoto J. Math. 10, 45–73 (1997)
McCoy, B.M., Tracy, C.A., Wu, T.T.: Painlevé functions of the third kind. J. Math. Phys. 18(5), 1058–1092 (1977)
Milne, A.E., Clarkson, P.A., Bassom, A.P.: Bäcklund transformations and solution hierarchies for the third Painlevé equation. Stud. Math. Appl. 98, 139–194 (1997)
Miwa, T.: Painlevé property of monodromy preserving deformation equations and the analyticity of τ-functions. Publ. RIMS, Kyoto Univ. 17, 703–712 (1981)
Mochizuki, T.: Asymptotic behaviour of tame harmonic bundles and an application to pure twistor \(\mathcal{D}\)-modules, Part 1. Mem. Am. Math. Soc. 185(869), xi+324 p. (2007)
Mochizuki, T.: Wild harmonic bundles and wild pure twistor D-modules. Astérisque 340, x+607 p. (2011)
Mochizuki, T.: Asymptotic behaviour of variation of pure polarized TERP structures. Publ. RIMS Kyoto Univ. 47(2), 419–534 (2011)
Murata, Y.: Classical solutions of the third Painlevé equation. Nagoya Math. J. 139, 37–65 (1995)
Niles, D.: The Riemann-Hilbert-Birkhoff inverse monodromy problem and connection formulae for the third Painlevé transcendents, vii+76 p. Dissertation, Purdue University, Indiana, 2009
Noumi, M.: Painlevé equations through symmetry. Translations of Math. Monographs, vol. 223. Amer. Math. Soc., Providence (2004)
Noumi, M., Takano, K., Yamada, Y.: Bäcklund transformations and the manifolds of Painlevé systems. Funkcial. Ekvac. 45(2), 237–258 (2002)
Noumi, M., Yamada, Y.: Symmetries in Painlevé equations. Sugaku Expositions 17, 203–218 (2004)
Ohyama, Y., Kawamuko, H., Sakai, H., Okamoto, K.: Studies on the Painlevé equations, V, third Painlevé equations of special type P III (D 7) and P III (D 8). J. Math. Sci. Univ. Tokyo 13, 145–204 (2006)
Ohyama, Y., Okumura, S.: A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations. J. Phys. A 39, 12129–12151 (2006)
Okamoto, K.: Sur les feuilletages associés aux equations du second ordre à points critiques fixes de P. Painlevé. Jpn. J. Math. 5, 1–79 (1979)
Okamoto, K.: Isomonodromic deformation and Painlevé equations and the Garnier system. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33, 575–618 (1986)
Okamoto, K.: Studies on the Painlevé equations. IV. Third Painlevé equation P III . Funkcial. Ekvac. 30, 305–332 (1987)
van der Put, M., Singer, M.F.: Galois theory of linear differential equations. Grundlehren der mathematischen Wissenschaften, vol. 328. Springer, New York (2003)
van der Put, M., Saito, M.-H.: Moduli spaces for linear differential equations and the Painlevé equations. Ann. Inst. Fourier (Grenoble) 59(7), 2611–2667 (2009)
van der Put, M., Top, J.: Geometric aspects of the Painlevé equations PIII(D 6) and PIII(D 7). SIGMA 10, 24 p. (2014)
Sabbah, C.: Déformations isomonodromiques et variétés de Frobenius. Savoirs Actuels, EDP Sciences, Les Ulis, 2002, Mathématiques. English translation: Isomonodromic deformations and Frobenius manifolds. Universitext, Springer and EDP Sciences, 2007
Sabbah, C.: Fourier-Laplace transform of a variation of polarized complex Hodge structure. J. Reine Angew. Math. 621, 123–158 (2008)
Sabbah, C.: Polarizable twistor \(\mathcal{D}\)-modules. Astérisque 300, vi+208 p. (2005)
Sabbah, C.: Introduction to Stokes structures. Lecture Notes in Mathematics, vol. 2060, xiv + 249 p. Springer, New York (2013)
Sakai, H.: Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Comm. Math. Phys. 220, 165–229 (2001)
Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)
Shioda, T., Takano, K.: On some Hamiltonian structures of Painlevé systems, I. Funkcial. Ekvac. 40, 271–291 (1997)
Sibuya, Y.: Perturbation of linear ordinary differential equations at irregular singular points. Funkcial. Ekvac. 11, 235–246 (1968)
Sibuya, Y.: Linear differential equations in the complex domain: problems of analytic continuation. Translations of Math. Monographs, vol. 82. Amer. Math. Soc., Providence (1990)
Takei, Y. (ed.): Algebraic, analytic and geometric aspects of complex differential equations. Painlevé hierarchies. RIMS Kôkyûroku Bessatsu B2. RIMS, Kyoto University (2007).
Temme, N.M.: Special Functions. An Introduction to the Classical Functions of Mathematical Physics. Wiley, New York (1996)
Terajima, H.: Families of Okamoto-Painlevé pairs and Painlevé equations. Ann. Mat. Pura Appl. (4) 186(1), 99–146 (2007)
Tsuda, T., Okamoto, K., Sakai, H.: Folding transformation of the Painlevé equations. Math. Annalen 331, 713–738 (2005)
Umemura, H.: Painlevé equations and classical functions. Sugaku Expositions 11, 77–100 (1998)
Umemura, H., Watanabe, H.: Solutions of the third Painlevé equation. I. Nagoya Math. J. 151, 1–24 (1998)
Umemura, H.: Painlevé equations in the past 100 years. Am. Math. Soc. Transl. (2) 204, 81–110 (2001)
Levi, D., Winternitz, P. (eds.): Painlevé transcendents. In: Their Asymptotics and Physical Applications. Nato ASI Series, Series B: Physics, vol. 278 (1992)
Widom, H.: On the solution of a Painlevé III equation. Math. Phys. Ana. Geom. (4) 3, 375–384 (2000)
Witte, N.S.: New transformations for Painlevé’s third transcendent. Proc. Am. Math. Soc. 132(6), 1649–1658 (2004)
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Guest, M.A., Hertling, C. (2017). Generalities on the Painlevé Equations. In: Painlevé III: A Case Study in the Geometry of Meromorphic Connections. Lecture Notes in Mathematics, vol 2198. Springer, Cham. https://doi.org/10.1007/978-3-319-66526-9_9
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