A Riemann–Hilbert approach to Painlevé IV

References

• P. Boalch, Symplectic Manifolds and Isomonodromic Deformations. Adv. in Math., 163 (2001) 137–205.
   Web of Science ®Google Scholar
• V.I. Gromak, I. Laine, S. Shimomura, Painlevé differential equations in the complex plane. de Gruyter Studies in Mathematics, 28. Walter de Gruyter & Co., Berlin, 2002.
   Google Scholar
• M. Inaba, Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence, Preprint, 2006, arXiv:math/0602004.
   Google Scholar
• M. Inaba, K. Iwasaki, and M.-H.Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I, Publ. Res. Inst. Math. Sci. 42 (2006), no. 4 987–1089.
   Web of Science ®Google Scholar
• M. Inaba, K. Iwasaki, and M.-H. Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II, Moduli spaces and arithmetic geometry (Tokyo), Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, pp. 387–432
   Google Scholar
• M. Inaba, K. Iwasaki and M.-H. Saito, Dynamics of the sixth Painlevé Equations, Thé ories Asymptotiques et Équations de Painlevé, Angers, Juin, 2004, ”Séminaires et Congrès” of the Société Mathématique de France (SMF) 14 2006, 103–167.
   Google Scholar
• M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function. Physica D, 2 (1981) 306–352.
   Google Scholar
• M. Jimbo, and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica D, 2 (1981) 407–448.
   Google Scholar
• M. Noumi, Painlevé equations through symmetry. Translated from the 2000 Japanese original by the author. Translations of Mathematical Monographs, 223. American Mathematical Society, Providence, RI, 2004.
   Google Scholar
• M. Noumi and Y. Yamada, Symmetries in the fourth Painlevé equation and Okamoto polynomials. Nagoya Math. J., 153 (1999) 53–86.
   Web of Science ®Google Scholar
• Y. Ohyama, S. Okumura, A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations. J. Phys. A 39 (2006), no. 39 12129–12151.
   Google Scholar
• K. Okamoto, 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.
   Google Scholar
• K. Okamoto, Isomonodromic deformation and Painlevé equations and the Garnier system. J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 33 (1986) 575–618.
   Google Scholar
• K. Okamoto, Studies on the Painlevé Equations III Second and Fourth Painlevé Equations, PII and PIV. Math. Ann., 275 (1986) 221–255.
   Web of Science ®Google Scholar
• K. Okamoto, The Hamiltonians associated to the Painlevé equations. The Painlevé property, 735–787, CRM Ser. Math. Phys. Springer, New York, 1999
   Google Scholar
• M. van der Put, Families of linear differential equations and the Painlevéequations, SMF Séminaires & Congrès 27 (2012), 203–220
   Google Scholar
• M. van der Put, Families of linear differential equations related to the second Painlevé equation in: Algebraic Methods in Dynamical Systems, Banach Center Publications, Volume 94 2011, 247–262
   Google Scholar
• M. van der Put and M-H. Saito, Moduli spaces for linear differential equations and the Painlevé equations.,Ann. Inst. Fourier 59 (2009), no. 7 2611–2667.
   Web of Science ®Google Scholar
• M. van der Put, M.F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der mathematischen Wissenschaften, Volume 328, Springer Verlag 2003.
   Google Scholar
• M.-H. Saito, T. Takebe, H. Terajima, Deformation of Okamoto-Painlevé pairs and Painlevé equations, J. Algebraic Geom. 11 (2002), no. 2 311–362
   Web of Science ®Google Scholar
• M-H. Saito, T. Takebe, Classification of Okamoto-Painlevé pairs, Kobe J. Math. 19 (2002) no. 1-2 21–50.
   Google Scholar
• M.-H. Saito, H. Umemura, Painlevé equations and deformations of rational surfaces with rational double points. Physics and combinatorics 1999 (Nagoya), 320–365, World Sci. Publishing, River Edge, NJ, 2001.
   Google Scholar
• M.-H. Saito, H. Terajima, Nodal curves and Riccati solutions of Painlevé equations, J. Math. Kyoto Univ.44 (2004), no. 3 529–568.
   Google Scholar
• H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Comm. Math. Phys., 220 (2001) 165–229.
   Web of Science ®Google Scholar
• H. Terajima, Families of Okamoto-Painlevé pairs and Painlevé equations, Ann. Mat. Pura Appl. (4), 186 (2007), no. 1 99–146.
   Web of Science ®Google Scholar