Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T04:19:11.005Z Has data issue: false hasContentIssue false
  • English
  • Français

Solutions of the third Painlevé equation I

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura  and
Humihiko Watanabe
Hiroshi Umemura
Affiliation:
Graduate School of Polymathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan, [email protected]
Humihiko Watanabe
Affiliation:
Graduate School of Mathematics, Kyushu University (at Ropponmatsu Branch), Chuo-ku, Fukuoka 810-8560, Japan
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We classify transcendental classical solutions of the third Painlevé equation. This result combined with the list of algebraic solutions in [11] gives a complete table of classical solutions of the third Painlevé equation.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 151 , September 1998 , pp. 1 - 24
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1998

References

[1] Bourbaki, N., Groupes et algèbres de Lie, Chapitres 4, 5, et 6, Masson.Google Scholar
[2] Gambier, B., Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes, Acta Math., 33 (1909), 1–55.Google Scholar
[3] Fokas, A. S. and Ablowitz, M. J., On a unified approach to transformations and elementary solutions of Painlevé equations, J. Math. Phys., 23 (1982), 2033–2042.Google Scholar
[4] Gromak, V. I., Solutions of the third Painlevé equation, Diff. Eq., 9 (1973), 1599–1600.Google Scholar
[5] Gromak, V. I., Theory of Painlevé’s equation, Diff. Eq., 11 (1975), 285–287.Google Scholar
[6] Gromak, V. I., One-parameter systems of solutions of Painlevé equations, Diff. Eq., 14 (1978), 1510–1513.Google Scholar
[7] Gromak, V. I., Reducibility of Painlevé equations, Diff. Eq., 20 (1984), 1191–1198.Google Scholar
[8] Gromak, V. I., Transformations of Painlevé equations, Dokl. Akad. Nauk. BSSR., 32 (1988), 395–398, (Russian).Google Scholar
[9] Gromak, V. I. and Lukashevich, N. A., Special classes of solutions of Painlevé’s equations, Diff. Eq., 18 (1982), 317–326.Google Scholar
[10] Lukashevich, N. A., On the theory of the third Painlevé equation, Diff. Eq., 3 (1967), 994–999.Google Scholar
[11] Murata, Y., Classical solutions of the third Painlevé equation, Nagoya math. J., 139 (1995), 37–65.CrossRefGoogle Scholar
[12] Nishioka, K., A note on the transcendency of Painlevé’s first transcendent, Nagoya Math. J., 109 (1988), 63–67.Google Scholar
[13] Okamoto, K., Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math., 5 (1979), 1–79.Google Scholar
[14] Okamoto, K., K, Isomonodromic deformation and Painlevé equations, and the Garnier system, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 33 (1986), 575–618.Google Scholar
[15] Okamoto, K., Studies on the Painlevé equations IV, third Painlevé equation PIII , Funk. Ekv., 30 (1987), 305–332.Google Scholar
[16] Painlevé, P., Mémoire sur les équations différentielles dont l’intégrale générale est uniforme, Bull. Soc. Math. France, 28 (1900), 201–261.Google Scholar
[17] Painlevé, P., Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale est uniforme, Acta Math., 25 (1900), 1–85.CrossRefGoogle Scholar
[18] Painlevé, P., Sur les équations différentielles du second ordre à points critiques fixes, C. R. Acad. Sci. Paris, 143 (1906), 1111–1117.Google Scholar
[19] Umemura, H., Birational automorphism groups and differential equations, Nagoya Math. J., 119 (1990), 1–80.Google Scholar
[20] Umemura, H., On the irreducibility of the first differential equation of Painlevé, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi NAGATA, Kinokuniya, Tokyo (1987), pp. 771–789.Google Scholar
[21] Umemura, H., Second proof of the irreducibility of first differential equation of Painlevé, Nagoya Math. J., 117 (1990), 125–171.Google Scholar
[22] Umemura, H., Differential Galois theory of infinite dimension, Nagoya Math. J., 144 (1996), 59–135.Google Scholar
[23] Umemura, H. and Watanabe, H., Solutions of the second and fourth Painlevé equations I, Nagoya Math. J., 148 (1997), 151–198.CrossRefGoogle Scholar
[24] Umemura, H. and Watanabe, H., Solutions of the third Painlevé equation II, in preparation.Google Scholar
[25] Watanabe, H., Solutions of the fifth Painlevé equation I, Hokkaido Math. J., 24 (1995), 231–267.Google Scholar