Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T19:34:47.028Z Has data issue: false hasContentIssue false
  • English
  • Français

Solutions of the second and fourth Painlevé equations, I

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura  and
Humihiko Watanabe
Hiroshi Umemura
Affiliation:
Graduate School of Polymathematics, Nagoya University, Nagoya 464-01, Japan, umemura@math.nagoya-u.ac.jp
Humihiko Watanabe
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka 810, 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.

A rigorous proof of the irreducibility of the second and fourth Painlevé equations is given by applying Umemura’s theory on algebraic differential equations ([26], [27], [28]) to the two equations. The proof consists of two parts: to determine a necessary condition for the parameters of the existence of principal ideals invariant under the Hamiltonian vector field; to determine the principal invariant ideals for a parameter where the principal invariant ideals exist. Our method is released from complicated calculation, and applicable to the proof of the irreducibility of the third, fifth and sixth equation (e.g. [32]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 148 , December 1997 , pp. 151 - 198
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1997

References

[1] Bourbaki, N., Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968.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.CrossRefGoogle Scholar
[3] Gromak, V. I., One-parameter systems of solutions of Painlevé’s equations, Diff. Eq., 14 (1978), 1510–1513.Google Scholar
[4] Gromak, V. I., Solutions of the second Painlevé equation, Diff. Eq., 18 (1982), 537–545.Google Scholar
[5] Gromak, V. I., Theory of the fourth Painlevé equation, Diff. Eq., 23 (1987), 506–513.Google Scholar
[6] Gromak, V. I., Transformations of Painlevé equations, Dokl. Akad. Nauk. BSSR., 32 (1988), 395–398, (in Russian).Google Scholar
[7] Gromak, V. I., Transformations of Painlevé equations (P 4) –(P 6), Dokl. Akad. Nauk. BSSR., 33 (1989), 17–20, (in Russian).Google Scholar
[8] Gromak, V. I., One-parameter families of solutions of the second Painlevé equation, Dokl. Akad. Nauk. BSSR., 33 (1989), 209–211, (in 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] Gromak, V. I. and Tsegel’nik, V. V., Solutions of Painlevé’s fourth equation and their applications, Diff. Eq., 27 (1991), 910–918.Google Scholar
[11] Ince, E. L., Ordinary differential equations, Dover.Google Scholar
[12] Kolchin, E. R., Differential Algebra and Algebraic groups, Academic Press, 1973.Google Scholar
[13] Lukashevich, N. A., Theory of the fourth Painlevé equation, Diff. Eq., 3 (1967), 395–399.Google Scholar
[14] Lukashevich, N. A., The second Painlevé equation, Diff. Eq., 7 (1971), 853–854.Google Scholar
[15] Murata, Y., Classical solutions of the third Painlevé equation, Nagoya Math. J., 139 (1995), 37–65.CrossRefGoogle Scholar
[16] Murata, Y., Rational solutions of the second and fourth Painlevé equations, Funk. Ekvac, 28 (1985), 1–32.Google Scholar
[17] Nishioka, K., A note on the transcendency of Painlevé’s first transcendent, Nagoya Math. J., 109 (1988), 63–67.CrossRefGoogle Scholar
[18] Noumi, M., Private communications (1986 and 1987).Google Scholar
[19] Noumi, M. and Okamoto, K., Irreducibility of the second and the fourth Painlevé equations, Funk. Ekvac, 40 (1997), 139–163.Google Scholar
[20] Okamoto, K., Studies on the Painlevé equations III, second and fourth Painlevé equations, PII and PIV , Math. Ann., 275 (1986), 221–255.CrossRefGoogle Scholar
[21] Okamoto, K., Private communication (1987).Google Scholar
[22] Painlevé, P., Mémoire sur les équations différentielles dont l’intégrale générale est uniforme, Bull. Soc. Math. France, 28 (1900), 201–261.CrossRefGoogle Scholar
[23] Painlevé, P., Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math., 25 (1900), 1–85.CrossRefGoogle Scholar
[24] Painlevé, P., C. R. Acad. Sci. Paris, 143 (1906), 1111–1117.Google Scholar
[25] Painlevé, P., Leçons de Stockholm, Œuvres de P. Painlevé I, Editions du C. N. R. S., Paris, 1972.Google Scholar
[26] Umemura, H., Birational automorphism groups and differential equations, Nagoya Math. J., 119 (1990), 1–80.CrossRefGoogle Scholar
[27] Umemura, H., On the irreducibility of the first differential equation of Painlevé, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi NAGATA (1987), Kinokuniya, Tokyo, 771–789.Google Scholar
[28] Umemura, H., Second proof of the irreducibility of first differential equation of Painlevé, Nagoya Math. J., 117 (1990), 125–171.CrossRefGoogle Scholar
[29] Umemura, H., Differential Galois theory of infinite dimension, Nagoya Math. J., 144 (1996), 59–135.CrossRefGoogle Scholar
[30] Umemura, H. and Watanabe, H., Solutions of the second and fourth Painlevé equations II, in preparation.Google Scholar
[31] Vorob’ev, A. P., On rational solutions of the second Painlevé equation, Diff. Eq., 1 (1965), 58–59.Google Scholar
[32] Watanabe, H., Solutions of the fifth Painlevé equation I, Hokkaido Math. J., 24 (1995), 231–267..CrossRefGoogle Scholar
[33] Yablonskii, A. I., On rational solutions of the second Painlevé equation, Vesti A. N. BSSR. Ser. Fiz-Tekh. Nauk., 3 (1959), 30–35, (in Russian).Google Scholar