Studies on the Painlevé equations. III: Second and fourth Painlevé equations, \(P_{II}\) and \(P_{IV}\). (English) Zbl 0589.58008
[Part II is to appear in Jap. J. Math., Part I in Ann. Mat. Pura Appl., IV. Ser.]
In this paper, which is the third part of the series of papers ”Studies on the Painlevé equations”, we study the second and the fourth Painlevé equations by means of the method of birational canonical transformations. We associate with each equation the nonautonomous Hamiltonian system (H), called the Painlevé system. The group of birational canonical transformations of (H) is investigated by the use of the notion of the affine Weyl group; we attach the root system (R) to each (H). We consider also the families of particular solutions of the Painlevé systems, written in terms of the Airy functions or the Hermite functions. In particular, the rational solutions of (H) are studied in detail. The \(\tau\)-functions related to (H) is the other main object of this article; it is shown that the sequence \(\{\tau_ n\); \(n\in {\mathbb{Z}}\}\) of \(\tau\)-functions of (H) satisfies the Toda equation: \(\delta^ 2\log \tau_ n=\tau_{n-1}\tau_{n+1}/\tau^ 2_ n\).
In this paper, which is the third part of the series of papers ”Studies on the Painlevé equations”, we study the second and the fourth Painlevé equations by means of the method of birational canonical transformations. We associate with each equation the nonautonomous Hamiltonian system (H), called the Painlevé system. The group of birational canonical transformations of (H) is investigated by the use of the notion of the affine Weyl group; we attach the root system (R) to each (H). We consider also the families of particular solutions of the Painlevé systems, written in terms of the Airy functions or the Hermite functions. In particular, the rational solutions of (H) are studied in detail. The \(\tau\)-functions related to (H) is the other main object of this article; it is shown that the sequence \(\{\tau_ n\); \(n\in {\mathbb{Z}}\}\) of \(\tau\)-functions of (H) satisfies the Toda equation: \(\delta^ 2\log \tau_ n=\tau_{n-1}\tau_{n+1}/\tau^ 2_ n\).
MSC:
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 14E05 | Rational and birational maps |
Digital Library of Mathematical Functions:
§32.10(ii) Second Painlevé Equation ‣ §32.10 Special Function Solutions ‣ Properties ‣ Chapter 32 Painlevé Transcendents§32.10(iv) Fourth Painlevé Equation ‣ §32.10 Special Function Solutions ‣ Properties ‣ Chapter 32 Painlevé Transcendents
§32.6(iii) Second Painlevé Equation ‣ §32.6 Hamiltonian Structure ‣ Properties ‣ Chapter 32 Painlevé Transcendents
§32.6(v) Other Painlevé Equations ‣ §32.6 Hamiltonian Structure ‣ Properties ‣ Chapter 32 Painlevé Transcendents
§32.7(viii) Affine Weyl Groups ‣ §32.7 Bäcklund Transformations ‣ Properties ‣ Chapter 32 Painlevé Transcendents
References:
| [1] | Airault, H.: Rational solutions of Painlevé equations. Studies in Appl. Math.61, 31-53 (1979) · Zbl 0496.58012 |
| [2] | Bourbaki, N.: Groupes et algèbres de Lie. Chaps. 4-6. Paris: Masson 1980 |
| [3] | Bureau, F.J.: Les équations différentielles du second ordre à points critiques fixes, I. Les intégrales de l’équation A2 de Painlevé. Bull. Cl. Sci. Acad. Roy. Belg.69, 80-104 (1983); II. Les intégrales de l’équation A4 de Painlevé, ibid Bull. Cl. Sci. Acad. Roy. Belg.69 397-433 (1983) · Zbl 0534.34021 |
| [4] | Darboux, G.: Leçons sur la théorie générale des surfaces. tII. 137. Chelsea 1972 |
| [5] | Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, II. Physica2 D, 407-448 (1981) · Zbl 1194.34166 |
| [6] | Kametaka, Y.: On poles of the rational solution of the Toda equation of Painlevé-II type. Proc. Japan Acad. Ser. A59, 358-360 (1983) · Zbl 0546.34042 · doi:10.3792/pjaa.59.358 |
| [7] | Kametaka, Y.: On poles of the rational solution of the Toda equation of Painlevé-IV type. Proc. Japan Acad. Ser. A59, 453-455 (1983) · Zbl 0562.34039 · doi:10.3792/pjaa.59.453 |
| [8] | Lukashevich, N.A.: Theory of the fourth Painlevé equation. Diffeer. Uravn.3, 395-399 (1967) · Zbl 0225.34001 |
| [9] | Lukashevich, N.A.: The second Painlevé equation. Differ. Uravn.7, 853-854 (1971) · Zbl 0271.34011 |
| [10] | Murata, Y.: Rational solutions of the second and the fourth Painlevé equations. Funkc. Ekvacioy. Ser. Int.28, 1-32 (1985) · Zbl 0597.34004 |
| [11] | Okamoto, K.: Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé. Jap. J. Math.5, 1-79 (1979) · Zbl 0426.58017 |
| [12] | Okamoto, K.: Polynomial Hamiltonians associated with Painlevé equations, I. Proc. Japan Acad. Ser. A56, 264-268 (1980); II, ibid. Proc. Japan Acad. Ser. A56 367-371 (1980) · Zbl 0476.34010 · doi:10.3792/pjaa.56.264 |
| [13] | Okamoto, K.: On the ?-function of the Painlevé equations. Physica2 D, 525-535 (1981) · Zbl 1194.34171 |
| [14] | Okamoto, K.: Studies on the Painlevé equations I, sixth Painlevé equationP VI. Ann. Mat.; II, fifth Painlevé equationP v. Jap. J. Math. 1986 |
| [15] | Okamoto, K.: Sur les échelles associées aux fonctions spéciales et léquation de Toda, preprint 1985 |
| [16] | Vorob’ev, A.P.: On rational solutions of the second Painlevé equation. Differ. Uravn.1, 58-59 (1965) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
