A certain expression of the first Painlevé hierarchy
Shun Shimomura
Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 80, Number 6
(2004), 105-109.
Abstract
We show that each equation in the first Painlevé hierarchy is equivalent to a system of nonlinear equations determined by a kind of generating function, and that it admits the Painlevé property. Our results are derived from the fact that the first Painlevé hierarchy follows from isomonodromic deformation of certain linear systems with an irregular singular point.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.pja/1116014786
Mathematical Reviews number (MathSciNet): MR2075451
Zentralblatt MATH identifier: 1061.34064
Digital Object Identifier: doi:10.3792/pjaa.80.105
References
Jimbo, M., Miwa, T., and Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function. Phys. D, 2, 306--352 (1981).
Mathematical Reviews (MathSciNet): MR630674
Digital Object Identifier: doi:10.1016/0167-2789(81)90013-0
Kawai, T., Koike, T., Nishikawa, Y., and Takei, Y.: On the Stokes geometry of higher order Painlevé equations. (To appear).
Kudryashov, N. A.: The first and second Painlevé equations of higher order and some relations between them. Phys. Lett. A, 224, 353--360 (1997).
Mathematical Reviews (MathSciNet): MR1431712
Digital Object Identifier: doi:10.1016/S0375-9601(96)00795-5
Kudryashov, N. A., and Soukharev, M. B.: Uniformization and transcendence of solutions for the first and second Painlevé hierarchies. Phys. Lett. A, 237, 206--216 (1998).
Mathematical Reviews (MathSciNet): MR1609399
Digital Object Identifier: doi:10.1016/S0375-9601(97)00850-5
Miwa, T.: Painlevé property of monodromy preserving deformation equations and the analyticity of $\tau$-functions. Publ. Res. Inst. Math. Sci., 17, 703--721 (1981).
Mathematical Reviews (MathSciNet): MR642657
Shimomura, S.: Painlevé property of a degenerate Garnier system of (9/2)-type and of a certain fourth order non-linear ordinary differential equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29, 1--17 (2000).
Mathematical Reviews (MathSciNet): MR1765543
Shimomura, S.: On the Painlevé I hierarchy. RIMS Kokyuroku, 1203, 46--50 (2001).
Mathematical Reviews (MathSciNet): MR1854796
Shimomura, S.: Poles and $\alpha$-points of meromorphic solutions of the first Painlevé hierarchy. Publ. Res. Inst. Math. Sci., 40, 471--485 (2004).
Mathematical Reviews (MathSciNet): MR2049643
Proceedings of the Japan Academy, Series A, Mathematical Sciences
