MR2493874 Yusuke SASANO Yusuke SASANO Symmetry in the Painlevé Systems and Their Extensions to Four-Dimensional Systems Funkcialaj Ekvacioj. Serio Internacia 51 2008 351--369 http://fe.math.kobe-u.ac.jp/FE/FullPapers/51-3/51_351.pdf http://www.ams.org/mathscinet-getitem?mr=MR2493874 We give a reformulation of the symmetry of the Painlevé equations $P_{V}$, $P_{IV}$, $P_{III}$ and $P_{II}$, respectively. Based on these constructions, we make natural extensions to fourth-order analogues for each of the Painlevé equations $P_{V}$ and $P_{III}$, respectively. Affine Weyl group, Birational symmetry, Coupled Painlevé system. 34M55, 34M45, 58F05, 32S65. 51-351 2008 Symmetry in the Painlevé Systems and Their Extensions to Four-Dimensional Systems Yusuke SASANO Yusuke SASANO
1 Fuji, K.; Suzuki, T. The sixth Painlevé equation arising from $D_4^{(1)}$ hierarchy J. Phys. A 39 2006 12073-12082 MR2266212
2 Kakei, S.; Kikuchi, T. Affine Lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction Int. Math. Res. Not. 2004 2004 4181-4209 MR2111361
3 Matano, T.; Matumiya, A.; Takano, K. On some Hamiltonian structures of Painlevé systems, II J. Math. Soc. Japan 51 1999 843-866 MR1705251
4 Noumi, M.; Yamada, Y. Affine Weyl Groups, Discrete Dynamical Systems and Painlevé Equations Comm. Math. Phys. 199 1998 281-295 MR1666847
5 Noumi, M.; Takano, K.; Yamada, Y. Bäcklund transformations and the manifolds of Painlevé systems Funkcial. Ekvac. 45 2002 237-258 MR1948601 1948601
6 Noumi, M.; Yamada, M. Symmetries in Painlevé equations Sugaku Expositions 17 2004 203-218 MR2095768
7 Noumi, M. Painlevé Equations Through Symmetry Translations of Mathematical Monographs, 223, American Mathematical Society, Providence, RI 2004 MR2044201 8 Okamoto, K. Introduction to Painlevé equations Sophia Kokyuroku in Math., 19, Sophia Univ. 1985, (in Japanese)
9 Okamoto, K. Studies on the Painlevé equations, I Ann. Mat. Pura Appl. 146 1987 337-381 MR0916698
10 Okamoto, K. Studies on the Painlevé equations, II Japan J. Math. 13 1987 47-76 MR0914314
11 Okamoto, K. Studies on the Painlevé equations, III Math. Ann. 275 1986 221-256 MR0854008
12 Okamoto, K. Studies on the Painlevé equations, IV Funkcial. Ekvac. 30 1987 305-332 MR0927186 0927186 13 Sasano, Y., Four-dimensional Painlevé systems of types $D_5^{(1)}$ and $B_4^{(1)}$, submitted
14 Sasano, Y.; Yamada, Y. Symmetry and holomorphy of Painlevé type systems RIMS Kokyuroku Bessatsu B2 2007 215-225 MR2310032
15 Shioda, T.; Takano, K. On some Hamiltonian structures of Painlevé systems I Funkcial. Ekvac. 40 1997 271-291 MR1480279 1480279
16 Suzuki, M.; Tahara, N.; Takano, K. Hierarchy of Bäcklund transformation groups of the Painlevé equations J. Math. Soc. Japan 56 2004 1221-1232 MR2092946
17 Takano, K. Confluence processes in defining manifolds for Painlevé systems Tohoku Math. J. (2) 53 2001 319-335 MR1829984
18 Tsuda, T.; Okamoto, K.; Sakai, H. Folding transformations of the Painlevé equations Math. Ann. 331 2005 713-738 MR2148794