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Nonlinear chains and Painlevé equations. (English) Zbl 0812.34030
Summary: ODE systems which define a periodically closed sequence of Bäcklund transformations (BT) and which are equivalent to the Painlevé equations $P\sb 2- P\sb 6$ are presented. Transformation properties of Painlevé equations can be easily derived from the discrete symmetries of these systems.

34C20Transformation and reduction of ODE and systems, normal forms
34L40Particular ordinary differential operators
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
Full Text: DOI
[1] . Leningrad math. J. 2, 377 (1991)
[2] Veselov, A. P.; Shabat, A. B.: Funct. anal. Appl.. 27, No. no. 2, 1 (1993)
[3] Flaschka, H.: J. math. Phys.. 21, 1016 (1980)
[4] Flaschka, H.; Newell, A.: Commun. math. Phys.. 76, 67 (1980)
[5] Calogero, F.; Degasperis, A.: Spectral transforms and solitons. (1982) · Zbl 0501.35072
[6] Lukashevich, N. A.: Diff. ur.. 3, 771 (1967)
[7] Lukashevich, N. A.: Diff. ur.. 7, 1124 (1971)
[8] Gromak, V. I.: Diff. ur.. 9, 2082 (1973)
[9] Gromak, V. I.: Diff. ur.. 11, 373 (1975)
[10] Gromak, V. I.: Diff. ur.. 12, 740 (1976)
[11] Fokas, A. S.; Yortsos, Y. C.: Lett. nuovo cimento. 30, 539 (1980)
[12] Gromak, V. I.; Lukashevich, N. A.: Diff. ur.. 18, 419 (1982)
[13] Airault, H.: Stud. appl. Math.. 61, 33 (1979)
[14] Boiti, M.; Pempinelli, F.: Nuovo cimento. 51 B, 70 (1979)
[15] Boiti, M.; Pempinelli, F.: Nuovo cimento. 56 B, 148 (1980)
[16] Fokas, A. S.; Ablowitz, M. J.: J. math. Phys.. 23, 2033 (1982)
[17] Bassom, A. P.; Clarkson, P. A.; Hicks, A. C.; Mcleod, J. B.: Proc. math. Phys. sci.. 437, 1 (1992)
[18] Ablowitz, M. J.; Segur, H.: Solitons and inverse scattering transform. (1980) · Zbl 0472.35002
[19] Gromak, V. I.; Lukashevich, N. A.: Analytical properties of the Painlevé equations solutions. (1990) · Zbl 0752.34003
[20] Mug\tilde{}an, U.; Fokas, A. S.: J. math. Phys.. 33, 2031 (1992)
[21] Adler, V. E.: Funct. anal. Appl.. 27, No. no. 2, 79 (1993)
[22] Zakharov, V. E.; Shabat, A. B.: Funct. anal. Appl.. 13, 13 (1979)
[23] Novikov, S. P.: Funct. anal. Appl.. 8, 54 (1974)
[24] Its, A. R.; Novokshenov, V. Yu.: The isomonodromic deformation method in the theory of Painlevé equation. Lecture notes in mathematics 1191 (1986) · Zbl 0592.34001
[25] Ince, E. L.: Ordinary differential equations. (1947) · Zbl 0063.02971
[26] Okamoto, K.: Math. ann.. 275, 221 (1986)