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Rational Solutions of Painlevé Equations

  • Yuan Wenjun (a1) and Li Yezhou (a2)
Abstract

Consider the sixth Painlevé equation (P6) below where α, β, γ and δ are complex parameters. We prove the necessary and sufficient conditions for the existence of rational solutions of equation (P6) in term of special relations among the parameters. The number of distinct rational solutions in each case is exactly one or two or infinite. And each of them may be generated by means of transformation group found by Okamoto [7] and Bäcklund transformations found by Fokas and Yortsos [4]. A list of rational solutions is included in the appendix. For the sake of completeness, we collected all the corresponding results of other five Painlevé equations (P1)−(P5) below, which have been investigated by many authors [1]–[7].

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References
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[1] Airault, H., Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1979), 3153.
[2] Erugin, N. P., The analytic theory and problems of the real theory of differential equations connected with the first method and with the methods of the analytic theory. Differential Equations 3 (1967), 943966.
[3] Fokas, A. S. and Ablotz, M. J., On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (1982), 20332042.
[4] Fokas, A. S. and Yortsos, Y. C., The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento 30 (1981), 539544.
[5] Gromak, V. I. and Lukashevich, N. A., Analytic properties of solutions of the Painlevé equations. Universitetskoe,Minsk, 1990, 1–157 (in Russian).
[6] Iwasaki, K., Kimura, H., Shimomura, S. and Yoshida, M., From Gauss to Painlevé. Vieweg, Braunschweig, 1991.
[7] Kitaev, A. V., Law, C. K. and McLeod, J. B., Rational solutions of the fifth Painlevé equation. Preprint.
[8] Okamoto, K., Studies of the Painlevé equations. Math. Ann. 275 (1986), 221255.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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