Geodesic
Rational Solutions of Painlevé Equations
Canadian journal of mathematics, Tome 54 (2002) no. 3, pp. 648-670

Voir la notice de l'article provenant de la source Cambridge University Press

Consider the sixth Painlevé equation $({{\text{P}}_{6}})$ below where $\alpha ,\beta ,\gamma$ and $\delta$ are complex parameters. We prove the necessary and sufficient conditions for the existence of rational solutions of equation $({{\text{P}}_{6}})$ 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 $({{\text{P}}_{1}})-({{\text{P}}_{5}})$ below, which have been investigated by many authors [1]–[7].
DOI : 10.4153/CJM-2002-024-0
Mots-clés : 30D35, 34A20, Painlevé differential equation, rational function, Bäcklund transformation 
Wenjun, Yuan; Yezhou, Li. Rational Solutions of Painlevé Equations. Canadian journal of mathematics, Tome 54 (2002) no. 3, pp. 648-670. doi: 10.4153/CJM-2002-024-0
@article{10_4153_CJM_2002_024_0,
     author = {Wenjun, Yuan and Yezhou, Li},
     title = {Rational {Solutions} of {Painlev\'e} {Equations}},
     journal = {Canadian journal of mathematics},
     pages = {648--670},
     year = {2002},
     volume = {54},
     number = {3},
     doi = {10.4153/CJM-2002-024-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2002-024-0/}
}
TY  - JOUR
AU  - Wenjun, Yuan
AU  - Yezhou, Li
TI  - Rational Solutions of Painlevé Equations
JO  - Canadian journal of mathematics
PY  - 2002
SP  - 648
EP  - 670
VL  - 54
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2002-024-0/
DO  - 10.4153/CJM-2002-024-0
ID  - 10_4153_CJM_2002_024_0
ER  - 
%0 Journal Article
%A Wenjun, Yuan
%A Yezhou, Li
%T Rational Solutions of Painlevé Equations
%J Canadian journal of mathematics
%D 2002
%P 648-670
%V 54
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2002-024-0/
%R 10.4153/CJM-2002-024-0
%F 10_4153_CJM_2002_024_0

[1] [1] Airault, H., Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1979), 31–53. Google Scholar

[2] [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), 943–966. Google Scholar

[3] [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), 2033–2042. Google Scholar

[4] [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), 539–544. Google Scholar

[5] [5] Gromak, V. I. and Lukashevich, N. A., Analytic properties of solutions of the Painlevé equations. Universitetskoe,Minsk, 1990, 1–157 (in Russian). Google Scholar

[6] [6] Iwasaki, K., Kimura, H., Shimomura, S. and Yoshida, M., From Gauss to Painlevé. Vieweg, Braunschweig, 1991. Google Scholar

[7] [7] Kitaev, A. V., Law, C. K. and McLeod, J. B., Rational solutions of the fifth Painlevé equation. Preprint. Google Scholar

[8] [8] Okamoto, K., Studies of the Painlevé equations. Math. Ann. 275 (1986), 221–255. Google Scholar

Cité par Sources :