Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 030, 20 pages      arXiv:1103.4906      https://doi.org/10.3842/SIGMA.2011.030

Rational Solutions of the Sasano System of Type A5(2)

Kazuhide Matsuda
Department of Engineering Science, Niihama National College of Technology, 7-1 Yagumo-chou, Niihama, Ehime, 792-8580, Japan

Received November 5, 2010, in final form March 17, 2011; Published online March 25, 2011

Abstract
In this paper, we completely classify the rational solutions of the Sasano system of type A5(2), which is given by the coupled Painlevé III system. This system of differential equations has the affine Weyl group symmetry of type A5(2).

Key words: affine Weyl group; rational solutions; Sasano system.

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References

  1. Airault H., Rational solutions of Painlevé equation, Stud. Appl. Math. 61 (1979), 31-53.
  2. Bassom A.P., Clarkson P.A., Hicks A.C., Bäcklund transformations and solution hierarchies for the fourth Painlevé equation, Stud. Appl. Math. 95 (1995), 1-71.
  3. Clarkson P.A., The third Painlevé equation and associated special polynomial, J. Phys. A: Math. Gen. 36 (2003), 9507-9532.
  4. Gambier B., Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critique fixes, Acta Math. 33 (1910), 1-55.
  5. Gromak V.I., Algebraic solutions of the third Painlevé equation, Dokl. Akad. Nauk BSSR 23 (1979), 499-502 (in Russian).
  6. Gromak V.I., Reducibility of the Painlevé equations, Differ. Equ. 20 (1983), 1191-1198.
  7. Hone A.N.W., Coupled Painlevé systems and quartic potentials, J. Phys. A: Math. Gen. 34 (2001), 2235-2246.
  8. Kitaev A.V., Law C.K., McLeod J.B., Rational solutions of the fifth Painlevé equation, Differential Integral Equations 7 (1994), 967-1000.
  9. Matsuda K., Rational solutions of the A4 Painlevé equation, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 5, 85-88.
  10. Matsuda K., Rational solutions of the A5(1) Painlevé equation, arXiv:0708.2960.
  11. Mazzoco M., Rational solutions of the Painlevé VI equation, J. Phys. A: Math. Gen. 34 (2001), 2281-2294, nlin.SI/0007036.
  12. Mazzoco M., Mo M.Y., The Hamiltonian structure of the second Painlevé hierarchy, Nonlinearity 20 (2007), 2845-2882, nlin.SI/0610066.
  13. Milne A.E., Clarkson P.A., Bassom A.P., Bäcklund transformations and solution hierarchies for the third Painlevé equation, Stud. Appl. Math. 98 (1997), 139-194.
  14. Murata Y., Rational solutions of the second and the fourth Painlevé equations, Funkcial. Ekvac. 28 (1985), 1-32.
  15. Murata Y., Classical solutions of the third Painlevé equation, Nagoya Math. J. 139 (1995), 37-65.
  16. Noumi M., Yamada Y., Higher order Painlevé equations of type Al(1), Funkcial. Ekvac. 41 (1998), 483-503, math.QA/9808003.
  17. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, PII and PIV, Math. Ann. 275 (1986), 221-255.
  18. Okamoto K., Studies on the Painlevé equations. I. Sixth Painlevé equation PVI, Ann. Mat. Pure Appl. (4) 146 (1987), 337-338.
  19. Okamoto K., Studies on the Painlevé equations. II. Fifth Painlevé equation PV, Japan. J. Math. (N.S.) 13 (1987), 47-76.
  20. Okamoto K., Studies on the Painlevé equations. IV. Third Painlevé equation PIII, Funkcial. Ekvac. 30 (1987), 305-332.
  21. Painlevé P., Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, Acta Math. 25 (1902), 1-85.
  22. Sasano Y., Higher order Painlevé equations of type Dl(1), RIMS Kokyuroku 1473 (2006), no. 1, 43-163.
  23. Sasano Y., Symmetries in the system of type A5(2), arXiv:0704.2327.
  24. Vorob'ev A.P., On rational solutions of the second Painlevé equation, Differ. Equ. 1 (1965), 58-59.
  25. Yablonskii A.I., On rational solutions of the second Painlevé equation, Vesti AN BSSR, Ser. Fiz.-Tech. Nauk (1959), no. 3, 30-35 (in Russian).
  26. Yuang W., Li Y., Rational solutions of Painlevé equations, Canad. J. Math. 54 (2002), 648-670.

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