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

SIGMA 7 (2011), 059, 15 pages      arXiv:1106.3384
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A6(1)

Mikio Murata
Department of Physics and Mathematics, College of Science and Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi, Kanagawa, 252-5258 Japan

Received February 07, 2011, in final form June 11, 2011; Published online June 17, 2011

An ultradiscrete system corresponding to the q-Painlevé equation of type A6(1), which is a q-difference analogue of the second Painlevé equation, is proposed. Exact solutions with two parameters are constructed for the ultradiscrete system.

Key words: Painlevé equations; ultradiscrete systems.

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  1. Doyon B., Two-point correlation functions of scaling fields in the Dirac theory on the Poincaré disk, Nuclear Phys. B 675 (2003), 607-630, hep-th/0304190.
  2. Dubrovin B., Mazzocco M., Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), 55-147, math.AG/9806056.
  3. Grammaticos B., Ohta Y., Ramani A., Takahashi D., Tamizhmani K.M., Cellular automata and ultra-discrete Painlevé equations, Phys. Lett. A 226 (1997), 53-58, solv-int/9603003.
  4. Grammaticos B., Ramani A., Papageorgiou V.G., Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), 1825-1828.
  5. Hamamoto T., Kajiwara K., Witte N.S., Hypergeometric solutions to the q-Painlevé equation of type (A1+A'1)(1), Int. Math. Res. Not. 2006 (2006), Art. ID 84619, 26 pages, nlin.SI/0607065.
  6. Isojima S., Konno T., Mimura N., Murata M., Satsuma J., Ultradiscrete Painlevé II equation and a special function solution, J. Phys. A: Math. Theor. 44 (2011), 175201, 10 pages.
  7. Jimbo M., Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982), 1137-1161.
  8. Joshi N., Lafortune S., How to detect integrability in cellular automata, J. Phys. A: Math. Gen. 38 (2005), L499-L504.
  9. Joshi N., Lafortune S., Integrable ultra-discrete equations and singularity analysis, Nonlinearity 19 (2006), 1295-1312.
  10. Joshi N., Nijhoff F.W., Ormerod C., Lax pairs for ultra-discrete Painlevé cellular automata, J. Phys. A: Math. Gen. 37 (2004), L559-L565.
  11. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Construction of hypergeometric solutions to the q-Painlevé equations, Int. Math. Res. Not. 2005 (2005), no. 24, 1441-1463, nlin.SI/0501051.
  12. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Hypergeometric solutions to the q-Painlevé equations, Int. Math. Res. Not. 2004 (2004), no. 47, 2497-2521, nlin.SI/0403036.
  13. Kajiwara K., Ohta Y., Satsuma J., Grammaticos B., Ramani A., Casorati determinant solutions for the discrete Painlevé-II equation, J. Phys. A: Math. Gen. 27 (1994), 915-922, solv-int/9310002.
  14. Mano T., Asymptotic behaviour around a boundary point of the q-Painlevé VI equation and its connection problem, Nonlinearity 23 (2010), 1585-1608.
  15. Mimura N., Isojima S., Murata M., Satsuma J., Singularity confinement test for ultradiscrete equations with parity variables, J. Phys. A: Math. Theor. 42 (2009), 315206, 7 pages.
  16. Murata M., Sakai H., Yoneda J., Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type E8(1), J. Math. Phys. 44 (2003), 1396-1414, nlin.SI/0210040.
  17. Nishioka K., A note on the transcendency of Painlevé's first transcendent, Nagoya Math. J. 109 (1988), 63-67.
  18. Nishioka S., Irreducibility of q-Painlevé equation of type A6(1) in the sense of order, J. Difference Equ. Appl., to appear.
  19. Nishioka S., Transcendence of solutions of q-Painlevé equation of type A6(1), Aequat. Math. 81 (2011), 121-134.
  20. Noumi M., Okamoto K., Irreducibility of the second and the fourth Painlevé equations, Funkcial. Ekvac. 40 (1997), 139-163.
  21. Ohyama Y., Analytic solutions to the sixth q-Painlevé equation around the origin, in Expansion of Integrable Systems, RIMS Kôkyûroku Bessatsu, Vol. B13, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, 45-52.
  22. Ormerod C.M., Hypergeometric solutions to an ultradiscrete Painlevé equation, J. Nonlinear Math. Phys. 17 (2010), 87-102, nlin.SI/0610048.
  23. Ramani A., Grammaticos B., Discrete Painlevé equations: coalescences, limits and degeneracies, Phys. A 228 (1996), 160-171, solv-int/9510011.
  24. Ramani A., Grammaticos B., Hietarinta J., Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), 1829-1832.
  25. Ramani A., Grammaticos B., Tamizhmani T., Tamizhmani K.M., Special function solutions of the discrete Painlevé equations, Comput. Math. Appl. 42 (2001), 603-614.
  26. Ramani A., Takahashi D., Grammaticos B., Ohta Y., The ultimate discretisation of the Painlevé equations, Phys. D 114 (1998), 185-196.
  27. Sakai H., Problem: discrete Painlevé equations and their Lax forms, in Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and Their Deformations. Painlevé Hierarchies, RIMS Kôkyûroku Bessatsu, Vol. B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 195-208.
  28. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  29. Takahashi D., Tokihiro T., Grammaticos B., Ohta Y., Ramani A., Constructing solutions to the ultradiscrete Painlevé equations, J. Phys. A: Math. Gen. 30 (1997), 7953-7966.
  30. Tokihiro T., Takahashi D., Matsukidaira J., Satsuma J., From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett. 76 (1996), 3247-3250.
  31. Umemura H., On the irreducibility of the first differential equation of Painlevé, in Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Kinokuniya, Tokyo, 1988, 771-789.

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