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Continuous, Discrete and Ultradiscrete Painlevé Equations

  • Nobutaka Nakazono
  • Yang Shi
  • Masataka Kanki
Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

We give an introductory lecture on the theory of Painlevé equations, which are one of the most important objects in the theory of integrable systems, and their discrete counterparts. The lecture is divided into three parts: the Painlevé equations, discrete Painlevé equations and ultradiscrete Painlevé equations.

Notes

Acknowledgements

Authors thank Profs. Tetsuji Tokihiro, Masatoshi Noumi and Akane Nakamura for helpful comments.

References

  1. 1.
    A.S. Fokas, B. Grammaticos, A. Ramani, From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180(2), 342–360 (1993)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    R. Fuchs, Sur quelques équations différentielles linéaires du second ordre. C. R. Acad. Sci. Paris 141(1), 555–558 (1905)MATHGoogle Scholar
  3. 3.
    B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes. Acta Math. 33(1), 1–55 (1910)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    G. Gasper, M. Rahman, Basic Hypergeometric Series, vol. 35. Encyclopedia of Mathematics and Its Applications (Cambridge University Press, Cambridge, 1990)MATHGoogle Scholar
  5. 5.
    B. Grammaticos, Y. Ohta, A. Ramani, H. Sakai, Degeneration through coalescence of the q-Painlevé VI equation. J. Phys. A 31(15), 3545–3558 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    B. Grammaticos, Y. Ohta, A. Ramani, D. Takahashi, K.M. Tamizhmani, Cellular automata and ultra-discrete Painlevé equations. Phys. Lett. A 226(1–2), 53–58 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    B. Grammaticos, A. Ramani, V. Papageorgiou, Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67(14), 1825–1828 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    B. Grammaticos, A. Ramani, J. Satsuma, R. Willox, A. Carstea, Reductions of integrable lattices. J. Nonlinear Math. Phys. 12(Suppl. 1), 363–371 (2005)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Hirota, Discrete analogue of a generalized Toda equation. J. Phys. Soc. Jpn. 50(11), 3785–3791 (1981)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    R. Hirota, S. Tsujimoto, T. Imai, Difference scheme of soliton equations. Sūrikaisekikenkyūsho Kōkyūroku 822, 144–152 (1993)MathSciNetMATHGoogle Scholar
  11. 11.
    J.E. Humphreys, Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29 (Cambridge University Press, Cambridge, 1990)Google Scholar
  12. 12.
    K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painlevé, vol. E16. Aspects of Mathematics (Friedrich Vieweg & Sohn, Braunschweig, 1991)Google Scholar
  13. 13.
    M. Jimbo, T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2(3), 407–448 (1981)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    M. Jimbo, H. Sakai, A q-analog of the sixth Painlevé equation. Lett. Math. Phys. 38(2), 145–154 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    N. Joshi, S. Lafortune, How to detect integrability in cellular automata. J. Phys. A 38(28), L499–L504 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    N. Joshi, N. Nakazono, Y. Shi, Reflection groups and discrete integrable systems. J. Intell. Syst. 1(1), xyw006 (2016)Google Scholar
  17. 17.
    K. Kajiwara, N. Nakazono, Hypergeometric solutions to the symmetric q-Painlevé equations. Int. Math. Res. Not. 2015(4), 1101–1140 (2015)MATHGoogle Scholar
  18. 18.
    K. Kajiwara, Y. Ohta, Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31(10), 2431–2446 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    K. Kajiwara, M. Noumi, Y. Yamada, A study on the fourth q-Painlevé equation. J. Phys. A 34(41), 8563–8581 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, 10 E 9 solution to the elliptic Painlevé equation. J. Phys. A 36, L263–L272 (2003)Google Scholar
  21. 21.
    K. Kajiwara, T. Masuda, Noumi, M., Ohta, Y., Yamada, Y.: Hypergeometric solutions to the q-Painlevé equations. Int. Math. Res. Not. 2004(47), 2497–2521 (2004)Google Scholar
  22. 22.
    K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Construction of hypergeometric solutions to the q-Painlevé equations. Int. Math. Res. Not. 2005(24), 1441–1463 (2005)CrossRefMATHGoogle Scholar
  23. 23.
    K. Kajiwara, M. Noumi, Yamada, Y.: Geometric aspects of Painlevé equations. J. Phys. A 50(7), 073001, 164 (2017)Google Scholar
  24. 24.
    M. Kanki, J. Mada, T. Tokihiro, Conserved quantities and generalized solutions of the ultradiscrete KdV equation. J. Phys. A 44(14), 145202, 13 (2011)Google Scholar
  25. 25.
    R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Online notes (1998)Google Scholar
  26. 26.
    D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. (5) 39(240), 422–443 (1895)Google Scholar
  27. 27.
    T. Masuda, Classical transcendental solutions of the Painlevé equations and their degeneration. Tohoku Math. J. (2) 56(4), 467–490 (2004)Google Scholar
  28. 28.
    N. Mimura, S. Isojima, M. Murata, J. Satsuma, Singularity confinement test for ultradiscrete equations with parity variables. J. Phys. A 42(31), 315206, 7 (2009)Google Scholar
  29. 29.
    M. Murata, Exact solutions with two parameters for an ultradiscrete Painlevé equation of type A6 (1). SIGMA Symmetry Integrability Geom. Methods Appl. 7, 15 pp. (2011). Paper 059Google Scholar
  30. 30.
    F.W. Nijhoff, V.G. Papageorgiou, Similarity reductions of integrable lattices and discrete analogues of the Painlevé II equation. Phys. Lett. A 153(6), 337–344 (1991)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    F.W. Nijhoff, A. Ramani, B. Grammaticos, Y. Ohta, On discrete Painlevé equations associated with the lattice KdV systems and the Painlevé VI equation. Stud. Appl. Math. 106(3), 261–314 (2001)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    M. Noumi, Painlevé Equations Through Symmetry, vol. 223. Translations of Mathematical Monograph (American Mathematical Society, Providence, RI, 2004)MATHGoogle Scholar
  33. 33.
    Y. Ohyama, H. Kawamuko, H. Sakai, K. Okamoto, Studies on the Painlevé equations. V. Third Painlevé equations of special type Piii(D7) and Piii(D8). J. Math. Sci. Univ. Tokyo 13(2), 145–204 (2006)Google Scholar
  34. 34.
    K. Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé. Jpn. J. Math. (N.S.) 5(1), 1–79 (1979)Google Scholar
  35. 35.
    K. Okamoto, Studies on the Painlevé equations. III. Second and fourth Painlevé equations, Pii and Piv. Math. Ann. 275(2), 221–255 (1986)Google Scholar
  36. 36.
    K. Okamoto, Studies on the Painlevé equations. I. Sixth Painlevé equation Pvi. Ann. Mat. Pura Appl. (4) 146, 337–381 (1987)Google Scholar
  37. 37.
    K. Okamoto, Studies on the Painlevé equations. II. Fifth Painlevé equation Pv. Jpn. J. Math. (N.S.) 13(1), 47–76 (1987)Google Scholar
  38. 38.
    K. Okamoto, Studies on the Painlevé equations. IV. Third Painlevé equation Piii. Funkcial. Ekvac. 30(2–3), 305–332 (1987)Google Scholar
  39. 39.
    C.M. Ormerod, Hypergeometric solutions to an ultradiscrete Painlevé equation. J. Nonlinear Math. Phys. 17(1), 87–102 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    C.M. Ormerod, Tropical geometric interpretation of ultradiscrete singularity confinement. J. Phys. A 46(30), 305204, 15 (2013)Google Scholar
  41. 41.
    C.M. Ormerod, Y. Yamada, From polygons to ultradiscrete Painlevé equations. SIGMA Symmetry Integrability Geom. Methods Appl. 11, 36 pp. (2015). Paper 056Google Scholar
  42. 42.
    P. Painlevé, Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme. Acta Math. 25(1), 1–85 (1902)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations. Phys. Lett. A 126(7), 419–421 (1988)ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    A. Ramani, B. Grammaticos, J. Hietarinta, Discrete versions of the Painlevé equations. Phys. Rev. Lett. 67(14), 1829–1832 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    A. Ramani, D. Takahashi, B. Grammaticos, Y. Ohta, The ultimate discretisation of the Painlevé equations. Phys. D 114(3–4), 185–196 (1998)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Commun. Math. Phys. 220(1), 165–229 (2001)ADSCrossRefMATHGoogle Scholar
  47. 47.
    D. Takahashi, On some soliton systems defined by using boxes and balls, in International Symposium on Nonlinear Theory and its Applications (NOLTA ’ 93 ), pp. 555–558 (1993)Google Scholar
  48. 48.
    D. Takahashi, J. Matsukidaira, Box and ball system with a carrier and ultradiscrete modified KdV equation. J. Phys. A 30(21), L733–L739 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    D. Takahashi, J. Satsuma, A soliton cellular automaton. J. Phys. Soc. Japan 59(10), 3514–3519 (1990)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    D. Takahashi, T. Tokihiro, B. Grammaticos, Y. Ohta, A. Ramani, Constructing solutions to the ultradiscrete Painlevé equations. J. Phys. A 30(22), 7953–7966 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    K. Takemura, T. Tsutsui, Ultradiscrete Painlevé VI with parity variables. SIGMA Symmetry Integrability Geom. Methods Appl. 9, 12 pp. (2013). Paper 070Google Scholar
  52. 52.
    T. Tokihiro, D. Takahashi, J. Matsukidaira, J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure. Phys. Rev. Lett. 76, 3247–3250 (1996)ADSCrossRefGoogle Scholar
  53. 53.
    T. Tokihiro, D. Takahashi, J. Matsukidaira, Box and ball system as a realization of ultradiscrete nonautonomous KP equation. J. Phys. A 33(3), 607–619 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    N.J. Zabusky, M.D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Physics and MathematicsAoyama Gakuin UniversitySagamihara, KanagawaJapan
  2. 2.School of Mathematics and StatisticsThe University of SydneySydneyAustralia
  3. 3.Faculty of Engineering ScienceKansai UniversityOsakaJapan

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