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On the relation between the continuous and discrete Painlevé equations

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Abstract

A method for deriving difference equations (the discrete Painlevé equations in particular) from the Bäcklund transformations of the continuous Painlevé equations is discussed. This technique can be used to derive several of the known discrete painlevé equations (in particular, the first and second discrete Painlevé equations and some of their alternative versions). The Painlevé equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painlevé equations. Exact solutions of the Painlevé equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.

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References

  1. E. Picard,C. R. Acad. Sci. Paris,104, 41–43 (1887).

    Google Scholar 

  2. E. L. Ince,Ordinary Differential Equations [in Russian], GNTI Ukraini, Khar'kov (1939); English transl., Dover, New York (1956).

    Google Scholar 

  3. M. J. Ablowitz and P. A. Clarkson,Solitons, Nonlinear Evolution Equations, and Inverse Scattering (LMS Lect. Notes Math., Vol. 149), Cambridge Univ. Press, Cambridge (1991).

    MATH  Google Scholar 

  4. H. Airault,Stud. Appl. Math.,61, 31–53 (1979).

    ADS  MathSciNet  MATH  Google Scholar 

  5. A. P. Bassom, P. A. Clarkson, and A. C. Hicks,Stud. Appl. Math.,95, 1–71 (1995).

    MathSciNet  MATH  Google Scholar 

  6. V. I. Gromak,Diff. Equat.,14, 1510–1513 (1979).

    Google Scholar 

  7. K. Kajiwara and T. Masuda, “A generalization of determinant formulas for the solutions of Painlevé II and XXXIV equations”, Preprint solv-int/9903014 (1999).

  8. K. Kajiwara and T. Masuda, “On the Umemura polynomials for the Painlevé III equation,” Preprint solvint/9903015 (1999).

  9. K. Kajiwara and Y. Ohta,J. Math. Phys.,37, 4393–4704 (1996).

    Article  MathSciNet  Google Scholar 

  10. K. Kajiwara and Y. Ohta,J. Phys. A.,31, 2431–2446 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. A. E. Milne, P. A. Clarkson, and A. P. Bassom,Stud. Appl. Math.,98, 139–194 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  12. Y. Murata,Funkc. Ekvacioj Ser. Int.,28, 1–32 (1985).

    MathSciNet  MATH  Google Scholar 

  13. Y. Murata,Nagoya Math. J.,139, 37–65 (1995).

    MathSciNet  MATH  Google Scholar 

  14. K. Okamoto,Ann. Math. Pure Appl.,146, 337–381 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Okamoto,Japan. J. Math.,13, 47–76 (1987).

    MathSciNet  MATH  Google Scholar 

  16. K. Okamoto,Math. Annal.,275, 221–255 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Okamoto,Funkc. Ekvacioj Ser. Int. 30, 305–332 (1987).

    MathSciNet  MATH  Google Scholar 

  18. A. S. Fokas and M. J. Ablowitz,J. Math. Phys.,23, 2033–2042 (1982).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. A. S. Fokas, U. Mugan, and M. J. Ablowitz,Physica D,30, 247–283 (1988).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. V. I. Gromak,Diff. Equat.,11, 285–287 (1976).

    MATH  Google Scholar 

  21. U. Mugan and A. S. Fokas,J. Math. Phys.,33, 2031–2045 (1992).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. H. Flaschka and A. C. Newell,Commun. Math. Phys.,76, 65–116 (1980).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. A. R. Its and V. Yu. Novokshenov,The Isomonodromic Deformation Method in the Theory of Painlevé Equations (Lect. Notes Math., Vol. 1191), Springer, Berlin (1986).

    MATH  Google Scholar 

  24. K. Okamoto,J. Fac. Sci. Univ. Tokyo Sec. IA Math.,33, 575–618 (1986).

    MathSciNet  MATH  Google Scholar 

  25. A. P. Magnus,J. Comput. Appl. Math.,57, 215–237 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  26. E. Brézin and V. A. Kazakov,Phys. Lett. B,236, 144–150 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  27. D. J. Gross and A. A. Migdal,Phys. Rev. Lett.,64, 127–130 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. V. Periwal and D. Shevitz,Phys. Rev. Lett.,64, 1326–1329 (1990).

    Article  ADS  Google Scholar 

  29. B. Grammaticos, A. Ramani, and V. Papageorgiou:Phys. Rev. Lett.,67, 1825–1828 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. A. Ramani, B. Grammaticos, and J. Hietarinta,Phys. Rev. Lett.,67, 1829–1832 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. J. Hietarinta and C. Viallet,Phys. Rev. Lett.,81, 325–328 (1998).

    Article  ADS  Google Scholar 

  32. M. Jimbo and H. Sakai,Lett. Math. Phys.,38, 145–154 (1996).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  33. M. Jimbo, H. Sakai, A. Ramani, and B. Grammaticos,Phys. Lett. A,217, 111–118 (1996).

    Article  ADS  MATH  Google Scholar 

  34. B. Grammaticos and A. Ramani, “The hunting for the discrete Painlevé VI is over,” Preprint solv-int/9901006 (1999).

  35. A. P. Bassom and P. A. Clarkson,Phys. Lett. A,194, 358–370 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. P. A. Clarkson and H. N. Webster, “Hierarchies of exact solutions for the discrete third Painlevé equation,”Chaos, Solitons, and Fractals (forthcoming).

  37. C. Cresswell and N. Joshi,J. Phys. A,32, 655–669 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. B. Grammaticos, F. W. Nijhoff, V. Papageorgiou, A. Ramani, and J. Satsuma,Phys. Lett. A,185, 446–452 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  39. B. Grammaticos, F. W. Nijhoff and A. Ramani, “Discrete Painlevé equations,” in:The Painlevé Property, One Century Later (R. Conte, ed.) (CRM Series in Math. Phys.), Springer, New York, pp. 413–416.

  40. J. Hietarinta and K. Kajiwara, “Rational solutions to d-PIV”, in:Symmetries and Integrability of Difference Equations (P. A. Clarkson and F. W. Nijhoff, eds.) (LMS Lect. Notes Series, Vol. 255), Cambridge Univ. Press, Cambridge (1999), pp. 206–216.

    Google Scholar 

  41. N. Joshi, A. Ramani, and B. Grammaticos,Phys. Lett. A,249, 59–62 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. K. Kajiwara, “The discrete Painlevé II equation and the classical special functions”, in:Symmetries and Integrability of Difference Eduations (P. A. Clarkson and F. W. Nijhoff, eds.) (LMS Lect. Notes Series, Vol. 255), Cambridge Univ. Press, Cambridge (1999), pp. 217–227.

    Google Scholar 

  43. K. Kajiwara, Y. Ohta, and J. Satsuma,J. Math. Phys.,36, 4162–4174 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. K. Kajiwara, Y. Ohta, J. Satsuma, B. Grammaticos, and A. Ramani,J. Phys. A,27, 915–922 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. K. Kajiwara, K. Yamamoto, and Y. Ohta,Phys. Lett. A,232, 189–199 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. F. W. Nijhoff, J. Satsuma, K. Kajiwara, B. Grammaticos, and A. Ramani,Inverse Problems,12, 697–716 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Y. Ohta, K. Kajiwara, and J. Satsuma, “Bilinear structure and exact solutions of the discrete Painlevé I equation,” in:Symmetries and Integrability of Difference Equations (P. Winternitz and D. Levi, eds.) (CRM Proc. Lect. Notes, Vol. 9), Am. Math. Soc., Providence, RI (1996), pp. 265–268.

    Google Scholar 

  48. Y. Ohta, A. Ramani, B. Grammaticos, and K. M. Tamizhmani,Phys. Lett. A.,216, 255–261 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  49. A. Ramani and B. Grammaticos,Physica A,228, 160–171 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. A. Ramani, B. Grammaticos, and J. Satsuma,J. Phys. A,28, 4655–4665 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. A. Ramani, Y. Ohta, J. Satsuma, and B. Grammaticos,Commun. Math. Phys.,192, 67–76 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. J. Satsuma, K. Kajiwara, B. Grammaticos, J. Hietarinta, and A. Ramani,J. Phys. A,28, 3541–3548 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. K. M. Tamizhmani, B. Grammaticos, and A. Ramani,Lett. Math. Phys.,29, 49–54 (1993).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  54. K. M. Tamizhmani, A. Ramani, B. Grammaticos, and K. Kajiwara,J. Phys. A,31, 5799–5810 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. K. M. Tamizhmani, A. Ramani, B. Grammaticos, and Y. Ohta,Lett. Math. Phys.,38, 289–296 (1996).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  56. A. S. Fokas, B. Grammaticos, and A. Ramani,J. Math. Anal. Appl.,180, 342–360 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  57. B. Grammaticos and A. Ramani, “Discrete Painlevé equations: derivation and properties,” in:Application of Analytic and Geometric Methods to Nonlinear Differential Equations (P. A. Clarkson, ed.) (NATO ASI Series C, Vol. 413), Kluwer, Dordrecht (1993), pp. 299–313.

    Google Scholar 

  58. B. Grammaticos and A. Ramani,J. Phys. A.,31, 5787–5798 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. V. I. Gromak and V. V. Tsegel'nik,Diff. Equat.,30, 1037–1043 (1994).

    MathSciNet  MATH  Google Scholar 

  60. V. I. Gromak and V. V. Tsegel'nik,Diff. Equat.,32, 1024–1029 (1996).

    MathSciNet  MATH  Google Scholar 

  61. V. V. Tsegel'nik,Theor. Math. Phys.,102, 265–266 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  62. V. V. Tsegel'nik,Diff. Equat.,32, 1433–1435 (1996).

    MathSciNet  MATH  Google Scholar 

  63. E. E. Whittaker and G. M. Watson,Modern Analysis (4th ed.), Cambridge Univ. Press, Cambridge (1927).

    MATH  Google Scholar 

  64. N. A. Lukashevich,Diff. Equat.,7, 853–854 (1971).

    Google Scholar 

  65. N. A. Lukashevich,Diff. Equat.,3, 395–399 (1967).

    Google Scholar 

  66. A. V. Kitaev, private communication (1991).

  67. A. S. Fokas, A. R. Its, and A. V. Kitaev,Commun. Math. Phys.,142, 313–344 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  68. A. P. Bassom, P. A. Clarkson, and A. C. Hicks,Adv. Diff. Equat.,1, 175–198 (1995).

    MathSciNet  Google Scholar 

  69. H. Umemura and H. Watanabe,Nagoya Math. J.,148, 151–198 (1997).

    MathSciNet  MATH  Google Scholar 

  70. P. A. Clarkson,Eur. J. Appl. Math.,1, 279–300 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  71. V. I. Gromak, “Backlund transformations of Painlevé equations and their applications,” in:The Painlevé Property, One Century Later (R. Conte, ed.) (CRM Series in Math. Phys.), Springer, New York, pp. 687–734.

  72. N. A. Lukashevich,Diff. Equat.,1, 561–564 (1965).

    Google Scholar 

  73. N. A. Lukashevich,Diff. Equat.,3, 994–999 (1967).

    Google Scholar 

  74. E. L. Mansfield and H. N. Webster,Stud. Appl. Math.,101, 321–341 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  75. M. Noumi, S. Okada, K. Okamoto, and H. Umemura, “Special polynomials associated with the Painlevé equation H,” in:Integrable Systems and Algebraic Geometry (M.-H. Saito, Y. Shimizu, and R. Ueno, eds.), World Scientific, Singapore (1998), pp. 349–372.

    Google Scholar 

  76. H. Umemura, “Special polynomials associated with the Painlevé equations I,” in:Proc. Workshop “Painlevé Transcendents” (Montreal, Canada, 1996) (forthcoming).

  77. H. Umemura and H. Watanabe,Nagoya Math. J.,151, 1–24 (1998).

    MathSciNet  MATH  Google Scholar 

  78. M. Noumi and Y. Yamada,Commun. Math. Phys.,199, 281–295 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  79. M. Noumi and Y. Yamada,Phys. Lett. A,247, 65–69 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  80. M. Noumi and Y. Yamada, “Symmetries in the fourth Painlevé equation and Okamoto polynomials,”Nagoya Math. J (forthcoming); Preprint q-alg/9708018 (1997).

  81. P. A. Clarkson, N. Joshi, and A. Pickering,Inverse Problems,15, 175–187 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Author notes

  1. H. N. Webster

    Present address: Meteorological Office, London Road, Bracknell, UK

Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of Kent, Canterbury, UK

    P. A. Clarkson, E. L. Mansfield & H. N. Webster

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  1. P. A. Clarkson
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  2. E. L. Mansfield
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 5–22, January, 1999.

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Clarkson, P.A., Mansfield, E.L. & Webster, H.N. On the relation between the continuous and discrete Painlevé equations. Theor Math Phys 122, 1–16 (2000). https://doi.org/10.1007/BF02551165

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