Letters in Mathematical Physics

, Volume 61, Issue 1, pp 1–14 | Cite as

Consistent Composition of Bäcklund Transformations Produces Confined Maps

  • C. Cresswell
  • N. Joshi
Article

Abstract

We prove that difference equations arising from consistent compositions of Bäcklund transformations of the continuous Painlevé equations possess the singularity confinement property.

Bäcklund transformations discrete equations integrability Painlevé analysis singularity confinement well-posedness 

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References

  1. 1.
    Ablowitz, M. J. and Clarkson, P. A.: Solitons, Nonlinear Evolution Equations, and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991.Google Scholar
  2. 2.
    Gross, D. J. and Migdal, A. A.: Phys. Rev. Lett. 64 (1990), 127-130.Google Scholar
  3. 3.
    Fokas, A. S., Its, A. R. and Kitaev, A. V.: Comm. Math. Phys. 142 (1991), 313-344; 147 (1992), 395-430.Google Scholar
  4. 4.
    Bessis, D., Itzykson, C. and Zuber, J.-B.: Adv. Math. 1 (1980), 109-157.Google Scholar
  5. 5.
    Shohat, J. A.: Duke Math. J. 5 (1939), 401-417.Google Scholar
  6. 6.
    Nijhoff, F. W. and Papageorgiou, V.: Phys. Lett. A 153 (1991), 337-344.Google Scholar
  7. 7.
    Papageorgiou, V., Nijhoff, F. W., Grammaticos, B. and Ramani, A.: Phys. Lett. A 164 (1992), 57-64.Google Scholar
  8. 8.
    Joshi, N., Burtonclay, D. and Halburd, R.: Lett. Math. Phys. 26 (1992), 123-131.Google Scholar
  9. 9.
    Grammaticos, B., Nijhoff, F. W. and Ramani, A.: In: The Painlevé Property, One Century Later, Springer, New York, 1999, pp. 413-516.Google Scholar
  10. 10.
    Grammaticos, B., Ramani, A. and Papageorgiou, V.: Phys. Rev. Lett. 67 (1991), 1825-1828.Google Scholar
  11. 11.
    Ramani, A., Grammaticos, B. and Hietarinta, J.: Phys. Rev. Lett. 67 (1991), 1829-1832.Google Scholar
  12. 12.
    Hietarinta, J. and Viallet, C.: Phys. Rev. Lett. 81 (1998), 325-328.Google Scholar
  13. 13.
    Joshi, N., Ramani, A. and Grammaticos, B.: Phys. Lett. A 249 (1998), 59-62.Google Scholar
  14. 14.
    Bassom, A. P., Clarkson, P. A. and Hicks, A. C.: Stud. Appl. Math. 95 (1995), 1-71.Google Scholar
  15. 15.
    Milne, A. E., Clarkson, P. A. and Bassom, A. P.: Stud. Appl. Math. 98, (1997), 139-194.Google Scholar
  16. 16.
    Fokas, A. S., Grammaticos, B. and Ramani, A.: J. Math. Anal., 180 (1993), 342-360.Google Scholar
  17. 17.
    Nijhoff, F., Satsuma, J., Kajiwara, K., Grammaticos, B. and Ramani, A.: Inverse Problems 12 (1996), 697-716.Google Scholar
  18. 18.
    Hietarinta, J. and Kruskal, M. D.: In: Painlevé Transcendents, NATO Adv. Sci. Inst. Ser. B Phys. 278, Plenum, New York, 1992, pp. 175-195.Google Scholar
  19. 19.
    Fokas, A. S. and Ablowitz, M. J.: J. Math. Phys. 23 (1982), 2033-2042.Google Scholar
  20. 20.
    Clarkson, P. A. and Bassom, A. P.: In: Symmetries and Integrability of Difference Equations, CRM Proc. Lectures Notes 9, Amer. Math. Soc., Providence, RI, 1996, pp. 63-77.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • C. Cresswell
    • 1
  • N. Joshi
    • 2
  1. 1.School of MathematicsUniversity of New South WalesUNSW SydneyAustralia
  2. 2.School of Mathematics and Statistics F07University of SydneyAustralia

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