Abstract
We consider the application of the Weiss-Tabor-Carnevale (WTC) Painlevé test to hierarchies of completely integrable evolution equations. A method of constructing the Painlevé index polynomial for such hierarchies is illustrated. For Burgers' hierarchy we are able to show that all WTC compatibility conditions are satisfied. This allows a simple construction of the Painlevé-Bäcklund transformation obtained from truncation of the principal Painlevé expansion.
Similar content being viewed by others
References
P. Painlevé, Bull. Soc. Math. France,28, 201 (1900).
P. Painlevé, Acta Math.,25, 1 (1902).
W. F. Osgood, “Topics in the Theory of Functions of Several Complex Variables” (“The Madison Colloquium, 1913, Part II”, American Mathematical Society, Rhode Island, 1914). Reprinted. Dover, New York, 1966.
R. S. Ward, Phys. Lett. A,102, 279 (1984).
J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys.,24, 522 (1983).
A. P. Fordy and A. Pickering, Phys. Lett. A,160, 347 (1991).
R. Conte, A. P. Fordy, and A. Pickering, Physica D,69, 33 (1993).
A. C. Newell, M. Tabor, and Y. B. Zeng, Physica D,29, 1 (1987).
P. J. Olver, J. Math. Phys.18, 1212 (1977).
B. Fuchssteiner, Nonlinear Anal. Theory Meth. Appl.,3, 849 (1979).
W. Strampp, Prog. Theor. Phys.,80, 384 (1988).
A. Pickering, J. Math. Phys.35, 821 (1994).
A. Pickering, “Painlevé indices for integrable hierarchies,” in preparation.
D. J. Kaup, Stud. Appl. Math.,62, 189 (1980).
A. P. Fordy and J. Gibbons, Phys. Lett. A,75, 325 (1980).
H. Yoshida, Cel. Mech.,31, 363 (1983).
A. P. Fordy and A. Pickering, in “Chaotic Dynamics: Theory and Practice,” ed. T. Bountis, Plenum, New York, 1992.
A. Pickering, Ph. D. thesis, University of Leeds, 1992.
D. V. Choodnovsky and G. V. Choodnovsky, Il Nuovo Cimento B,40, 339 (1977).
A. S. Fokas, Ph. D. thesis, California Institute of Technology, 1979.
Additional information
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 509–516, June, 1994.
Rights and permissions
About this article
Cite this article
Pickering, A. Integrable hierarchies: Painlevé indices and compatibility conditions. Theor Math Phys 99, 761–766 (1994). https://doi.org/10.1007/BF01017065
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01017065