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

SIGMA 3 (2007), 073, 6 pages      arXiv:0706.2719      https://doi.org/10.3842/SIGMA.2007.073

Do All Integrable Evolution Equations Have the Painlevé Property?

K.M. Tamizhmani a, Basil Grammaticos b and Alfred Ramani c
a) Departement of Mathematics, Pondicherry University, Kalapet, 605014 Puducherry, India
b) IMNC, Université Paris VII-Paris XI, CNRS, UMR 8165, Bât. 104, 91406 Orsay, France
c) Centre de Physique Théorique, Ecole Polytechnique, CNRS, 91128 Palaiseau, France

Received June 12, 2007; Published online June 19, 2007

Abstract
We examine whether the Painlevé property is necessary for the integrability of partial differential equations (PDEs). We show that in analogy to what happens in the case of ordinary differential equations (ODEs) there exists a class of PDEs, integrable through linearisation, which do not possess the Painlevé property. The same question is addressed in a discrete setting where we show that there exist linearisable lattice equations which do not possess the singularity confinement property (again in analogy to the one-dimensional case).

Key words: integrability; linearisability; Painlevé property; singularity confinement.

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References

  1. Zabusky N.J., Kruskal M.D., Interaction of solitons in a collisionless plasma and the recurrence of ther initial state, Phys. Rev. Lett. 15 (1965), 240-243.
  2. Calogero F., Degasperis A., Spectral transforms and solitons, North-Holland, Amsterdam 1982.
  3. Ablowitz M.J., Ramani A., Segur H., Nonlinear evolution equations and ordinary differential equations of Painlevé type, Lett. Nuov. Cim. 23 (1978), 333-338.
  4. Weiss J., Tabor M., Carnevale G., The Painlevé property for partial differential equations, J. Math. Phys. 24 (1983), 522-526.
  5. Jimbo M., Kruskal M.D., Miwa T., Painlevé test for self-dual Yang-Mills equation, Phys. Lett. A 92 (1982), 59-60.
  6. Grammaticos B., Ramani A., Papageorgiou V., Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), 1825-1828.
  7. Ablowitz M.J., Halburd R., Herbst B., On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), 889-905.
  8. Hietarinta J., Viallet C.M., Singularity confinement and chaos in discrete systems, Phys. Rev. Lett. 81 (1991), 325-328.
  9. Grammaticos B., Tamizhmani T., Ramani A., Tamizhmani K.M., Growth and integrability in discrete systems, J. Phys. A: Math. Gen. 34 (2001), 3811-3821.
  10. Hietarinta J., Viallet C.M., Searching for integrable lattice maps using factorization, arXiv:0705.1903.
  11. Ramani A., Grammaticos B., Tremblay S., Integrable systems without the Painlevé property, J. Phys. A: Math. Gen. 33 (2000), 3045-3052.
  12. Chazy J., Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale à ses points critiques fixes, Acta Math. 34 (1911), 317-385.
  13. Ince E.L., Ordinary differential equations, Dover, London, 1956.
  14. Ramani A., Grammaticos B., Tamizhmani K.M., Lafortune S., Again, linearisable mappings, Physica A 252 (1998), 138-150.
  15. Tsuda T., Ramani A., Grammaticos B., Takenawa T., A class of integrable and nonintegrable mappings and their dynamics, Preprint, 2007.
  16. Ramani A., Grammaticos B., Karra G., Linearisable mappings, Physica A 181 (1992), 115-127.
  17. Calogero F., Why are certain nonlinear PDEs both widely applicable and integrable?, in What is Integrability?, Editor V.E. Zakharov, Springer, New York, 1990, 1-62.

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