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


SIGMA 4 (2008), 051, 9 pages      arXiv:0806.1466      http://dx.doi.org/10.3842/SIGMA.2008.051

Quantum Painlevé Equations: from Continuous to Discrete

Hajime Nagoya a, Basil Grammaticos b and Alfred Ramani c
a) Graduate School of Mathematical Sciences, The University of Tokyo, Japan
b) IMNC, Université Paris VII & XI, CNRS, UMR 8165, Bât. 104, 91406 Orsay, France
c) Centre de Physique Théorique, Ecole Polytechnique, CNRS, 91128 Palaiseau, France

Received March 05, 2008, in final form May 03, 2008; Published online June 09, 2008

Abstract
We examine quantum extensions of the continuous Painlevé equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlevé equations II, IV and V. From their auto-Bäcklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlevé equations.

Key words: discrete systems; quantization; Painlevé equations.

pdf (190 kb)   ps (138 kb)   tex (12 kb)

References

  1. Fokas A., Grammaticos B., Ramani A., From continuous to discrete Painlevé equations, J. Math. Anal. Appl. 180 (1993), 342-360.
  2. Grammaticos B., Nijhoff F.W., Papageorgiou V., Ramani A., Satsuma J., Linearization and solutions of the discrete Painlevé III equation, Phys. Lett. A 185 (1994), 446-452, solv-int/9310003.
  3. Grammaticos B., Ramani A., Papageorgiou V., Discrete dressing transformations and Painlevé equations, Phys. Lett. A 235 (1997), 475-479.
  4. Grammaticos B., Ramani A., Papageorgiou V., Nijhoff F., Quantization and integrability of discrete systems, J. Phys. A: Math. Gen. 25 (1992), 6419-6427.
  5. Grammaticos B., Ramani A., From continuous Painlevé IV to the asymmetric discrete Painlevé I, J. Phys. A: Math. Gen. 31 (1998), 5787-5798.
  6. Hietarinta J., Classical versus quantum integrability, J. Math. Phys. 25 (1984), 1833-1840.
  7. Hietarinta J., Grammaticos B., On the h2-correction terms in quantum integrability, J. Phys. A: Math. Gen. 22 (1989), 1315-1322.
  8. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  9. Nagoya H., Quantum Painlevé systems of type Al(1), Internat. J. Math. 15 (2004), 1007-1031, math.QA/0402281.
  10. Nagoya H., Quantum Painlevé systems of type An-1(1) with higher degree Lax operators, Internat. J. Math. 18 (2007), 839-868.
  11. Noumi M., Yamada Y., Higher order Painlevé equations of type Al(1), Funkcial. Ekvac. 41 (1998), 483-503, math.QA/9808003.
  12. Novikov S.P., Quantization of finite-gap potentials and a nonlinear quasiclassical approximation that arises in nonperturbative string theory, Funct. Anal. Appl. 24 (1990), 296-306.
  13. Quispel G.R.W., Nijhoff F.W., Integrable two-dimensional quantum mappings, Phys. Lett. A 161 (1992), 419-422.
  14. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations. II, Phys. D 34 (1989), 183-192.
  15. Ramani A., Willox R., Grammaticos B., Carstea A.S., Satsuma J., Limits and degeneracies of discrete Painlevé equations: a sequel, Phys. A 347 (2005), 1-16.
  16. Ramani A., Tamizhmani T., Grammaticos B., Tamizhmani K.M., The extension of integrable mappings to non-commuting variables, J. Nonlinear Math. Phys. 10 (2003), suppl. 2, 149-165.

Previous article   Next article   Contents of Volume 4 (2008)