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


SIGMA 4 (2008), 051, 9 pages      arXiv:0806.1466      https://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.

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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.

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