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

SIGMA 7 (2011), 049, 13 pages      arXiv:1101.5422
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

Symmetries in Connection Preserving Deformations

Christopher M. Ormerod
La Trobe University, Department of Mathematics and Statistics, Bundoora VIC 3086, Australia

Received January 31, 2011, in final form May 18, 2011; Published online May 24, 2011

We wish to show that the root lattice of Bäcklund transformations of the q-analogue of the third and fourth Painlevé equations, which is of type (A2+A1)(1), may be expressed as a quotient of the lattice of connection preserving deformations. Furthermore, we will show various directions in the lattice of connection preserving deformations present equivalent evolution equations under suitable transformations. These transformations correspond to the Dynkin diagram automorphisms.

Key words: q-Painlevé; Lax pairs; q-Schlesinger transformations; connection; isomonodromy.

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  1. Adams R.C., On the linear ordinary q-difference equation, Ann. of Math. 30 (1928), 195-205.
  2. Bellon M.P., Viallet C.-M., Algebraic entropy, Comm. Math. Phys. 204 (1999), 425-437, chao-dyn/9805006.
  3. Birkhoff G.D., The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. Amer. Acad. 49 (1913), 512-568.
  4. Birkhoff G.D., Guenther P.E., Note on a canonical form for the linear q-difference system, Proc. Nat. Acad. Sci. USA 27 (1941), 218-222.
  5. Birkhoff G.D., Trjitzinsky W.J., Analytic theory of singular difference equations, Acta Math. 60 (1933), 1-89.
  6. Forrester P.J., Ormerod C.M., Witte N.S., Connection preserving deformations and q-semi-classical orthogonal polynomials, arXiv:0906.0640.
  7. Grammaticos B., Ramani A., Papageorgiou V., Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), 1825-1828.
  8. Hay M., Hietarinta J., Joshi N., Nijhoff F., A Lax pair for a lattice modified KdV equation, reductions to q-Painlevé equations and associated Lax pairs, J. Phys. A: Math. Theor. 40 (2007), F61-F73.
  9. Its A.R., Kitaev A.V., Fokas A.S., An isomonodromy approach to the theory of two-dimensional quantum gravity, Russian Math. Surveys 45 (1990), no. 6, 155-157.
  10. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  11. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III, Phys. D 4 (1981), 26-46.
  12. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function, Phys. D 2 (1981), 306-352.
  13. Jimbo M., Sakai H., A q-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145-154, chao-dyn/9507010.
  14. Kajiwara K., Kimura K., On a q-difference Painlevé III equation. I. Derivation, symmetry and Riccati type solutions, J. Nonlinear Math. Phys. 10 (2003), 86-102, nlin.SI/0205019.
  15. Kajiwara K., Noumi M., Yamada Y., A study on the fourth q-Painlevé equation, J. Phys. A: Math. Gen. 34 (2001), 8563-8581, nlin.SI/0012063.
  16. Murata M., Lax forms of the q-Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 115201, 17 pages, arXiv:0810.0058.
  17. Noumi M., Yamada Y., Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys. 199 (1998), 281-295, math.QA/9804132.
  18. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, PII and PIV, Math. Ann. 275 (1986), 221-255.
  19. Okamoto K., Studies on the Painlevé equations. I. Sixth Painlevé equation PVI, Ann. Mat. Pura Appl. (4) 146 (1987), 337-381.
  20. Okamoto K., Studies on the Painlevé equations. II. Fifth Painlevé equation PV, Japan. J. Math. (N.S.) 13 (1987), 47-76.
  21. Okamoto K., Studies on the Painlevé equations. IV. Third Painlevé equation PIII, Funkcial. Ekvac. 30 (1987), 305-332.
  22. Ormerod C., A study of the associated linear problem for q-PV, J. Phys. A: Math. Theor. 44 (2011), 025201, 26 pages, arXiv:0911.5552.
  23. Ormerod C.M., The lattice structure of connection preserving deformations for q-Painlevé equations I, SIGMA 7 (2011), 045, 22 pages, arXiv:1010.3036.
  24. Papageorgiou V.G., Nijhoff F.W., Grammaticos B., Ramani A., Isomonodromic deformation problems for discrete analogues of Painlevé equations, Phys. Lett. A 164 (1992), 57-64.
  25. Ramani A., Grammaticos B., Hietarinta J., Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), 1829-1832.
  26. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  27. Sauloy J., Galois theory of Fuchsian q-difference equations, Ann. Sci. École Norm. Sup. (4) 36 (2003), 925-968, math.QA/0210221.
  28. Sauloy J., Algebraic construction of the Stokes sheaf for irregular linear q-difference equations, in Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. I, Astérisque (2004), no. 296, 227-251.
  29. van der Put M., Reversat M., Galois theory of q-difference equations, Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), 665-718, math.QA/0507098.

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