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

SIGMA 7 (2011), 045, 22 pages      arXiv:1010.3036      https://doi.org/10.3842/SIGMA.2011.045

The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I

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

Received November 26, 2010, in final form May 03, 2011; Published online May 07, 2011

Abstract
We wish to explore a link between the Lax integrability of the q-Painlevé equations and the symmetries of the q-Painlevé equations. We shall demonstrate that the connection preserving deformations that give rise to the q-Painlevé equations may be thought of as elements of the groups of Schlesinger transformations of their associated linear problems. These groups admit a very natural lattice structure. Each Schlesinger transformation induces a Bäcklund transformation of the q-Painlevé equation. Each translational Bäcklund transformation may be lifted to the level of the associated linear problem, effectively showing that each translational Bäcklund transformation admits a Lax pair. We will demonstrate this framework for the q-Painlevé equations up to and including q-PVI.

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

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References

  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., General theory of linear difference equations, Trans. Amer. Math. Soc. 12 (1911), 243-284.
  4. 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.
  5. 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.
  6. Carmichael R.D., The general theory of linear q-difference equations, Amer. J. Math. 34 (1912), 147-168.
  7. Di Vizio L., Ramis J.-P., Sauloy J., Zhang C., Équations aux q-différences, Gaz. Math. 96 (2003), 20-49.
  8. Fuchs R., Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen, Math. Ann. 63 (1907), 301-321.
  9. Fuchs R., Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen, Math. Ann. 70 (1911), 525-549.
  10. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge, 1990.
  11. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  12. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III, Phys. D 4 (1981), 26-46.
  13. 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.
  14. Jimbo M., Sakai H., A q-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145-154, chao-dyn/9507010.
  15. Joshi N., Burtonclay D., Halburd R.G., Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem, Lett. Math. Phys. 26 (1992), 123-131.
  16. Le Caine J., The linear q-difference equation of the second order, Amer. J. Math. 65 (1943), 585-600.
  17. Murata M., Lax forms of the q-Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 115201, 17 pages, arXiv:0810.0058.
  18. Noumi M., An introduction to birational Weyl group actions, in Symmetric Functions 2001: Surveys of Developments and Perspectives, NATO Sci. Ser. II Math. Phys. Chem., Vol. 74, Kluwer Acad. Publ., Dordrecht, 2002, 179-222.
  19. Noumi M., Yamada Y., A new Lax pair for the sixth Painlevé equation associated with ^so(8), in Microlocal Analysis and Complex Fourier Analysis, World Sci. Publ., River Edge, NJ, 2002, 238-252, math-ph/0203029.
  20. Ormerod C.M., A study of the associated linear problem for q-PV, arXiv:0911.5552.
  21. 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.
  22. Ramani A., Grammaticos B., Hietarinta J., Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), 1829-1832.
  23. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  24. Sakai H., A q-analog of the Garnier system, Funkcial. Ekvac. 48 (2005), 273-297.
  25. Sakai H., Lax form of the q-Painlevé equation associated with the A2(1) surface, J. Phys. A: Math. Gen. 39 (2006), 12203-12210.
  26. Sauloy J., Galois theory of Fuchsian q-difference equations, Ann. Sci. École Norm. Sup. (4) 36 (2003), 925-968, math.QA/0210221.
  27. Shioda T., Takano K., On some Hamiltonian structures of Painlevé systems. I, Funkcial. Ekvac. 40 (1997), 271-291.
  28. Trjitzinsky W.J., Analytic theory of linear q-difference equations, Acta Math. 61 (1933), 1-38.
  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.
  30. van der Put M., Singer M.F., Galois theory of difference equations, Lecture Notes in Mathematics, Vol. 1666, Springer-Verlag, Berlin, 1997.
  31. Yamada Y., Lax formalism for q-Painlevé equations with affine Weyl group symmetry of type En(1), Int. Math. Res. Not., to appear, arXiv:1004.1687.
  32. Yamada Y., A Lax formalism for the elliptic difference Painlevé equation, SIGMA 5 (2009), 042, 15 pages, arXiv:0811.1796.

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