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

SIGMA 10 (2014), 002, 19 pages      arXiv:1308.4233      https://doi.org/10.3842/SIGMA.2014.002
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation

Christopher M. Ormerod
Department of Mathematics, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA

Received September 19, 2013, in final form December 28, 2013; Published online January 03, 2014

Abstract
We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlevé equation with $E_6^{(1)}$ symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlevé equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlevé equation.

Key words: difference equations; integrability; reduction; isomonodromy.

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References

  1. Ablowitz M.J., Segur H., Exact linearization of a Painlevé transcendent, Phys. Rev. Lett. 38 (1977), 1103-1106.
  2. Arinkin D., Borodin A., Moduli spaces of $d$-connections and difference Painlevé equations, Duke Math. J. 134 (2006), 515-556, math.AG/0411584.
  3. Birkhoff G.D., General theory of linear difference equations, Trans. Amer. Math. Soc. 12 (1911), 243-284.
  4. Boll R., Classification of 3D consistent quad-equations, J. Nonlinear Math. Phys. 18 (2011), 337-365, arXiv:1009.4007.
  5. Borodin A., Isomonodromy transformations of linear systems of difference equations, Ann. of Math. 160 (2004), 1141-1182, math.CA/0209144.
  6. Clarkson P.A., Painlevé equations - nonlinear special functions, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Springer, Berlin, 2006, 331-411.
  7. Clarkson P.A., Kruskal M.D., New similarity reductions of the Boussinesq equation, J. Math. Phys. 30 (1989), 2201-2213.
  8. Doliwa A., Non-commutative lattice-modified Gel'fand-Dikii systems, J. Phys. A: Math. Theor. 46 (2013), 205202, 14 pages, arXiv:1302.5594.
  9. Dzhamay A., Sakai H., Takenawa T., Discrete Hamiltonian structure of Schlesinger transformations, arXiv:1302.2972.
  10. Flaschka H., Newell A.C., Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), 65-116.
  11. Hay M., Hierarchies of nonlinear integrable $q$-difference equations from series of Lax pairs, J. Phys. A: Math. Theor. 40 (2007), 10457-10471.
  12. Hay M., Hietarinta J., Joshi N., Nijhoff F.W., 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.
  13. Hay M., Howes P., Shi Y., A systematic approach to reductions of type-Q ABS equations, arXiv:1307.3390.
  14. Hay M., Kajiwara K., Masuda T., Bilinearization and special solutions to the discrete Schwarzian KdV equation, J. Math-for-Ind. 3A (2011), 53-62, arXiv:1102.1829.
  15. Hone A.N.W., van der Kamp P.H., Quispel G.R.W., Tran D.T., Integrability of reductions of the discrete Korteweg-de Vries and potential Korteweg-de Vries equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), 20120747, 23 pages, arXiv:1211.6958.
  16. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  17. Jimbo M., Sakai H., A $q$-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145-154, chao-dyn/9507010.
  18. Kajiwara K., The hypergeometric solutions of the additive Painlevé equations with $E$-type affine Weyl symmetry, Reports of RIAM Symposium No. 19ME-S2 (in Japanese).
  19. 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.
  20. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., ${}_{10}E_9$ solution to the elliptic Painlevé equation, J. Phys. A: Math. Gen. 36 (2003), L263-L272, nlin.SI/0303032.
  21. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Construction of hypergeometric solutions to the $q$-Painlevé equations, Int. Math. Res. Not. 2005 (2005), 1441-1463, nlin.SI/0501051.
  22. 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.
  23. Kruskal M.D., Tamizhmani K.M., Grammaticos B., Ramani A., Asymmetric discrete Painlevé equations, Regul. Chaotic Dyn. 5 (2000), 273-280.
  24. Murata M., New expressions for discrete Painlevé equations, Funkcial. Ekvac. 47 (2004), 291-305, nlin.SI/0304001.
  25. Murata M., Lax forms of the $q$-Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 115201, 17 pages, arXiv:0810.0058.
  26. Murata M., Sakai H., Yoneda J., Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type $E^{(1)}_8$, J. Math. Phys. 44 (2003), 1396-1414, nlin.SI/0210040.
  27. Nijhoff F., Capel H., The discrete Korteweg-de Vries equation, Acta Appl. Math. 39 (1995), 133-158.
  28. Nijhoff F., Hone A., Joshi N., On a Schwarzian PDE associated with the KdV hierarchy, Phys. Lett. A 267 (2000), 147-156, solv-int/9909026.
  29. Nijhoff F., Joshi N., Hone A., On the discrete and continuous Miura chain associated with the sixth Painlevé equation, Phys. Lett. A 264 (2000), 396-406, solv-int/9906006.
  30. Nijhoff F.W., Papageorgiou V.G., Similarity reductions of integrable lattices and discrete analogues of the Painlevé II equation, Phys. Lett. A 153 (1991), 337-344.
  31. Nijhoff F.W., Quispel G.R.W., Capel H.W., Direct linearization of nonlinear difference-difference equations, Phys. Lett. A 97 (1983), 125-128.
  32. Nijhoff F.W., Ramani A., Grammaticos B., Ohta Y., On discrete Painlevé equations associated with the lattice KdV systems and the Painlevé VI equation, Stud. Appl. Math. 106 (2001), 261-314, solv-int/9812011.
  33. Nijhoff F.W., Walker A.J., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasg. Math. J. 43A (2001), 109-123, nlin.SI/0001054.
  34. Noumi M., Special functions arising from discrete Painlevé equations: a survey, J. Comput. Appl. Math. 202 (2007), 48-55.
  35. Ormerod C.M., The lattice structure of connection preserving deformations for $q$-Painlevé equations I, SIGMA 7 (2011), 045, 22 pages, arXiv:1010.3036.
  36. Ormerod C.M., A study of the associated linear problem for $q$-${\rm P}_{\rm V}$, J. Phys. A: Math. Theor. 44 (2011), 025201, 26 pages, arXiv:0911.5552.
  37. Ormerod C.M., Reductions of lattice mKdV to $q$-${\rm P}_{\rm VI}$, Phys. Lett. A 376 (2012), 2855-2859, arXiv:1112.2419.
  38. Ormerod C.M., van der Kamp P.H., Hietarinta J., Quispel G.R.W., Twisted reductions of integrable lattice equations, and their Lax representations, arXiv:1307.5208.
  39. Ormerod C.M., van der Kamp P.H., Quispel G.R.W., Discrete Painlevé equations and their Lax pairs as reductions of integrable lattice equations, J. Phys. A: Math. Theor. 46 (2013), 095204, 22 pages, arXiv:1209.4721.
  40. Ormerod C.M., Witte N.S., Forrester P.J., Connection preserving deformations and $q$-semi-classical orthogonal polynomials, Nonlinearity 24 (2011), 2405-2434, arXiv:0906.0640.
  41. Papageorgiou V.G., Nijhoff F.W., Capel H.W., Integrable mappings and nonlinear integrable lattice equations, Phys. Lett. A 147 (1990), 106-114.
  42. 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.
  43. Praagman C., Fundamental solutions for meromorphic linear difference equations in the complex plane, and related problems, J. Reine Angew. Math. 369 (1986), 101-109.
  44. Rains E.M., An isomonodromy interpretation of the hypergeometric solution of the elliptic Painlevé equation (and generalizations), SIGMA 7 (2011), 088, 24 pages, arXiv:0807.0258.
  45. Ramani A., Carstea A.S., Grammaticos B., On the non-autonomous form of the $Q_4$ mapping and its relation to elliptic Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 322003, 8 pages.
  46. Ramani A., Grammaticos B., Tamizhmani T., Tamizhmani K.M., Special function solutions of the discrete Painlevé equations, Comput. Math. Appl. 42 (2001), 603-614.
  47. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  48. Sakai H., Hypergeometric solution of $q$-Schlesinger system of rank two, Lett. Math. Phys. 73 (2005), 237-247.
  49. Sakai H., Lax form of the $q$-Painlevé equation associated with the $A^{(1)}_2$ surface, J. Phys. A: Math. Gen. 39 (2006), 12203-12210.
  50. Tran D.T., van der Kamp P.H., Quispel G.R.W., Involutivity of integrals of sine-Gordon, modified KdV and potential KdV maps, J. Phys. A: Math. Theor. 44 (2011), 295206, 13 pages, arXiv:1010.3471.
  51. van der Kamp P.H., Quispel G.R.W., The staircase method: integrals for periodic reductions of integrable lattice equations, J. Phys. A: Math. Theor. 43 (2010), 465207, 34 pages, arXiv:1005.2071.
  52. Witte N.S., The correspondence between the Askey table of orthogonal polynomial systems and the Sakai scheme of discrete Painlevé equations, in preparation.
  53. Witte N.S., Ormerod C.M., Construction of a Lax pair for the $E_6^{(1)}$ $q$-Painlevé system, SIGMA 8 (2012), 097, 27 pages, arXiv:1207.0041.
  54. Yamada Y., Padé method to Painlevé equations, Funkcial. Ekvac. 52 (2009), 83-92.
  55. Yamada Y., Lax formalism for $q$-Painlevé equations with affine Weyl group symmetry of type $E^{(1)}_n$, Int. Math. Res. Not. 2011 (2011), 3823-3838, arXiv:1004.1687.

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