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

SIGMA 6 (2010), 034, 14 pages      arXiv:1004.2945
Contribution to the Proceedings of the Eighth International Conference Symmetry in Nonlinear Mathematical Physics

The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces

Oksana Ye. Hentosh
Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3B Naukova Str., Lviv, 79060, Ukraine

Received November 16, 2009, in final form February 24, 2010; Published online April 17, 2010

The Hamiltonian representation for the hierarchy of Lax-type flows on a dual space to the Lie algebra of shift operators coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is found by means of a specially constructed Bäcklund transformation. The Hamiltonian description for the corresponding set of squared eigenfunction symmetry hierarchies is represented. The relation of these hierarchies with Lax integrable (2+1)-dimensional differential-difference systems and their triple Lax-type linearizations is analysed. The existence problem of a Hamiltonian representation for the coupled Lax-type hierarchy on a dual space to the central extension of the shift operator Lie algebra is solved also.

Key words: Lax integrable differential-difference systems; Bäcklund transformation; squared eigenfunction symmetries.

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  1. Ablowitz M.J., Segur H., Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981.
  2. Adler M., On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de Vries type equations, Invent. Math. 50 (1979), 219-248.
  3. Aratyn H., Nissimov E., Pacheva S., Supersymmetric Kadomtsev-Petviashvili hierarchy: "ghost" symmetry structure, reductions, and Darboux-Bäcklund solutions, J. Math. Phys. 40 (1999), 2922-2932, solv-int/9801021.
  4. Blaszak M., Marciniak K., r-matrix approach to lattice integrable systems, J. Math. Phys. 35 (1994), 4661-4682.
  5. Blaszak M., Multi-Hamiltonian theory of dynamical systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1998.
  6. Blaszak M., Szum A., Prykarparsky A., Central extension approach to integrable field and lattice-field systems in (2+1)-dimensions, Rep. Math. Phys. 44 (1999), 37-44.
  7. Blaszak M., Szum A., Lie algebraic approach to the construction of (2+1)-dimensional lattice-field and field integrable Hamiltonian equations, J. Math. Phys. 42 (2001), 225-259.
  8. Blaszak M., Szablikowski B.M., Classical R-matrix theory for bi-Hamiltonian field systems, J. Phys. A: Math. Theor. 42 (2009), 404002, 35 pages, arXiv:0902.1511.
  9. Bobenko A.I., Matveev V.B., Sall' M.A., Nonlocal analogues of the Korteweg-de Vries and Kadomtsev-Petviashvili equations, Dokl. Akad. Nauk SSSR 265 (1982), 1357-1360 (in Russian).
  10. Bogoyavlensky O.I., On perturbations of the periodic Toda lattice, Comm. Math. Phys. 51 (1976), 201-209.
  11. Bogoyavlensky O.I., Overturned solitons, Nauka, Moscow, 1991 (in Russian).
  12. Carillo S., Oevel W., Squared eigenfunction symmetries for soliton equations. I, J. Math. Anal. Appl. 217 (1998), 161-178.
  13. Carillo S., Oevel W., Squared eigenfunction symmetries for soliton equations. II, J. Math. Anal. Appl. 217 (1998), 179-199.
  14. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987.
  15. Fuchssteiner B., Fokas A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981), 47-66.
  16. Hentosh O.Ye., Lax integrable supersymmetric hierarchies on extended phase spaces, SIGMA 2 (2006), 001, 11 pages, nlin.SI/0601007.
  17. Hentosh O., Prytula M., Prykarpatsky A., Differential-geometric and Lie-algebraic foundations of studying integrable nonlinear dynamical systems on functional manifolds, Lviv National University, Lviv, 2006 (in Ukrainian).
  18. Hentosh O.Ye., Prykarpatsky A.K., Integrable three-dimensional coupled nonlinear dynamical systems related with centrally extended operator Lie algebras, Opuscula Math. 27 (2007), 231-244.
  19. Hentosh O.Ye., Lax integrable supersymmetric hierarchies on extended phase spaces of two anticommuting variables, in Modern Analysis and Applications (The Mark Krein Centenary Conference), Vol. 2, Differential Operators and Mechanics, Oper. Theory Adv. Appl., Vol. 191, Birkhäuser Verlag, Basel, 2009, 365-379.
  20. Kajiwara K., Satsuma J., The conserved quantities and symmetries of two-dimensional Toda lattice hierarchy, J. Math. Phys. 32 (1991), 506-514.
  21. Kostant B., The solution to a generalized Toda lattice and representation theory, Adv. in Math. 34 (1979), 195-338.
  22. Kupershmidt B.A. Discrete Lax equations and differential-difference calculus, Astérisque (1985), no. 123, 21 pages.
  23. Lax P.D., Periodic solutions of the KdV equation, Comm. Pure Appl. Math. 28 1975, 141-188.
  24. Ma W.-X., Zhou X., Binary symmetry constraints of N-wave iteraction in 1+1 and 2+1 dimensions, J. Math. Phys. 42 (2001), 4345-4382, nlin.SI/0105061.
  25. Ma W.-X., Geng X., Bäcklund transformations of soliton systems from symmetry constraints, nlin.SI/0107071.
  26. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.
  27. Morosi C., Pizzocchero L., r-matrix theory, formal Casimirs and the periodic Toda lattice, J. Math. Phys. 37 (1996), 4484-4513.
  28. Nissimov E., Pacheva S., Symmetries of supersymmetric integrable hierarchies of KP type, J. Math. Phys. 43 (2002), 2547-2586, nlin.SI/0102010.
  29. Oevel W., R structures, Yang-Baxter equations, and related involution theorems, J. Math. Phys. 30 (1989), 1140-1149.
  30. Oevel W., Strampp W., Constrained KP hierarchy and bi-Hamiltonian structures, Comm. Math. Phys. 157 (1993), 51-81.
  31. Oevel W., Poisson brackets for integrable lattice systems, in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., Vol. 26, Birkhäuser Boston, Boston, MA, 1997, 261-283.
  32. Ogawa Yu., On the (2+1)-dimensional extention of 1-dimensional Toda lattice hierarchy, J. Nonlinear Math. Phys. 15 (2008), 48-65.
  33. Perelomov A.M., Integrable systems of classical mechanics and Lie algebras, Nauka, Moscow, 1990 (in Russian).
  34. Prykarpatsky A.K., Hentosh O.E., Samoilenko V.Hr., The Lie-algebraic structure of Lax type integrable nonlocal differential-difference equations, Nonlinear Oscillations 3 (2000), no. 1, 84-94.
  35. Prykarpatsky A.K., Mykytiuk I.V., Algebraic integrability of nonlinear dynamical systems on manifolds. Classical and quantum aspects,Mathematics and its Applications, Vol. 443, Kluwer Academic Publishers Group, Dordrecht, 1998.
  36. Prykarpatsky A.K., Hentosh O.Ye., The Lie-algebraic structure of (2+1)-dimensional Lax-type integrable nonlinear dynamical systems, Ukrainian Math. J. 56 (2004), 1117-1126.
  37. Reiman A.G., Semenov-Tyan-Shanskii M.A., Integrable systems. Theoretical-group approach, Institute for Computer Investigations, Moscow - Izhevsk, 2003 (in Russian).
  38. Samoilenko A.M., Prykarpatsky Y.A., Prykarpatsky A.K., The spectral and differential geometric aspects of a generalized de Rham-Hodge theory related with Delsarte transmutation operators in multidimension and its applications to spectral and soliton problems, Nonlinear Anal. 65 (2006), 395-4322.
  39. Sato M., Soliton equations as dynamical systems on an infinite dimensional Grassmann manifolds, in Random Systems and Dynamical Systems (Proc. Symp., Kyoto 1981), RIMS Kokyuroku 439 (1981), 30-46.
  40. Semenov-Tyan-Shanskii M.A., What is a classical r-matrix?, Funct. Anal. Appl. 17 (1983), no. 4, 259-272.
  41. Suris Yu., Miura transformations of Toda-type integrable systems with applications to the problem of integrable discretizations, solv-int/9902003.
  42. Symes W.W., Systems of Toda type, inverse spectral problem, and representation theory, Invent. Math. 59 (1980), 13-51.
  43. Tamizhmani K.M., Kanaga Vel S., Differential-difference Kadomtsev-Petviashvili equation: properties and integrability, J. Indian Inst. Sci. 78 (1998), 311-372.
  44. Yao Y., Liu X., Zeng Y., A new extended discrete KP hierarchy and a generalized dressing method, J. Phys. A: Math. Theor. 42 (2009), 454026, 10 pages, arXiv:0907.2783.
  45. Zeng Y.B., Shao Y.J., Xue W.M., Positon solutions of the KdV equation with self-consistent sources, Theoret. and Math. Phys. 137 (2003), 1622-1631.

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