Integrable three-dimensional coupled nonlinear dynamical systems related to centrally extended operator Lie algebras and their Lax type three-linearization

J. Golenia; O. Hentosh; A. Prykarpatsky

Open Mathematics (2007)

  • Volume: 5, Issue: 1, page 84-104
  • ISSN: 2391-5455

Abstract

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The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied.

How to cite

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J. Golenia, O. Hentosh, and A. Prykarpatsky. "Integrable three-dimensional coupled nonlinear dynamical systems related to centrally extended operator Lie algebras and their Lax type three-linearization." Open Mathematics 5.1 (2007): 84-104. <http://eudml.org/doc/269031>.

@article{J2007,
abstract = {The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied.},
author = {J. Golenia, O. Hentosh, A. Prykarpatsky},
journal = {Open Mathematics},
keywords = {Three-dimensional Lax-type flows; R-matrix approach; integrable Hamiltonian systems; Backlund transformation; Lax-type flows; -matrix approach; Bäcklund transformation; Davey-Stewartson type systems},
language = {eng},
number = {1},
pages = {84-104},
title = {Integrable three-dimensional coupled nonlinear dynamical systems related to centrally extended operator Lie algebras and their Lax type three-linearization},
url = {http://eudml.org/doc/269031},
volume = {5},
year = {2007},
}

TY - JOUR
AU - J. Golenia
AU - O. Hentosh
AU - A. Prykarpatsky
TI - Integrable three-dimensional coupled nonlinear dynamical systems related to centrally extended operator Lie algebras and their Lax type three-linearization
JO - Open Mathematics
PY - 2007
VL - 5
IS - 1
SP - 84
EP - 104
AB - The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied.
LA - eng
KW - Three-dimensional Lax-type flows; R-matrix approach; integrable Hamiltonian systems; Backlund transformation; Lax-type flows; -matrix approach; Bäcklund transformation; Davey-Stewartson type systems
UR - http://eudml.org/doc/269031
ER -

References

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  1. [1] M. Adler: “On a Trace Functional for Formal Pseudo-Differential perators and the Symplectic Structures of a Korteweg-de Vries Equation”, Invent. Math., 1979, Vol. 50(2), pp. 219–248. 
  2. [2] V.I. Arnold: Mathematical Methods of Classical Mechanics, Nauka, Moscow, 1989 (in Russian). 
  3. [3] M. Blaszak: Multi-Hamiltonian Theory of Dynamical Systems, Springer, Verlag-Berlin-Heidelberg, 1998. 
  4. [4] L. Dickey: Soliton equations and Hamiltonian systems, World Scientific, Vol. 42, 1991. 
  5. [5] O.Ye. Hentosh: “Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces”, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 1, (2005), p. 11 (to be published). 
  6. [6] P.D. Lax: “Periodic Solutions of the KdV Equation”, Commun. Pure Appl. Math., Vol. 28, (1975), pp. 141–188. http://dx.doi.org/10.1002/cpa.3160280105 Zbl0295.35004
  7. [7] S.V. Manakov: “The Method of Inverse Scattering Problem and Two-Dimensional Evolution Equations”, Adv. Math. Sci., Vol. 31(5), (1976), pp. 245–246. 
  8. [8] Yu.I. Manin and A.O. Radul: “A Supersymmetric Extension of the Kadomtsev-Petviashvili Hierarchy”, Comm. Math. Phys., Vol. 28, (1985), pp. 65–77. http://dx.doi.org/10.1007/BF01211044 Zbl0607.35075
  9. [9] V.B. Matveev and M.I. Salle: Darboux-Bäcklund transformations and applications, Springer, New York, 1993. 
  10. [10] E. Nissimov and S. Pacheva: “Symmetries of Supersymmetric Integrable Hierarchies of KP Type”, J. Math. Phys., Vol. 43, (2002), pp. 2547–2586. http://dx.doi.org/10.1063/1.1466533 Zbl1059.37054
  11. [11] S.P. Novikov (Ed.): Soliton Theory: Method of the Inverse Problem, Nauka, Moscow, 1980 (in Russian). 
  12. [12] W. Oevel: “R-Structures, Yang-Baxter Equations and Related Involution Theorems”, J. Math. Phys., Vol. 30, (1989), pp. 1140–1149. http://dx.doi.org/10.1063/1.528333 Zbl0689.35077
  13. [13] W. Oevel and Z. Popowicz: “The bi-Hamiltonian Structure of Fully Supersymmetriń Korteweg-de Vries Systems”, Comm. Math. Phys., Vol. 139, (1991), pp. 441–460. http://dx.doi.org/10.1007/BF02101874 Zbl0742.35063
  14. [14] W. Oevel, W. Strampp and K.P. Constrained: “Hierarchy and bi-Hamiltonian Structures”, Comm. Math. Phys., Vol. 157, (1993), pp. 51–81. http://dx.doi.org/10.1007/BF02098018 Zbl0793.35095
  15. [15] A.K. Prykarpatsky and O.Ye. Hentosh: “The Lie-Algebraic Structure of (2+1)-Dimensional Lax Type Integrable Nonlinear Dynamical Systems”, Ukrainian Math. J., Vol. 56, (2004), pp. 939–946. http://dx.doi.org/10.1023/B:UKMA.0000031706.91337.bd Zbl1075.37026
  16. [16] A.K. Prykarpatsky and I.V. Mykytiuk: Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Classical and Quantum Aspects, Kluwer Academic Publishers, Dordrecht-Boston-London, 1998. Zbl0937.37055
  17. [17] A.K. Prykarpatsky and D. Blackmore: “Versal deformations of a Dirac type differential operator”, J. Nonlin. Math. Phys., Vol. 6(3), (1999), pp. 246–254. Zbl1068.37046
  18. [18] A.K. Prykarpatsky, V.Hr. Samoilenko, R.I. Andrushkiw, Yu.O. Mitropolsky and M.M. Prytula: “Algebraic Structure of the Gradient-Holonomic Algorithm for Lax Integrable Nonlinear Systems. I”, J. Math. Phys., Vol. 35, (1994), pp. 1763–1777. http://dx.doi.org/10.1063/1.530569 Zbl0801.58023
  19. [19] A.M. Samoilenko and A.K. Prykarpatsky: “The spectral and differential-geometric aspects of a generalized de Rham-Hodge theory related with Delsarte transmutation operators in multi-dimension and its applications to spectral and soliton problems”, Nonlinear Analysis TMA, Vol. 65, (2006), pp. 395–432, 395–432. http://dx.doi.org/10.1016/j.na.2005.07.039 Zbl1093.58012
  20. [20] Y.A. Prykarpatsky: “The structure of integrable Lax type flows on nonlocal manifolds: dynamical systems with sources”, Math. Methods Phys.-Mech. Fields., Vol. 40(4), (1997), pp. 106–115. Zbl0911.34056
  21. [21] A.M. Samoilenko, A.K. Prykarpatsky and V.G. Samoylenko: “The structure of Darboux-type binarytransformations and their applications in soliton theory”, Ukr. Math. J., Vol. 55(12), (2003), pp. 1704–1723 (in Ukrainian). 
  22. [22] Y.A. Prykarpatsky, A.M. Samoilenko and A.K. Prykarpatsky: “The multi-dimensional Delsarte transmutation operators, their differential-geometric structure and applications. Part.1”, Opuscula Math., Vol. 23, (2003), pp. 71–80. Zbl1101.35003
  23. [23] Y.A. Prykarpatsky, A.M. Samoilenko and A.K. Prykarpatsky: “The de Rham-Hodge-Skrypnik theory of Delsarte transmutation operators in multi-dimension and its applications”, Rep. Math. Phys., Vol. 55(3), (2005), pp. 351–363. http://dx.doi.org/10.1016/S0034-4877(05)80051-5 Zbl1085.58029
  24. [24] J. Golenia, Y.A. Prykarpatsky, A.M. Samoilenko and A.K. Prykarpatsky: “The general differential-geometric structure of multi-dimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory Part 2”, Opuscula Math., Vol. 24, (2004), pp. 71–83. Zbl1102.35006
  25. [25] A.G. Reiman: ”Semenov-Tian-Shansky M.A”, The Integrable Systems, Computer Science Institute Publisher, Moscow-Izhevsk, 2003 (in Russian). Zbl1030.37048
  26. [26] A.G. Reiman and M.A. Semenov-Tian-Shansky: “The Hamiltonian Structure of Kadomtsev-Petviashvili Type Equations”, In: LOMI Proceedings, Vol. 164, Nauka, Leningrad, 1987, pp. 212–227 (in Russian). Zbl0546.58031
  27. [27] A.M. Samoilenko and Y.A. Prykarpatsky: Algebraic-analytic aspects of completely integrable dynamical systems and their perturbations, Institute of Mathematics Publisher, Vol. 41, Kyiv, 2002 (in Ukrainian). 
  28. [28] A.M. Samoilenko, A.K. Prykarpatsky and Y.A. Prykarpatsky: “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., Vol. 65, (2006), pp. 395–432. http://dx.doi.org/10.1016/j.na.2005.07.039 Zbl1093.58012
  29. [29] A.M. Samoilenko, V.G. Samoilenko, Yu.M. Sydorenko: “The Kadomtsev-Petviashvili Equation Hierarchy with Nonlocal Constraints: Multi-Dimensional Generalizations and Exact Solutions of Reduced Systems”, Ukrainian Math. J., Vol. 49, (1999), pp. 78–97 (in Ukrainian). http://dx.doi.org/10.1007/BF02487409 Zbl0935.37035
  30. [30] M. Sato: “Soliton Equations as Dynamical Systems on Infinite Grassmann Manifolds”, RIMS Kokyuroku, Kyoto Univ., Vol. 439, (1981), pp. 30–40. 
  31. [31] M.A. Semenov-Tian-Shansky: “What is the R-Matrix”, Funct. Anal. Appl., Vol. 17(4), (1983), pp. 17–33 (in Russian). 
  32. [32] L.A. Takhtadjian and L.D. Faddeev: Hamiltonian Approach in Soliton Theory, Springer, USA, 1986. 
  33. [33] Zakharov B. E., Integrable Systems in Multi-Dimensional Spaces, Lect. Notes Phys., Vol. 153, (1983), 190–216. http://dx.doi.org/10.1007/3-540-11192-1_38 
  34. [34] L.P. Nizhnik: Inverse Scattering Problems for Hyperbolic Equations, Kiev, Nauk. Dumka Publ., 1991 (in Russian). Zbl0791.35142
  35. [35] M.M. Prytula: Lie-algebraic structure of nonlinear dynamical systems on augmented functional manifolds, Ukrainian Math. Zh., Vol. 49(11), (1997), pp. 1512–1518. http://dx.doi.org/10.1007/BF02487508 
  36. [36] B. Konopelchenko, Yu. Sidorenko and W. Strampp: “(1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems”, Phys. Lett. A., Vol. 157, (1991), pp. 17–21. http://dx.doi.org/10.1016/0375-9601(91)90402-T 
  37. [37] J.C.C. Nimmo: “Darboux tarnsformations from reductions of the KP-hierarchy”, In: V.G. Makhankov, A.R. Bishop and D.D. Holm: Nonlinear evolution equations and dynamical systems (NEEDS’94), World Scient. Publ., 1994. 

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