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2007 | 5 | 1 | 84-104
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
5
Numer
1
Strony
84-104
Opis fizyczny
Daty
wydano
2007-03-01
online
2007-03-01
Twórcy
autor
autor
  • Department of Nonlinear Mathematical Analysis of NAS, ohen@net.ua
Bibliografia
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  • [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
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  • [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
  • [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
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  • [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.
  • [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
  • [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
  • [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.
  • [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] 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.
  • [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
  • [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.
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  • [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).
  • [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] 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
  • [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
  • [30] M. Sato: “Soliton Equations as Dynamical Systems on Infinite Grassmann Manifolds”, RIMS Kokyuroku, Kyoto Univ., Vol. 439, (1981), pp. 30–40.
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  • [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] L.P. Nizhnik: Inverse Scattering Problems for Hyperbolic Equations, Kiev, Nauk. Dumka Publ., 1991 (in Russian).
  • [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] 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] 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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-006-0038-1
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