Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników


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



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.










Opis fizyczny




  • The AGH University of Science and Technology
  • Department of Nonlinear Mathematical Analysis of NAS
  • The AGH University of Science and Technology


  • [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] V.I. Arnold: Mathematical Methods of Classical Mechanics, Nauka, Moscow, 1989 (in Russian).
  • [3] M. Blaszak: Multi-Hamiltonian Theory of Dynamical Systems, Springer, Verlag-Berlin-Heidelberg, 1998.
  • [4] L. Dickey: Soliton equations and Hamiltonian systems, World Scientific, Vol. 42, 1991.
  • [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] P.D. Lax: “Periodic Solutions of the KdV Equation”, Commun. Pure Appl. Math., Vol. 28, (1975), pp. 141–188.
  • [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] Yu.I. Manin and A.O. Radul: “A Supersymmetric Extension of the Kadomtsev-Petviashvili Hierarchy”, Comm. Math. Phys., Vol. 28, (1985), pp. 65–77.
  • [9] V.B. Matveev and M.I. Salle: Darboux-Bäcklund transformations and applications, Springer, New York, 1993.
  • [10] E. Nissimov and S. Pacheva: “Symmetries of Supersymmetric Integrable Hierarchies of KP Type”, J. Math. Phys., Vol. 43, (2002), pp. 2547–2586.
  • [11] S.P. Novikov (Ed.): Soliton Theory: Method of the Inverse Problem, Nauka, Moscow, 1980 (in Russian).
  • [12] W. Oevel: “R-Structures, Yang-Baxter Equations and Related Involution Theorems”, J. Math. Phys., Vol. 30, (1989), pp. 1140–1149.
  • [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.
  • [14] W. Oevel, W. Strampp and K.P. Constrained: “Hierarchy and bi-Hamiltonian Structures”, Comm. Math. Phys., Vol. 157, (1993), pp. 51–81.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [25] A.G. Reiman: ”Semenov-Tian-Shansky M.A”, The Integrable Systems, Computer Science Institute Publisher, Moscow-Izhevsk, 2003 (in Russian).
  • [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.
  • [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).
  • [30] M. Sato: “Soliton Equations as Dynamical Systems on Infinite Grassmann Manifolds”, RIMS Kokyuroku, Kyoto Univ., Vol. 439, (1981), pp. 30–40.
  • [31] M.A. Semenov-Tian-Shansky: “What is the R-Matrix”, Funct. Anal. Appl., Vol. 17(4), (1983), pp. 17–33 (in Russian).
  • [32] L.A. Takhtadjian and L.D. Faddeev: Hamiltonian Approach in Soliton Theory, Springer, USA, 1986.
  • [33] Zakharov B. E., Integrable Systems in Multi-Dimensional Spaces, Lect. Notes Phys., Vol. 153, (1983), 190–216.
  • [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.
  • [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.
  • [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



Identyfikator YADDA

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.