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, Tome 5 (2007), p. 84-104 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2007-01-01

Zbl 1126.35054

EUDML-ID : urn:eudml:doc:269031

Bibliographie

[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. http://dx.doi.org/10.1002/cpa.3160280105 | Zbl 0295.35004

[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. http://dx.doi.org/10.1007/BF01211044 | Zbl 0607.35075

[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. http://dx.doi.org/10.1063/1.1466533 | Zbl 1059.37054

[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. http://dx.doi.org/10.1063/1.528333 | Zbl 0689.35077

[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 | Zbl 0742.35063

[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 | Zbl 0793.35095

[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 | Zbl 1075.37026

[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. | Zbl 0937.37055

[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. | Zbl 1068.37046

[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 | Zbl 0801.58023

[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 | Zbl 1093.58012

[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. | Zbl 0911.34056

[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. | Zbl 1101.35003

[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 | Zbl 1085.58029

[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. | Zbl 1102.35006

[25] A.G. Reiman: ”Semenov-Tian-Shansky M.A”, The Integrable Systems, Computer Science Institute Publisher, Moscow-Izhevsk, 2003 (in Russian). | Zbl 1030.37048

[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). | Zbl 0546.58031

[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 | Zbl 1093.58012

[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 | Zbl 0935.37035

[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. 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). | Zbl 0791.35142

[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.