Opuscula Math. 27, no. 2 (2007), 231-244

Opuscula Mathematica

# Integrable three-dimensional coupled nonlinear dynamical systems related with centrally extended operator Lie algebras

Oksana Ye. Hentosh
Anatoliy K. Prykarpatsky

Abstract. A hierarchy of Lax-type flows on a dual space to the centrally extended Lie algebra of integral-differential operators with matrix-valued coefficients is considered. By means of a specially constructed Backlund transformation the Hamiltonian representations for these flows coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems are obtained. The Hamiltonian description of the corresponding set of additional symmetry hierarchies is represented. The relation of these hierarchies with Lax integrable $$(3+1)$$-dimensional nonlinear dynamical systems and their triple Lax-type linearizations is analysed.

Keywords: centrally extended operator Lie algebra, Lax-type flows, Backlund transformation, "ghost" symmetries.

Mathematics Subject Classification: 37K05, 37K10, 37K30, 37K35, 37K15.

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• Oksana Ye. Hentosh
• NAS of Ukraine, Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., Lviv 79060, Ukraine
• Anatoliy K. Prykarpatsky
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland