Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 034, 14 pages      arXiv:1004.2945      https://doi.org/10.3842/SIGMA.2010.034
Contribution to the Proceedings of the Eighth International Conference Symmetry in Nonlinear Mathematical Physics

The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces

Oksana Ye. Hentosh
Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3B Naukova Str., Lviv, 79060, Ukraine

Received November 16, 2009, in final form February 24, 2010; Published online April 17, 2010

Abstract
The Hamiltonian representation for the hierarchy of Lax-type flows on a dual space to the Lie algebra of shift operators coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is found by means of a specially constructed Bäcklund transformation. The Hamiltonian description for the corresponding set of squared eigenfunction symmetry hierarchies is represented. The relation of these hierarchies with Lax integrable (2+1)-dimensional differential-difference systems and their triple Lax-type linearizations is analysed. The existence problem of a Hamiltonian representation for the coupled Lax-type hierarchy on a dual space to the central extension of the shift operator Lie algebra is solved also.

Key words: Lax integrable differential-difference systems; Bäcklund transformation; squared eigenfunction symmetries.

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