Hidden Symmetries of Lax Integrable Nonlinear Systems

Denis Blackmore, Yarema Prykarpatsky, Jolanta Golenia, Anatoli Prykapatski

Department of Applied Mathematics, AGH University of Science and Technology, Krakow, Poland.
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, USA.
Department of Mathematics, Agriculture University, Krakow, Poland.

DOI: 10.4236/am.2013.410A3013                

Abstract

Recently devised new symplectic and differential-algebraic approaches to studying hidden symmetry properties of nonlinear dynamical systems on functional manifolds and their relationships to Lax integrability are reviewed. A new symplectic approach to constructing nonlinear Lax integrable dynamical systems by means of Lie-algebraic tools and based upon the Marsden-Weinstein reduction method on canonically symplectic manifolds with group symmetry, is described. Its natural relationship with the well-known Adler-Kostant-Souriau-Berezin-Kirillov method and the associated R-matrix method [1,2] is analyzed in detail. A new modified differential-algebraic approach to analyzing the Lax integrability of generalized Riemann and Ostrovsky-Vakhnenko type hydrodynamic equations is suggested and the corresponding Lax representations are constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of these generalized Riemann type hierarchies are discussed by means of the symplectic, gradientholonomic and geometric methods.

Keywords

Lie-Algebraic Approach; Marsden-Weinstein Reduction Method; R-Matrix Structure; Poissonian Manifold; Differential-Algebraic Methods; Gradient Holonomic Algorithm; Lax Integrability; Symplectic Structures; Compatible Poissonian Structures; Lax Representation

Conflicts of Interest

The authors declare no conflicts of interest.

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