Hidden Symmetries of Lax Integrable Nonlinear Systems


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.

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D. Blackmore, Y. Prykarpatsky, J. Golenia and A. Prykapatski, "Hidden Symmetries of Lax Integrable Nonlinear Systems," Applied Mathematics, Vol. 4 No. 10C, 2013, pp. 95-116. doi: 10.4236/am.2013.410A3013.

Conflicts of Interest

The authors declare no conflicts of interest.


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