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Hidden Symmetries of Lax Integrable Nonlinear Systems

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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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Applied Mathematics*, Vol. 4 No. 10C, 2013, pp. 95-116. doi: 10.4236/am.2013.410A3013.

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