TITLE:
Hidden Symmetries of Lax Integrable Nonlinear Systems
AUTHORS:
Denis Blackmore, Yarema Prykarpatsky, Jolanta Golenia, Anatoli Prykapatski
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
JOURNAL NAME:
Applied Mathematics,
Vol.4 No.10C,
October
23,
2013
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.