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.
Highlights
It is well known that hidden symmetry properties, related to symplectic, differential-geometric, differential-algebraic (D-A) or analytical structures of nonlinear Hamiltonian dynamical systems on functional manifolds, such as an infinite hierarchy of conservation laws and compatible Poissonian structures, often give rise to their Lax integrability
Concerning the D -structure determining Equation (69) one can anticipate that a study of its solutions would describe a set of nonlinear dynamical systems on the reduced phase space M possessing an infinite hierarchy of mutually commuting conservation laws
We shall prove by means of gradientholonomic and differential algebraic tools Proposition 4, stating the Lax integrability of the dynamical system (97), in particular, we will devise an effective approach for constructing its exact Lax representation and related compatible Poissonian structures
Summary
It is well known that hidden symmetry properties, related to symplectic, differential-geometric, differential-algebraic (D-A) or analytical structures of nonlinear Hamiltonian dynamical systems on functional manifolds, such as an infinite hierarchy of conservation laws and compatible Poissonian structures, often give rise to their Lax integrability. When studying integrability properties of infinite so called Riemann type hydrodynamical hierarchies, a new direct approach to testing the Lax integrability of a priori given nonlinear dynamical systems with special structure, based on treating the related symplectic and differential-algebraic structures of differentiations, was suggested [8] and devised in [9] By means of this technique the direct integrability problem was effectively reduced to the classical one of finding the corresponding compatible representations in suitably constructed differential rings. K X : M T M on the phase space M : KX T , c;l, k d dt g t Tg t 1 , c g t 1 gx t ,T (10)
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