Hidden Symmetries of Lax Integrable Nonlinear Systems
Denis Blackmore , Yarema A. Prykarpatskyy , Jolanta Golenia , Anatoli Prykarpatsky
AbstractRecently 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.
|Journal series||Applied mathematics, ISSN 2152-7385, (0 pkt)|
|Publication size in sheets||1.05|
|Keywords in English||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|
|Score|| = 0.0, 26-07-2017, ArticleFromJournal|
= 5.0, 26-07-2017, ArticleFromJournal
|Finansowanie||Denis Blackmore acknowledges the National Science Foundation (Grant CMMI-1029809) and Anatoli Prykapatski and Yarema Prykarpatsky acknowledge the Scientific and Technological Research Council of Turkey (TUBITAK/NASU-111T558 Project) for partial support of their research.|
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