Hidden Symmetries of Lax Integrable Nonlinear Systems

Denis Blackmore , Yarema A. Prykarpatskyy , Jolanta Golenia , Anatoli Prykarpatsky

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.
Author Denis Blackmore
Denis Blackmore,,
-
, Yarema A. Prykarpatskyy (FoEEaLS / DoAM)
Yarema A. Prykarpatskyy,,
- Department of Applied Mathematics
, Jolanta Golenia
Jolanta Golenia,,
-
, Anatoli Prykarpatsky
Anatoli Prykarpatsky,,
-
Journal seriesApplied mathematics, ISSN 2152-7385, (0 pkt)
Issue year2013
Vol4
No10C
Pages95-116
Publication size in sheets1.05
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
DOIDOI:10.4236/am.2013.410A3013
URL http://file.scirp.org/pdf/AM_2013102315281890.pdf
Internal identifierWIŚIG/2013/17
Languageen angielski
Score (nominal)5
ScoreMinisterial score = 0.0, 26-07-2017, ArticleFromJournal
Ministerial score (2013-2016) = 5.0, 26-07-2017, ArticleFromJournal
Citation count*
Additional fields
FinansowanieDenis 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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