Hidden Symmetries of Lax Integrable Nonlinear Systems
| Autor: | Anatoli Prykapatski, Yarema A. Prykarpatsky, Denis Blackmore, Jolanta Golenia |
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| Rok vydání: | 2013 |
| Předmět: |
Pure mathematics
Dynamical systems theory Integrable system Lax equivalence theorem General Medicine Symplectic representation Algebra Riemann hypothesis symbols.namesake Nonlinear Sciences::Exactly Solvable and Integrable Systems Lax pair symbols Symplectomorphism Mathematics::Symplectic Geometry Symplectic geometry Mathematics |
| Zdroj: | Applied Mathematics. :95-116 |
| ISSN: | 2152-7393 2152-7385 |
| DOI: | 10.4236/am.2013.410a3013 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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