O. Hentosh, M. Prytula and A. Prykarpatsky, “Differential-Geometric and Lie-Algebraic Foundations of Investigating Nonlinear Dynamical Systems on Functional Manifolds,” 2nd Edition, Lviv University Publishing, Lviv, 2006. (in Ukrainian)

has been cited by the following article:

• 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.