Hidden Symmetries of Lax Integrable Nonlinear Systems

• 作者: Anatoli Prykapatski Denis Blackmore Jolanta Golenia Yarema Prykarpatsky
• 来源期刊: 应用数学(英文)
• 学科: 数学
• 关键词: 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
• 年，卷（期）: 2013,（10）
• 页码范围: 95-116
• 页数: 22页
• 分类号: O1

摘要

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.