R-matrices and higher Poisson brackets for integrable systems

DOI10.1016/0378-4371(89)90398-1zbMath0717.35081OpenAlexW1982945117MaRDI QIDQ2638444

Walter Oevel, Orlando Ragnisco

Publication date: 1989

Published in: Physica A (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0378-4371(89)90398-1

zbMATH Keywords

Boussinesq equationPoisson bracketLie-Poisson structure

Mathematics Subject Classification ID

Related Items (49)

INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEM: LOCAL EQUATIONS OF MOTION AND THEIR HAMILTONIAN PROPERTIES ⋮ Multi-Hamiltonian structures for a class of degenerate completely integrable systems ⋮ Bi-Hamiltonian structures of the coupled AKNS hierarchy and the coupled Yajima–Oikawa hierarchy ⋮ Lax–Nijenhuis operators for integrable systems ⋮ The algebra of q-pseudodifferential symbols and the q-WKP(n) algebra ⋮ R-matrix theory, formal Casimirs and the periodic Toda lattice ⋮ An extension of the Kadomtsev-Petviashvili hierarchy and its Hamiltonian structures ⋮ R-MATRICES AND HAMILTONIAN STRUCTURES FOR CERTAIN LAX EQUATIONS ⋮ A tri-Hamiltonian formulation of the full Kostant-Toda lattice ⋮ Relativistic Toda systems ⋮ Super Toda lattices ⋮ Hamiltonian structures for Ito's equation ⋮ A trihamiltonian extension of the Toda lattice ⋮ The R-matrix theory and the reduction of Poisson manifolds ⋮ A Camassa-Holm equation interacted with the Degasperis-Procesi equation ⋮ Super-Toda lattices ⋮ Fermionic one- and two-dimensional Toda lattice hierarchies and their bi-Hamiltonian structures ⋮ Lax pairs, symmetries and recursion operators for Toda lattices ⋮ A note on quadratic Poisson brackets on \(\text{gl}(n,\mathbb{R})\) related to Toda lattices ⋮ Classical double, \(R\)-operators, and negative flows of integrable hierarchies ⋮ Smooth multisoliton solutions of a 2-component peakon system with cubic nonlinearity ⋮ Separation of variables for the classical elliptic reflection equation algebra ⋮ R-matrix approach to lattice integrable systems ⋮ Multiple Hamiltonian structures for Toda-type systems ⋮ A generalized super Camassa-Holm equation ⋮ Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky lattices ⋮ The extended supersymmetrization of the nonlinear Schrödinger equation ⋮ On bi-Hamiltonian structure of two-component Novikov equation ⋮ 4-dimensional Frobenius manifolds and Painlevé VI ⋮ Separation of variables for quadratic algebras: Algebras of Maillet and reflection-equation algebras ⋮ The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems ⋮ On deformations of standard R-matrices for integrable infinite-dimensional systems ⋮ Two-component generalizations of the Novikov equation ⋮ Lie algebraic approach to the construction of (2+1)-dimensional lattice-field and field integrable Hamiltonian equations ⋮ COMPATIBLE POISSON STRUCTURES OF TODA TYPE DISCRETE HIERARCHY ⋮ Bi-Hamiltonian structures for integrable systems on regular time scales ⋮ The sine-Gordon model with integrable defects revisited ⋮ Double extended cubic peakon equation ⋮ On the bi-Hamiltonian structure, dressing transformation and constrained flows for equations in \((2+1)\) dimensions ⋮ Local and non-local multiplicative Poisson vertex algebras and differential-difference equations ⋮ Hamiltonian structures of fermionic two-dimensional Toda lattice hierarchies ⋮ Bihamiltonian structure of the two-component Kadomtsev–Petviashvili hierarchy of type B ⋮ On the Continuous Limit of Integrable Lattices II. Volterra Systems and SP(N) Theories ⋮ r-Matrices for Relativistic Deformations of Integrable Systems ⋮ Constrained KP hierarchy and bi-Hamiltonian structures ⋮ The negative Toda hierarchy and rational Poisson brackets ⋮ An \(r\)-matrix approach to nonstandard classes of integrable equations ⋮ Completely integrable bi-Hamiltonian systems ⋮ The Hamiltonian structures of the two-dimensional Toda lattice and \(R\)-matrices

Cites Work

• Unnamed Item
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• Unnamed Item
• The local structure of Poisson manifolds
• What is a classical r-matrix?
• The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev-Petviashvili equation
• Symplectic structures, their Bäcklund transformations and hereditary symmetries
• Symmetry properties and bi-Hamiltonian structure of the Toda lattice
• Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications
• Nijenhuis G-manifolds and Lenard bicomplexes: A new approach to KP systems
• The solution to a generalized Toda lattice and representation theory
• The Schouten bracket and Hamiltonian operators
• Theory of nonlinear lattices
• Systems of Toda type, inverse spectral problems, and representation theory
• Fractional powers of operators and Hamiltonian systems
• On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-deVries type equations
• Mathematics of dispersive water waves
• Korteweg-de Vries equation: a completely integrable Hamiltonian system
• Relativistic Toda systems
• Mastersymmetries, Higher Order Time-Dependent Symmetries and Conserved Densities of Nonlinear Evolution Equations
• R structures, Yang–Baxter equations, and related involution theorems
• Dimensional deformations of integrable systems: I. An approach to integrability in multidimensions
• Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States
• The Hamiltonian structure of the nonabelian Toda hierarchy
• The periodic relativistic Toda lattice: direct and inverse problem
• Mastersymmetries, angle variables, and recursion operator of the relativistic Toda lattice
• Scattering and inverse scattering for first order systems
• Nonlinear evolution equations, rescalings, model PDES and their integrability: I
• The inverse scattering transform: Semi-infinite interval
• A simple model of the integrable Hamiltonian equation
• Method for Solving the Korteweg-deVries Equation
• The Toda lattice. II. Existence of integrals
• Integrals of nonlinear equations of evolution and solitary waves
• Dressing transformations and Poisson group actions
• Solutions of the classical Yang-Baxter equation for simple Lie algebras

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