A family of Hamiltonian structures, hierarchy of Hamiltonians, and reduction for first-order matrix differential operators

DOI10.1007/BF01086570zbMath0455.58016MaRDI QIDQ1150197

A. G. Reyman, Michaael A. Semenov-Tian-Shansky

Publication date: 1980

Published in: Functional Analysis and its Applications (Search for Journal in Brave)

zbMATH Keywords

Poisson bracketcoadjoint actionfirst-order matrix operatorsHamiltonian structures in a space of differential operators in a single variablehierarchy of HamiltoniansLie group of smooth maps on the circle with values in a real semi-simple Lie group

Mathematics Subject Classification ID

Infinite-dimensional Lie (super)algebras (17B65) Infinite-dimensional Lie groups and their Lie algebras: general properties (22E65) Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems (37J99) Partial differential equations on manifolds; differential operators (58J99) Geometric quantization (53D50) Manifolds of mappings (58D15)

Related Items (13)

Hamiltonian differencing of fluid dynamics ⋮ Lax–Nijenhuis operators for integrable systems ⋮ Generalized Knizhnik–Zamolodchikov equations and isomonodromy quantization of the equations integrable via the Inverse Scattering Transform: Maxwell–Bloch system with pumping ⋮ A new integrable lattice hierarchy associated with a discrete \(3\times 3\) matrix spectral problem: \(N\)-fold Darboux transformation and explicit solutions ⋮ New hierarchies of integrable lattice equations and associated properties: Darboux transformation, conservation laws and integrable coupling ⋮ Modifying Lax equations and the second Hamiltonian structure ⋮ Symmetries and Integrability ⋮ N-fold Darboux transformation and soliton solutions for Toda lattice equation ⋮ N-SOLITON SOLUTIONS AND CONSERVATION LAWS OF THE MODIFIED TODA LATTICE EQUATION ⋮ Stability in bihamiltonian systems and multidimensional rigid body ⋮ On the spectral theory of the integro-differential operator Lambda generating nonlinear evolution equations ⋮ Singularities of bi-Hamiltonian systems ⋮ Hierarchy of Poisson brackets for elements of a scattering matrix

Cites Work

• On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-deVries type equations
• A simple model of the integrable Hamiltonian equation

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