Constrained KP hierarchy and bi-Hamiltonian structures
type: Item
instance of: scholarly article
title: Constrained KP hierarchy and bi-Hamiltonian structures
zbMATH Open document ID: 0793.35095
DOI: 10.1007/BF02098018
published in: Communications in Mathematical Physics
publication date: 1994-08-22T00:00:00Z
review text: The Kadomtsev-Petviashvili (KP) hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions. Constraining the KP flows in terms of squared eigenfunctions one obtains 1+1-dimensional integrable equations with scattering problems given by pseudodifferential Lax operators. The bi-Hamiltonian nature of these systems is shown by a systematic construction of two general Poisson brackets on the algebra of associated Lax operators. Gauge transformations provide Miura links to modified equations. These systems are constrained flows of the modified KP hierarchy, for which again a general description of their bi-Hamiltonian nature is given. The gauge transformations are shown to be Poisson maps relating the bi-Hamiltonian structures of the constrained KP hierarchy and the modified KP hierarchy. The simplest realization of this scheme yields the AKNS hierarchy and its Miura link to the Kaup-Broer hierarchy.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 58J72
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 487180
zbMATH Keywords: Kadomtsev-Petviashvili equation
zbMATH Keywords: pseudodifferential Lax operators
zbMATH Keywords: gauge transformations
zbMATH Keywords: bi-Hamiltonian structures
zbMATH Keywords: KP hierarchy
author: W. Oevel
author: Walter Strampp
reviewed by: Valeriy Yumaguzhin
MaRDI profile type: Publication
cites work: Rational and elliptic solutions of the korteweg-de vries equation and a related many-body problem
cites work: The periodic problem for the Korteweg-de Vries equation
cites work: GEOMETRY AND ACTION-ANGLE VARIABLES OF MULTI SOLITON SYSTEMS
cites work: Additional symmetries for integrable equations and conformal algebra representation
cites work: The AKNS hierarchy as symmetry constraint of the KP hierarchy
cites work: Symmetry constraints of the KP hierarchy
cites work: Gauge transformation and bi-Hamiltonian structure of finite-dimensional integrable system reduced from a soliton equation
cites work: Formation and interaction of sonic-Langmuir solitons -- inverse scattering method.
cites work: Q3308741
cites work: Landau-Lifshitz equation: solitons, quasi-periodic solutions and infinite-dimensional Lie algebras
cites work: Solitons and infinite dimensional Lie algebras
cites work: (1+1)-dimensional Hamiltonian systems as symmetry constraints of the Kadomtsev-Petviashvili equation
cites work: Constraints of the Kadomtsev–Petviashvili hierarchy
cites work: Fractional powers of operators and Hamiltonian systems
cites work: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-deVries type equations
cites work: What is a classical r-matrix?
cites work: Mathematics of dispersive water waves
cites work: A remark on the commuting flows defined by Lax equations.
cites work: GAUGE TRANSFORMATIONS AND RECIPROCAL LINKS IN 2 + 1 DIMENSIONS
cites work: An \(r\)-matrix approach to nonstandard classes of integrable equations
cites work: Nonlinear Poisson structures and r-matrices
cites work: R-matrices and higher Poisson brackets for integrable systems
description: scientific article; zbMATH DE number 487180