Linear integral transformations and hierarchies of integrable nonlinear evolution equations
type: Item
instance of: scholarly article
title: Linear integral transformations and hierarchies of integrable nonlinear evolution equations
zbMATH Open document ID: 0654.35086
DOI: 10.1016/0167-2789(88)90003-6
author: Frank Nijhoff
published in: Physica D
publication date: 1988-01-01T00:00:00Z
review text: Integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations. These are (Riemann-Hilbert type of) integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an (indefinite) parameter k. The potential matrices in these equations obey a set of nonlinear recursion relations, leading to a hierarchy, associated with Heisenberg ferromagnet type of equations. Central in the treatment is an embedding of the hierarchy into an infinite-matrix structure, which is constructed on the basis of the integral equations. In terms of this infinite-matrix structure the equations governing the hierarchies become quite simple. Furthermore, it leads in a straightforward way to various generalizations, such as to other types of linear spectral problems, multicomponent system and lattice equations. Generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment. Finally, some results on conserved densities and the Hamiltonian structure are briefly discussed.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35G20
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 4067488
zbMATH Keywords: Integrable hierarchies
zbMATH Keywords: nonlinear evolution equations
zbMATH Keywords: linear integral equations
zbMATH Keywords: nonlinear recursion
zbMATH Keywords: Heisenberg ferromagnet
zbMATH Keywords: infinite-matrix structure
MaRDI profile type: Publication
full work available at URL: 0167-2789(88)90003-6
OpenAlex ID: W1967009566
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description: scientific article; zbMATH DE number 4067488