Reduction in soliton hierarchies and special points of classical \(r\)-matrices
type: Item
instance of: scholarly article
title: Reduction in soliton hierarchies and special points of classical \(r\)-matrices
zbMATH Open document ID: 1431.37055
published in: Journal of Geometry and Physics
publication date: 2018-06-06T00:00:00Z
review text: The article is a good example of a well-written paper which can be interesting not only for the specialists in the field but also for graduate as well as doctoral students. The paper starts with a quite deep introduction, which contains the state-of-the-art of the modern theory of integrable soliton equations in two dimensions in the setting of the algebraic \(r\)-matrix approach. The main purpose of the paper is to develop a general method of classification of hierarchies of soliton equations in dimension two. The second section discusses the definition of the classical \(r\)-matrix, degenerated \(r\)-matrix and singular point of the \(r\)-matrix and presents several good examples and remarks. The third section is devoted to the theory of \(r\)-matrix algebras. Here we can find the definitions of the one- and many-pole algebras, as well as several examples and corresponding theorems. The general scheme of the construction of the one-pole and many-pole infinite-dimensional Lie algebras associated with classical \(r\)-matrices is described in the last section. Here it is shown that if at least one of the poles coincide with a special point of the classical \(r\)-matrix then the structure of the corresponding algebra is changed. This leads to the change in the structure of Lax pairs and to a reduction in the corresponding integrable hierarchy. Besides, this section proposes several examples of explicit constructions of integrable hierarchies corresponding to some special examples of classical \(r\)-matrices.
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K30
Mathematics Subject Classification ID: 16T25
Mathematics Subject Classification ID: 17B80
zbMATH DE Number: 6881419
zbMATH Keywords: integrable hierarchies
zbMATH Keywords: soliton equations
zbMATH Keywords: classical \(r\)-matrices
zbMATH Keywords: Lie algebras
author: T. Skrypnyk
reviewed by: Yarema A. Prykarpatsky
MaRDI profile type: Publication
full work available at URL: j.geomphys.2018.03.023
OpenAlex ID: W2795545237
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DOI: 10.1016/J.GEOMPHYS.2018.03.023
description: scientific article; zbMATH DE number 6881419