Reduction in soliton hierarchies and special points of classical \(r\)-matrices
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.
scientific article; zbMATH DE number 6881419
scientific article