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Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy. (English) Zbl 0571.35103
This paper is the last one of a series of papers on transformation groups for soliton equations (see the preceding reviews). The main conclusion drawn from these papers may be stated as follows: The space of $$\tau$$ functions for a hierarchy of soliton equations is the orbit of the vacuum vector for the Fock representation of an infinite dimensional Lie algebra.
In this paper the authors present a detailed study of reduction problems using three kinds of equations: the KP equation, the BKP equation, and the two-component BKP equation. Several new series of soliton equations were obtained along with the explicit forms of N-soliton solutions. These soliton solutions and the corresponding Euclidean Lie algebras has been listed in a table.
This paper is organized as follows: Section 1 presents the structure of infinite dimensional Lie algebras which govern the KP hierarchy, etc. The reductions were discussed in section 2. In section 3 the Hirota bilinear equations were studied from the viewpoint of the representation theory of Kac-Moody Lie algebras.
Reviewer: L.-Y.Shih

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 17B65 Infinite-dimensional Lie (super)algebras 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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