Principal fiber bundle interpretation of the KP-hierarchy
type: Item
instance of: scholarly article
title: Principal fiber bundle interpretation of the KP-hierarchy
zbMATH Open document ID: 0802.58006
author: Jacek Szmigielski
author: Josef F. Dorfmeister
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1994-12-19T00:00:00Z
review text: The authors analyze the Grassmannian and Lax pair approaches to the Kadomtsev-Petviashvili hierarchy in the framework of principal fiber bundles. In \S1 they recall the definition of the Grassmannian manifold \(X\) and the associated \(\tau\)-functions. In \S2 they consider a set \({\mathcal W}_ 0\) that will turn out in \S4 to be a principal \(G \ell (H_ +)\)-fiber bundle over \(X\) and a map \(\widehat \varphi\) from \({\mathcal W}_ 0\) to generalized pseudo-differential operators. In Theorem 2.6 they prove \(\widehat \varphi_{Wb} = \widehat \varphi_ b \widehat \varphi_ W\), \(W \in {\mathcal W}\), \(b \in G \ell (H_ +)\). This property of \(\widehat \varphi\) will be crucial for the principal fiber bundle aspect of this paper. In \S3 the authors consider the usual flow \(\Gamma_ +\) on the Grassmannian \(X\) and introduce two generalized pseudo-differential operators \(\widehat \psi_ W^ +\) and \(\widehat \psi_ W^ -\), \(W \in {\mathcal W}_ 0\). Here \(\widehat \psi_ W^ -\) is the pseudo-differential operator associated with the Baker function \(\psi_ W\) and \(\widehat \psi_ W^ +\) is an infinite differential operator. One of the main results of the present paper is the ``splitting'' \(\widehat \psi^ +_ W \circ \widehat \psi_ W = \widehat \psi^ -_ W\). It allows us to define principal \(G \ell (H_ +)\) fiber bundles of pseudo-differential operators in \S4. The main contents of \S4 is to show that \(E = G {\underset {P} \times} G \ell (H_ +)\) is isomorphic with \({\mathcal W}_ 0\) as a principal fiber bundle and also with three (different but isomorphic) \(G \ell (H_ +)\)- principal fiber bundles of generalized pseudo-differential operators. The base of these bundles is given by \(\{\widehat \psi^ -_ W; W \in {\mathcal W}_ 0\}\) and the total space is given in one bundle as \(\{\widehat \varphi_ W; W \in {\mathcal W}_ 0\}\), whereas it is given in another bundle as \(\{\widehat \psi^ +_ W; W \in {\mathcal W}_ 0\}\). This gives an interpretation of \(\widehat \ps…
Mathematics Subject Classification ID: 58B25
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35Q15
zbMATH DE Number: 597454
zbMATH Keywords: Kadomtsev-Petviashvili equation
zbMATH Keywords: principal fiber bundles
reviewed by: Vasil G. Angelov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195168205
OpenAlex ID: W1996998588
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DOI: 10.2977/PRIMS/1195168205
description: scientific article; zbMATH DE number 597454