Principal fiber bundle interpretation of the KP-hierarchy
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 \(\widehat \psi^ +_ W\) and \(\widehat \psi^ -_ W\) as fiber and base elements in a principal fiber bundle. Finally, they discuss differential equations associated with generalized Baker functions.
scientific article; zbMATH DE number 597454
scientific article