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# Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting

Published:2001-04-01
Printed: Apr 2001
• G. F. Helminck
• J. W. van de Leur
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## Abstract

In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\KP$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau$-functions is given.
 Keywords: KP hierarchy, Darboux transformation, Grassmann manifold
 MSC Classifications: 22E65 - Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05] 22E70 - Applications of Lie groups to physics; explicit representations [See also 81R05, 81R10] 35Q53 - KdV-like equations (Korteweg-de Vries) [See also 37K10] 35Q58 - Other completely integrable equations (See also 37J35, 37K10)58B25 - Group structures and generalizations on infinite-dimensional manifolds [See also 22E65, 58D05]