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 KP hierarchy, Darboux transformation, Grassmann manifold

MSC Classifications:

22E65, 22E70, 35Q53, 35Q58, 58B25 show english descriptions Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05]
Applications of Lie groups to physics; explicit representations [See also 81R05, 81R10]
KdV-like equations (Korteweg-de Vries) [See also 37K10]
Other completely integrable equations (See also 37J35, 37K10)
Group structures and generalizations on infinite-dimensional manifolds [See also 22E65, 58D05] 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]

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