Geometric Bäcklund–Darboux Transformations for the KP Hierarchy Klein-Gordon equations is studied in the Sobolev space Hs = Hs(

In this paper it is shown that, if you have two planes in the Sato Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by Bäcklund–Darboux (shortly BD-)transformations. The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from so-called elementary BD-transformations and is given here in a closed form. The corresponding action on the tau-function, associated to a plane in the Grassmannian, is also determined.