Geometric Bäcklund--Darboux transformations for the KP hierarchy

Abstract

We shown that, if you have two planes in the Segal-Wilson Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by Bäcklund-Darboux transformations (BDT). The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from elementary BDT's and is given here in a closed form. The geometric description of elementary DBT's requires that one has a geometric interpretation of the dual wavefunctions involved. This is done here with the help of a suitable algebraic characterization of the wavefunction. The BDT's also induce transformations of the tau-function associated to a plane in the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric characterization of the BDT'ss that preserves these subsystems of the KP hierarchy. This generalizes the classical Darboux-transformations. we also determine an explicit expression for the squared eigenfunction potentials. Next a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy. It is shown that infinite flags in the Grassmannian yield solutions of the latter hierarchy. these flags can be constructed by means of BDT's, starting from some plane. Other applications of these BDT's are a geometric way to characterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy and the construction of a vast collection of orthogonal polynomials, playing a role in matrix models.