ON THE GEOMETRY OF DARBOUX TRANSFORMATIONS FOR THE KP HIERARCHY AND ITS CONNECTION WITH THE DISCRETE KP HIERARCHY

We tackle the problem of interpreting the Darboux transformation for t he KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equa tions and obtain a new hierarchy of equations, which we call the Darbo ux-KP hierarchy (DKP). We employ the DKP equations to discuss the rela tionships among the KP equations, the modified KP equations, and the d iscrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invaria nt submanifold, and that the discrete KP equations are obtained as ite rations of the DKP ones.