Bi-Hamiltonian formalism and the Darboux - Crum method: I. From the KP to the mKP hierarchy

We consider the Darboux - Crum method for the KP hierarchy and its relation with other key objects in soliton theory such as the Miura map and the mKP hierarchy. Our study takes place in a very privileged system of coordinates, namely the Laurent series whose components are the moments of KP. In these geometrically important coordinates, the KP turns out to be a subsystem of the mKP. We show that associated to each solution of the mKP hierarchy one obtains two solutions of the KP. One such solution is obtained by the Miura map and the other solution of KP is related to the first one by a Darboux - Crum transformation. Furthermore, in this context the Miura map is just a linear projection with a one-dimensional kernel. We also discuss the problem of reductions to the Gelfand - Dickey hierarchies.