We present a novel approach to the Kadomtsev-Petviashvili (KP) hierarchy and its modified counterpart, the mKP hierarchy based on factorizations of formal pseudo-differential operators and a matrix-valued Lax operator for the mKP hierarchy. As a result of this framework we obtain new B\"acklund transformations for the KP hierarchy and the possibility of transferring classes of KP solutions into those of mKP solutions, and vice versa. As an application of our techniques we provide a new derivation of soliton solutions of the KP and mKP equation.
"On the (modified) Kadomtsev-Petviashvili hierarchy." Differential Integral Equations 8 (4) 797 - 812, 1995.