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By using pseudo-differential operators containing two derivations, we extendthe Kadomtsev-Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. Forthe KP-mKP hierarchy, we obtain its B\"{a}cklund transformations, bilinearequations of Baker-Akhiezer functions and Hirota equations of tau functions.Moreover, we show that this hierarchy is equivalent to a subhierarchy of thedispersive Whitham hierarchy associated to the Riemann sphere with its infinitypoint and one movable point marked.

A KP-mKP Hierarchy via Operators with Two Derivations