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The explicit expression of the flow equations of the noncommutativeKadomtsev-Petviashvili(ncKP) hierarchy is derived. Compared with the flowequations of the KP hierarchy, our result shows that the additional terms inthe flow equations of the ncKP hierarchy indeed consist of commutators ofdynamical coordinates \{$u_i$\}. The recursion operator for the flow equationsunder $n$-reduction is presented. Further, under 2-reduction, we calculate anonlocal recursion operator $\Phi(2)$ of the noncommutative Korteweg-deVries(ncKdV) hierarchy, which generates a hierarchy of local, higher-orderflows. Thus we solve the open problem proposed by P.J. Olver and V.V.Sokolov(Commun.Math.Phys. 193 (1998), 245-268).

Explicit Flow Equations and Recursion Operator of the ncKP hierarchy