The extended versions of the noncommutative KP and mKP equations and Miura transformation

Extended versions of the noncommutative(nc) KP equation and the nc mKP equation are constructed in a unified way, for which two types of quasideterminant solutions are also presented. In commutative setting, the quasideterminant solutions provide the known and unknown Wronskian and Grammian solutions for the bilinear KP equation with self-consistent sources and the bilinear mKP equation with self-consistent sources, respectively. Miura transformation is established for the extended nc KP and nc mKP equations.