On noncommutative modified KP systems

We investigate the gauge transformations of the noncommutative modified KP hierarchy and the noncommutative constrained modified KP hierarchy in the Kupershmidt–Kiso version. By introducing the Orlov–Schulman operator, which depends on time variables and dressing operators, we construct the quantum torus symmetry for the noncommutative modified KP hierarchy. We also give a recursion operator for the noncommutative constrained modified KP hierarchy. We present an extended noncommutative modified KP hierarchy. The compatibility equations between the noncommutative modified KP flows and the extended noncommutative modified KP flows are constructed.