Abstract
We reduce the \(q\) -KP hierarchy and the \(q\) -mKP hierarchy. By using the link between the \(q\) -KP hierarchy and the \(q\) -mKP hierarchy, we obtain a relation between the \(q\) -KdV and the \(q\) -mKdV hierarchies, as well as a relation between the \(q\) -Boussinesq and \(q\) -mBoussinesq equations. The connection between them can also be established by a gauge transformation. We verify that when in the limit \(q\to 1\) , these relations correspond to the results for classical systems.
