The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons

Based on the eigenfunction symmetry constraint of the q-deformed modified KP hierarchy, a q-deformed mKP hierarchy with self-consistent sources ($q$-mKPHSCSs) is constructed. The q-mKPHSCSs contains two types of q-deformed mKP equation with self-consistent sources. By combination of the dressing method and the method of variation of constants, a generalized dressing approach is proposed to solve the q-deformed KP hierarchy with self-consistent sources ($q$-KPHSCSs). Using the gauge transformation between the $q$-KPHSCSs and the $q$-mKPHSCSs, the q-deformed Wronskian solutions for the $q$-KPHSCSs and the $q$-mKPHSCSs are obtained. The one-soliton solutions for the q-deformed KP (mKP) equation with a source are given explicitly.