A new constrained mKP hierarchy and the generalized Darboux   transformation for the mKP equation with self-consistent sources
  The mKP equation with self-consistent sources (mKPESCS) is treated in the framework of the constrained mKP hierarchy. We introduce a new constrained mKP hierarchy which may be viewed as the stationary hierarchy of the mKP hierarchy with self-consistent sources. This offers a natural way to obtain the Lax representation for the mKPESCS. Based on the conjugate Lax pairs, we construct the generalized Darboux transformation with arbitrary functions in time $t$ for the mKPESCS which, in contrast with the Darboux transformation for the mKP equation, provides a non-auto-B\"{a}cklund transformation between two mKPESCSs with different degrees. The formula for $n$-times repeated generalized Darboux transformation is proposed and enables us to find the rational solutions (including the lump solutions), soliton solutions and the solutions of breather type of the mKPESCS. 
Ting Xiao and Yunbo Zeng