Miura and Darboux transformations in the SUSY KP hierarchies

The Miura links between the KP and modified KP hierarchies are extended to the SUSY KP (SKP) and SUSY modified KP hierarchies (SmKP) of Manin Radul and Jacobian types. The corresponding Darboux transformations in the SUSY case are constructed by using Miura links. In this sense, one can better understand why one step Darboux transformation of the SKP hierarchy can not keep the odd flows. In addition, the Darboux transformations in the SmKP case can be obtained naturally by the Miura links. Then the squared eigenfunction symmetry of the SmKP hierarchy is obtained by using Miura link from the SKP hierarchy. At last, starting from the results in the SKP hierarchy, the constrained SmKP hierarchy of Manin Radul type can also be constructed by Miura link, that is, by adding appropriate squared eigenfunction symmetry in odd flow parts. The Darboux transformations of the constrained SKP and SmKP hierarchies are also discussed by using the Miura links.