The fully supersymmetric AKNS problem: Darboux transformations and discrete systems

The fully supersymmetric AKNS spectral problem with four superfields proposed by Morosi and Pizzocchero is considered. This is a hybrid problem of the classical AKNS and supersymmetric AKNS spectral problems. Under the assumption that Darboux matrices are linear with respect to the spectral parameter, three Darboux transformations and the corresponding Backlund transformations are constructed. Through proper reductions, we obtain Darboux transformations for the coupled supersymmetric nonlinear Schrodinger and coupled supersymmetric modified Korteweg-de Vries equations, and recover the Darboux transformations for Manin-Radul supersymmetric Korteweg-de Vries equation and supersymmetric AKNS hierarchy, respectively. The first Darboux-Backlund transformation is further applied to construct integrable discrete super systems, and both semi-discrete and fully discrete systems are obtained. The associated continuum limits are studied as well.