Effect of a potential step or impurity on the Bose-Einstein condensate mean field

The full set of stationary states of the mean field of a Bose-Einstein condensate in the presence of a potential step or pointlike impurity are presented in closed analytic form. The nonlinear Schroedinger equation in one dimension is taken as a model. The nonlinear analogs of the continuum of stationary scattering states, as well as evanescent waves, are discussed. The solutions include asymmetric soliton trains and other wave functions of intriguing form, such as a pair of dark solitons bound by an impurity