Nonlinear transformation for a class of gauged Schrِdinger equations with complex nonlinearities

In the present contribution we consider a class of Schrödinger equations containing complex nonlinearities, describing systems with conserved norm |ψ|2 and minimally coupled to an abelian gauge field. We introduce a nonlinear transformation which permits the linearization of the source term in the evolution equations for the gauge field, and transforms the nonlinear Schrödinger equations into another one with real nonlinearities. We show that this transformation can be performed either on the gauge field Aμ or, equivalently, on the matter field ψ. Since the transformation does not change the quantities |ψ|2 and Fμν, it can be considered as a generalization of the gauge transformation of third kind introduced some years ago by other authors.