  We consider factorizations of the stationary and non-stationary Schroedinger
equation in R^n which are based on appropriate Dirac operators. These
factorizations lead to a Miura transform which is an analogue of the classical
one-dimensional Miura transform but also closely related to the Riccati
equation. In fact, the Miura transform is a nonlinear Dirac equation. We give
an iterative procedure which is based on fix-point principles to solve this
nonlinear Dirac equation. The relationship to nonlinear Schroedinger equations
like the Gross-Pitaevskii equation are highlighted.
