  Transformations performing on the dependent and/or the independent variables
are an useful method used to classify PDE in class of equivalence. In this
paper we consider a large class of U(1)-invariant nonlinear Schr\"odinger
equations containing complex nonlinearities. The U(1) symmetry implies the
existence of a continuity equation for the particle density
$\rho\equiv|\psi|^2$ where the current ${\bfm j}_{_\psi}$ has, in general, a
nonlinear structure. We introduce a nonlinear gauge transformation on the
dependent variables $\rho$ and ${\bfm j}_{\psi}$ which changes the evolution
equation in another one containing only a real nonlinearity and transforms the
particle current ${\bfm j}_{_\psi}$ in the standard bilinear form. We extend
the method to U(1)-invariant coupled nonlinear Schr\"odinger equations where
the most general nonlinearity is taken into account through the sum of an
Hermitian matrix and an anti-Hermitian matrix. By means of the nonlinear gauge
transformation we change the nonlinear system in another one containing only a
purely Hermitian nonlinearity. Finally, we consider nonlinear Schr\"odinger
equations minimally coupled with an Abelian gauge field whose dynamics is
governed, in the most general fashion, through the Maxwell-Chern-Simons
equation. It is shown that the nonlinear transformation we are introducing can
be applied, in this case, separately to the gauge field or to the matter field
with the same final result. In conclusion, some relevant examples are presented
to show the applicability of the method.
