  It is shown that the density-potential mapping and the ${\cal
V}$-representability problems in the time-dependent current density functional
theory (TDCDFT) are reduced to the solution of a certain many-body nonlinear
Schr\"odinger equation (NLSE). The derived NLSE for TDCDFT adds a link which
bridges the earlier NLSE-based formulations of the time-dependent deformation
functional theory (TDDefFT) and the time-dependent density functional theory
(TDDFT). We establish close relations between the nonlinear many-body problems
which control the existence of TDCDFT, TDDFT, and TDDefFT, and thus develop a
unified point of view on the whole family of the TDDFT-like theories.
