Localized exact solutions of $\mathcal{P}\mathcal{T}$ symmetric nonlinear Schr\"odinger equation with space and time modulated nonlinearities

Using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schr"odinger equation (NLSE) with space and time modulated nonlinearity and in the presence of an external potential depending on space and time. In particular we obtain exact solutions of NLSE in the presence of a number of non Hermitian $\mathcal{P}\mathcal{T}$ symmetric external potentials.