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Using canonical transformations we obtain localized (in space) exactsolutions of the nonlinear Schr\"odinger equation (NLSE) with space and timemodulated nonlinearity and in the presence of an external potential dependingon space and time. In particular we obtain exact solutions of NLSE in thepresence of a number of non Hermitian $\mathcal{PT}$ symmetric externalpotentials.

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