By using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schrödinger equation (NLSE) with cubic and quintic space and time modulated nonlinearities and in the presence of time-dependent and inhomogeneous external potentials and amplification or absorption (source or drain) coefficients. We obtain a class of wide localized exact solutions of NLSE in the presence of a number of non-Hermitian parity-time $\left(\mathcal{PT}\right)$-symmetric external potentials, which are constituted by a mixing of external potentials and source or drain terms. The exact solutions found here can be applied to theoretical studies of ultrashort pulse propagation in optical fibers with focusing and defocusing nonlinearities. We show that, even in the presence of gain or loss terms, stable solutions can be found and that the $\mathcal{PT}$ symmetry is an important feature to guarantee the conservation of the average energy of the system.

• Received 22 May 2014
• Revised 30 September 2014

DOI:https://doi.org/10.1103/PhysRevE.91.013205