Localized nonlinear optical modes and the corresponding support structures: Exact solutions to the nonlinear Schrödinger equation with external potentials

Two analytical techniques for the generation of wide classes of exact solutions of the nonlinear Schrödinger equation (NLSE) containing an external potential are proposed. Both methods are illustrated by a variety of localized solutions, including solitary optical vortices, for both the self-focusing and self-defocusing nonlinearities. The stability of solutions was tested by direct numerical simulations of the NLSE; the existence of stable localized modes was confirmed through the simulation.