A self-dual Bogomolʹnyi formulation of the nonlinear Schrödinger equation

We obtain a self-dual formulation of the conventional nonlinear Schrödinger equation (NLSE) in the 1+1 dimension by studying the dimensional reduction of the self-dual Chern-Simons nonlinear Schrödinger model (NLSM) in the 2+1 dimension. It is found that this self-dual formulation allows us to find not only the well-known solitions from the Bogomolʹnyi bound and the Galilean boost, but also other soliton solutions in the presence of the background sources.