In this talk, we will revisit the squared-eigenfunctions-based perturbation theory of the scalar defocusing nonlinear Schrodinger equation on a nonzero background, which we developed so that it can correctly predict the slow-time evolution of the dark soliton parameters, as well as the radiation shelf emerging on the sides of the soliton. Proof of the completeness of the squared eigenfunctions is provided. Our closure relation accounts for the singularities of the scattering data at the branch points of the continuous spectrum, which leads to the correct discrete eigenfunctions of the closure relation. Using the one-soliton closure relation and its correct discrete eigenmodes, the slow-time evolution equations of the soliton parameters are determined. Moreover, the first-order correction integral to the dark soliton is shown to contain a pole due to singularities of the scattering data at the branch points. Analysis of this first-order correction integral leads to predictions for the height, velocity and phase gradient of the shelves, as well as a formula for the slow time evolution of the soliton's phase, which in turn allows one to determine the slow-time dependence of the soliton center. We will also provide comparisons of our results with direct numerical simulations, and with earlier results.