Integrable perturbation theory for dark solitons of the defocusing nonlinear Schrodinger equation

The goal of this work is to revisit the eigenfunction-expansion-based perturbation theory of the defocusing nonlinear Schrdinger equation on a nonzero background, and develop it to correctly predict the slow-time evolution of the dark soliton parameters, as well as the radiation shelf emerging on the soliton sides.Proof of the closure of the squared eigenfunctions is provided, and the complete set of eigenfunctions of the linearization operator is used to expand the first-order perturbation solution.Our closure/completeness relation accounts for the singularities of the scattering data at the branch points of the continuous spectrum, which leads to the correct discrete eigenfunctions.Using the onesoliton 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 integral leads to predictions for 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.All the results are corroborated by direct numerical simulations, and compared with earlier results.