Exact Bogoliubov-de Gennes Solutions for Grey Soliton Backgrounds
  We derive and discuss the complete set of exact solutions to the one-dimensional Bogoliubov-de Gennes equations for small amplitude excitations around general grey soliton solutions to the cubic nonlinear Schr\"odinger equation. Our results extend the previously known case of the motionless dark soliton background. We derive our non-zero frequency solutions using a variant of the factorization method for Schr\"odinger equations with reflectionless potentials. We also discuss the zero mode solutions at length. 
Philip B. Walczak, James R. Anglin
quant-ph