Stability of stationary states in the cubic nonlinear Schroedinger   equation: applications to the Bose-Einstein condensate
  The stability properties and perturbation-induced dynamics of the full set of stationary states of the nonlinear Schroedinger equation are investigated numerically in two physical contexts: periodic solutions on a ring and confinement by a harmonic potential. Our comprehensive studies emphasize physical interpretations useful to experimentalists. Perturbation by stochastic white noise, phase engineering, and higher order nonlinearity are considered. We treat both attractive and repulsive nonlinearity and illustrate the soliton-train nature of the stationary states. 
Lincoln D. Carr, J. Nathan Kutz, and William P. Reinhardt