  We consider a one-dimensional defocusing Gross--Pitaevskii equation with a
parabolic potential. Dark solitons oscillate near the center of the potential
trap and their amplitude decays due to radiative losses (sound emission). We
develop a systematic asymptotic multi-scale expansion method in the limit when
the potential trap is flat. The first-order approximation predicts a uniform
frequency of oscillations for the dark soliton of arbitrary amplitude. The
second-order approximation predicts the nonlinear growth rate of the
oscillation amplitude, which results in decay of the dark soliton. The results
are compared with the previous publications and numerical computations.
