Gray solitons and phonons in repulsive Bose-Einstein condensates confined in one-dimensional nonharmonic traps

We numerically solve the time-dependent Gross-Pitaevskii equation (GPE) that describes the evolution of an elongated dilute repulsive atomic Bose-Einstein condensate trapped in a one-dimensional (1D) nonharmonic potential. We find that the gray solitons, which are propagative solutions of the 1D GPE, traveling at an initial constant velocity, smaller than the speed of sound, oscillate through the trapped condensate, but that this oscillatory motion is accompanied by a spontaneous emission of small sound waves. By examining the gray soliton trajectory and its velocity in the trapped repulsive Bose-Einstein condensate, we show that the oscillatory motion is uniform and nondissipative except at the returning points of the gray soliton, where it exhibits a slight radiative acceleration (antidamping). Our numerical results are in good agreement with previous theoretical predictions, but show the need to take radiation emission into account