The propagation of solitary waves in clouds of Bose--Einstein condensed atoms

The Gross--Pitaevskii equation which describes the motion of a Bose--Einstein condensed (BEC) atom in an elongated trap is solved analytically, and a solitary-wave solution is obtained in the low-density case without neglecting the effect of the interatomic interaction on the transverse function. It is shown that this effect leads to the velocity of the solitary wave slowing down and the profile of the solitary wave widening.