  We theoretically investigate the dynamics of a matter-wave soliton created in
a harmonic potential, which is attractive in the transverse direction but
expulsive in the longitudinal direction. This Bose-Einstein-condensate (BEC)
bright soliton made of $^7$Li atoms has been observed in a recent experiment
(Science {\bf 296}, 1290 (2002)). We show that the non-polynomial Schr\"odinger
equation, an effective one-dimensional equation we derived from the
three-dimensional Gross-Pitaevskii equation, is able to reproduce the main
experimental features of this BEC soliton in an expulsive potential.
