  A Lagrangian based method is used to derive an analytical model for studying
the dynamics of matter-wave bright soliton created in a harmonic potential
which is attractive in the transverse direction and expulsive in the
longitudinal direction. By means of sech trial functions and a Ritz
optimization procedure, evolution eqautions are constructed for width,
amplitude and nonlinear frequency chirp of the propagating soliton of the
atomic condensate. Our eqaution for the width is an exact agreement with that
of Carr and Castin $[ Phys. Rev. A {\bf{66}}, 063602 (2002)]$, obtained by more
detailed analysis. In agreement with the experiment of Paris group $[ Science
{\bf{296}}, 1290 (2002)]$, the expulsive potential is found to cause the
soliton to explode for $N|a_s|=0.98$, $N$ being the number of atoms in the
condensate and $a_s$, the scattering length of the atom-atom interaction.
