  We examine the dynamics of a bright solitary wave in the presence of a
repulsive or attractive localized ``impurity'' in Bose-Einstein condensates
(BECs). We study the generation and stability of a pair of steady states in the
vicinity of the impurity as the impurity strength is varied. These two new
steady states, one stable and one unstable, disappear through a saddle-node
bifurcation as the strength of the impurity is decreased. The dynamics of the
soliton is also examined in all the cases (including cases where the soliton is
offset from one of the relevant fixed points). The numerical results are
corroborated by theoretical calculations which are in very good agreement with
the numerical findings.
