  We investigate bright matter-wave solitons in the presence of a spatially
varying nonlinearity. It is demonstrated that a translation mode is excited due
to the spatial inhomogeneity and its frequency is derived analytically and also
studied numerically. Both cases of purely one-dimensional and ``cigar-shaped''
condensates are studied by means of different mean-field models, and the
oscillation frequencies of the pertinent solitons are found and compared with
the results obtained by the linear stability analysis.Numerical results are
shown to be in very good agreement with the corresponding analytical
predictions.
