  We consider the dynamics of dark matter solitons moving through non-uniform
cigar-shaped Bose-Einstein condensates described by the mean field
Gross-Pitaevskii equation with generalized nonlinearities, in the case when the
condition for the modulation stability of the Bose-Einstein condensate is
fulfilled. The analytical expression for the frequency of the oscillations of a
deep dark soliton is derived for nonlinearities which are arbitrary functions
of the density, while specific results are discussed for the physically
relevant case of a cubic-quintic nonlinearity modeling two- and three-body
interactions, respectively. In contrast to the cubic Gross-Pitaevskii equation
for which the frequencies of the oscillations are known to be independent of
background density and interaction strengths, we find that in the presence of a
cubic-quintic nonlinearity an explicit dependence of the oscillations frequency
on the above quantities appears. This dependence gives rise to the possibility
of measuring these quantities directly from the dark soliton dynamics, or to
manage the oscillation via the changes of the scattering lengths by means of
Feshbach resonance. A comparison between analytical results and direct
numerical simulations of the cubic-quintic Gross-Pitaevskii equation shows good
agreement which confirms the validity of our approach.
