  We consider the persistence and stability of dark solitons in the
Gross-Pitaevskii (GP) equation with a small decaying potential. We show that
families of black solitons with zero speed originate from extremal points of an
appropriately defined effective potential and persist for sufficiently small
strength of the potential. We prove that families at the maximum points are
generally unstable with exactly one real positive eigenvalue, while families at
the minimum points are generally unstable with exactly two complex-conjugated
eigenvalues with positive real part. This mechanism of destabilization of the
black soliton is confirmed in numerical approximations of eigenvalues of the
linearized GP equation and full numerical simulations of the nonlinear GP
equation with cubic nonlinearity. We illustrate the monotonic instability
associated with the real eigenvalues and the oscillatory instability associated
with the complex eigenvalues and compare the numerical results of evolution of
a dark soliton with the predictions of Newton's particle law for its position.
