We reveal that even weak inherent discreteness of a nonlinear model can lead
to instabilities of the localized modes it supports. We present the first
example of an oscillatory instability of dark solitons, and analyse how it may
occur for dark solitons of the discrete nonlinear Schrodinger and generalized
Ablowitz-Ladik equations.Comment: 11 pages, 4 figures, to be published in Physical Review Letter
