  In the present work we consider a model that has been proposed at the
continuum level for self-defocusing nonlinearities in atomic BECs in order to
capture phenomenologically the loss of condensate atoms to thermal ones. We
explore the model at the discrete level, using this prototypical setting
combining dispersion, nonlinearity and gain/loss to illustrate the idea that
modes associated with negative "energy" (mathematically: negative Krein
signature) can give rise to instability of excited states when gain/loss terms
are introduced in a nonlinear dynamical lattice. We showcase this idea by
considering one-, two- and three-site discrete modes, exploring their stability
semi-analytically, and corroborating their continuation over the gain/loss
parameter numerically, as well as manifesting through direct numerical
simulations their unstable nonlinear dynamics.
