Instabilities via Negative Krein Signature in a Non-Conservative DNLS Model

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.