Ground state and elementary excitations of disordered Gross-Pitaevskii lattices

We examine the one-dimensional Gross-Pitaevskii lattice at zero temperature in the presence of disorder. We obtain analytical expressions for the thermodynamic properties of the ground state, i.e. the chemical potential and the participation number density, and compare them with direct numerical calculations. For small ground state density, we identify a Lifshits regime where disorder dominates over the interactions. For large ground-state density, the interaction dominates and screens the disorder. The fluctuations above the ground state yield Bogoliubov modes (BM). We compute their localization properties by measuring participation numbers and localization length. The localization length diverges at zero energy. In the strong interaction regime, a novel finite energy BM anomaly develops with a strong increase of localization length.