Synchronization of coupled oscillators on finite-dimensional lattices

Frequency synchronization is a phenomenon that occurs in many areas, from condensed matter to biological systems. We present a simple model of a lattice of phase oscillators with nearest-neighbor couplings, which may be derived from complex Ginzburg-Landau equations for a driven-dissipative Bose-Einstein condensate in a periodic potential. This is a generalization of the Kuramoto model to include an additional cosine term in the coupling. We show that, due to this additional coupling term, the oscillators exhibit global frequency synchronization on extended lattices in dimensions d<4. We do this by connecting the model to the quantum description of localization of a particle in a random potential through a mapping to a modified Kardar-Parisi-Zhang equation. We use this approach to derive the critical coupling strength for synchronization, and explain the phase patterns in the synchronized state.  These results are supported by numerical simulations in one and two dimensions. Our findings indicate that a general class of locally coupled oscillators can support global synchronized states that are robust against static disorder.