Moving embedded lattice solitons

It was recently proved that isolated unstable "embedded lattice solitons" (ELS) may exist in discrete systems. The discovery of these ELS gives rise to relevant questions such as the following: are there continuous families of ELS?, can ELS be stable?, is it possible for ELS to move along the lattice?, how do ELS interact?. The present work addresses these questions by showing that a novel differential-difference equation (a discrete version of a complex mKdV equation) has a two-parameter continuous family of exact ELS. The numerical tests reveal that these solitons are stable and robust enough to withstand collisions. The model may apply to the description of a Bose-Einstein condensate with dipole-dipole interactions between the atoms, trapped in a deep optical-lattice potential.