Radiationless Travelling Waves In Saturable Nonlinear Schr\"odinger Lattices

The longstanding problem of moving discrete solitary waves in nonlinear Schr{"o}dinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. {\em Genuinely localised travelling waves} are computed as a function of the system parameters {\it for the first time}. The relevant solutions exist only for finite velocities.