Localized structures in nonlinear lattices with diffusive coupling and external driving

We study the stabilization of localized structures by discreteness in one-d imensional lattices of diffusively coupled nonlinear sites. We find that in an external driving field these structures may lose their stability by eit her relaxing to a homogeneous state or nucleating a pair of oppositely movi ng fronts. The corresponding bifurcation diagram demonstrates a cusp singul arity. The obtained analytic results are in good quantitative agreement wit h numerical simulations.