Localized Structures in Nonlinear Lattices with Diffusive Coupling and   External Driving
  We study the stabilization of localized structures by discreteness in one-dimensional lattices of diffusively coupled nonlinear sites. We find that in an external driving field these structures may lose their stability by either relaxing to a homogeneous state or nucleating a pair of oppositely moving fronts. The corresponding bifurcation diagram demonstrates a cusp singularity. The obtained analytic results are in good quantitative agreement with numerical simulations. 
Igor Mitkov, Konstantin Kladko, and A. R. Bishop