Compactlike breathers: Bridging the continuous with the anticontinuous limit

We consider discrete nonlinear lattices characterized by on-site nonlinear potentials and nonlinear dispersive interactions that, in the continuous limit, support exact compacton solutions. We show that the compact support feature of the solutions in the continuous limit persists all the way to the anticontinuous limit. While in the large coupling regime the compact discrete breather solution retains the essential simple cosinelike compacton shape, in the close vicinity of the anticontinuous limit it acquires a spatial shape characterized by a fast stretched exponential decay, preserving thus its essentially compact nature. The discrete compact breathers in the anticontinuous limit are generated through a numerically exact procedure and are shown to be generally stable.