Fronts and interfaces in bistable extended mappings

Abstract

We study the interfaces' time evolution in one-dimensional bistable extended dynamical systems with discrete time. The dynamics is governed by the competition between a local piece-wise affine bistable mapping and any couplings given by the convolution with a function of bounded variation. We prove the existence of travelling wave interfaces, namely fronts, and the uniqueness of the corresponding selected velocity and shape. This selected velocity is shown to be the propagating velocity for any interface, to depend continuously on the couplings and to increase with the symmetry parameter of the local nonlinearity. We apply the results to several examples including discrete and continuous couplings, and the planar fronts' dynamics in multi-dimensional Coupled Map Lattices. We eventually emphasize on the extension to other kinds of fronts and to a more general class of bistable extended mappings for which the couplings are allowed to be nonlinear and the local map to be smooth.