Interaction of noise supported Ising–Bloch fronts with Dirichlet boundaries

We investigate the behavior of fronts in bistable activator–inhibitor systems near Dirichlet boundaries. In particular, analytical and numerical investigations are performed for a FitzHugh–Nagumo dynamics. Conditions for a bound state and for a rebound of fronts are formulated. We also show fronts which oscillate between two reflecting boundaries. If additive noise is applied, nucleation of pairwise fronts near the boundary is observed. The front running towards the boundary is reflected there, and a pulse-like sequence of fronts propagating away from the boundary is established. Thus, noise and the boundary play the role of a pacemaker of a permanent progression of fronts. The sequence becomes highly ordered at optimal noise level. We also present examples of a two dimensional generalization of this noisy pacemaker.

► Front solutions in bistable activator–inhibitor systems near Dirichlet boundaries.
► Bound, rebound and oscillatory states of fronts near boundaries.
► Nucleation of pairwise fronts near the boundary induced by additive noise.
► Noisy pacemaker near boundary of highly ordered at optimal noise level.
► Examples of two dimensional generalization of noisy pacemakers near impurities.