Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

Abstract

We consider a system of lattice Nagumo equations with cubic nonlinearity, but with a negative discrete diffusion coefficient. We are interested in the existence, uniqueness, stability, and nonexistence of the traveling wavefront solutions of this system, and we shall call this problem the antidiffusion lattice Nagumo problem. By rewriting this system as a spatially periodic system with inhomogeneous but positive periodic diffusion coefficients and periodic nonlinearities, we uncover a rich solution behavior that includes many possible connecting orbits in the antidiffusion case. Second, we observe the presence of bistable and monostable dynamics. In the bistable region, we study the phenomenon of propagation of failure while in the monostable region, we compute the minimum wave speed.