Traveling Wave Solutions for Bistable Differential-Difference Equations with Periodic Diffusion

We consider traveling wave solutions to spatially discrete reaction-diffusion equations with nonlocal variable diffusion and bistable nonlinearities. To find the traveling wave solutions we introduce an ansatz in which the wave speed depends on the underlying lattice as well as on time. For the case of spatially periodic diffusion we obtain analytic solutions for the traveling wave problem using a piecewise linear nonlinearity. The formula for the wave forms is implicitly defined in the general periodic case and we provide an explicit formula for the case of period two diffusion. We present numerical studies for time t = 0 fixed and for the time evolution of the traveling waves. When t = 0 we study the cases of homogeneous, period two, and period four diffusion coefficients using a cubic nonlinearity, and uncover, numerically, a period doubling bifurcation in the wave speed versus detuning parameter relation. For the time evolution case we also discover a detuning parameter dependent bifurcation in observed phenomena, which is a product of both the nonlocal diffusion operator and the spinodal effects of the nonlinearity.