Traveling Wave Solutions for Bistable Differential-Difference Equations With Periodic Diffusion
C E. Elmer, E S. Vanleck
We consider traveling wave solutions to spatially discrete reaction-diffusion equations with nonlocal variable diffusion and bistable nonlinearities. 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 results for 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.