Problems that are modeled by nonlinear spatially discrete diffusion equations occur in the material sciences (crystal growth, interface motion in crystalline materials, spinoidal decomposition, grain boundary movement in thin films), biology (the bidomain model for cardiac tissue, tissue filtration, gas exchange in lungs, calcium waves), etc. The authors provide an analytical solution using the Fourier transform for the case when the bistable nonlinearity is represented by the piecewise linear function. Numerical studies for the time evolution of the traveling waves are presented. Some results concerning type of bifurcation are obtained.