Stability of Concatenated Traveling Waves: Alternate Approaches

We consider a reaction-diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is asymptotically stable. As in [25], [24], and [14], we show that the sequence of traveling waves is itself asymptotically stable: as t → ∞, the solution approaches the concatenated wave pattern, with different shifts of each wave allowed. Our proof is similar to that of [14] in that it is based on spatial dynamics, Laplace transform, and exponential dichotomies, but it incorporates a number of modifications.