Colliding dissipative pulses—the shooting manifold

We study multi-pulse solutions in excitable media. Under the assumption that a single pulse is asymptotically stable, we show that there is a well-defined ”shooting manifold”, consisting of two pulses traveling towards each other. In the phase space, the two-dimensional manifold is a graph over the manifold of linear superpositions of two pulses located at x1 and x2, with x1 − x2 1. It is locally invariant under the dynamics of the reaction-diffusion system and uniformly asymptotically attracting with asymptotic phase. The main difficulty in the proof is the fact that the linearization at the leading order approximation is strongly non-autonomous since pulses approach each other with speed of order one.