Abstract
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 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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 59-79 |
| Number of pages | 21 |
| Journal | Journal of Differential Equations |
| Volume | 245 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jul 1 2008 |
Bibliographical note
Funding Information:This work was partially supported by the National Science Foundation through grant NSF DMS-0203301 (A.S.). J.D.W. would like to thank the School of Mathematics at the University of Minnesota, where he was employed during the writing of this paper. The authors would also like to thank the referee for pointing out references [4,5,8,12].