Separating Dissipative Pulses: The Exit Manifold

We prove that if a reaction-diffusion equation (in one space dimension) has asymptotically stable, exponentially localized traveling wave solutions then there are solutions of the system which are nearly the linear superposition of two such pulses moving in opposite directions away from one another. Moreover, such solutions are themselves asymptotically stable. This result is meant to complement analytic or numeric studies into interactions of such pulses over finite times which might result in the scenario treated here. Since the pulses are moving in opposite directions, it is not possible to put the problem into a moving reference frame which renders the linear problem autonomous. We overcome this difficulty by embedding the original system in a larger one wherein the linear part can be written as a time independent piece plus another piece which, even though it is non-autonomous and large, has certain properties which allow us to treat it as if it were a small perturbation.