Adiabatic stability under semi-strong interactions: The weakly damped regime

Abstract

We rigorously derive multi-pulse interaction laws for the semi-strong interactions in a family of singularly-perturbed and weakly- damped reaction-diffusion systems in one space dimension. Most sig- nificantly, we show the existence of a manifold of quasi-steadyN-pulse solutions and identify a normal-hyperbolicity condition which bal- ances the asymptotic weakness of the linear damping against the alge- braic evolution rate of the multi-pulses. Our main result isthe adiabatic stability of the manifolds subject to this normal hyperbolicity condi- tion. More specifically, the spectrum of the linearization about a fixed N-pulse configuration contains an essential spectrum that is asymptot- ically close to the origin, as well as semi-strong eigenvalues which move at leading order as the pulse positions evolve. We characterize the semi- strong eigenvalues in terms of the spectrum of an explicitN×N matrix, and rigorously bound the error between the N-pulse manifold and the evolution of the full system, in a polynomially weighted space, so long as the semi-strong spectrum remains strictly in the left-half complex plane, and the essential spectrum is not too close to the origin.