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

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 significantly, we show the existence of a manifold of quasi-steady N-pulse solutions and identify a "normal-hyperbolicity" condition which balances the asymptotic weakness of the linear damping against the algebraic evolution rate of the multi-pulses. Our main result is the adiabatic stability of the manifolds subject to this normal hyperbolicity condition. More specifically, the spectrum of the linearization about a fixed N-pulse configuration contains essential spectrum that is asymptotically 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 explicit N by 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.