We examine a singularly perturbed, coupled, weakly damped, reaction-diffusion system in one space dimension. This system is examined in the semi-strong pulse interaction regime. We rigorously construct a slow manifold of N-pulse solutions. We identify neutral modes and uncouple them. We solve this reduced nonlinear N-dimensional system with a fixed point method, which generates an equilibrium solution for the reduced system. We turn the coupling back on and continue the slow manifold back to the original system. After analyzing the eigenvalue problem and using renormalization group methods, we show the approximate invariant manifold for the full system is adiabatically stable. We also derive an explicit formula for the pulse dynamics. This work is the first rigorous analysis of the weakly damped regime, in which the essential spectrum approaches the origin.
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