The authors study the stationary one-pulse solutions of the Gray-Scott model, which consists of a coupled pair of singularly perturbed reaction-diffusion equations. They analyze the perturbed equations using the Evans function and an associated topological invariant, known as the stability index. Since the fast singular limits of the solutions are approximated by strongly unstable solutions of Fisher’s equation, one would expect that there will always be an unstable eigenvalue; nevertheless there exists a broad region in parameter space in which the solutions are stable. The authors analyze and explain this apparently paradoxical behaviour by showing that in the fast-slow decomposition of both the Evans function and the stability index calculation the NLEP (nonlinear eigenvalue problem) possesses a singularity which cancels the contribution from the apparent Fisher eigenvalue. The NLEP properly measures the coupling of the slow field to the fast field, and the strength of this coupling plays a central role in the solution’s stability.
In order to determine the precise multiplicity of critical eigenvalues, i.e.eigenvalues, which are close to the origin in the singular limit, it is necessary to perform index calculations based on an analytic continuation of the Evans function to a two-sheeted Riemann surface. This is due to the presence of a branch point of the usual Evans function which lies in the essential spectrum of the pulse and which converges toward the origin of the singular limit. The authors present a new and general construction of the stability index, which is valid on the entire domain of analyticity of the Evans function including the essential spectrum, where the usual construction fails.
The authors present their powerful methods in a quite compact form. These methods should not only be useful for the specific model and similar equations, but for a large class of singularly perturbed equations.