The Gray-Scott equations form a system of two singularly perturbed reaction-diffusion equations which exhibit several interesting and novel spatio-temporal patterns. This talk is concerned with the stability analysis of stationary homoclinic pulse solutions near the singular limit. These solutions are (rigorously) obtained from a matched asymptotic expansion in a previous study by Doelman, Kaper, and Zegeling, in which one component has a singular "spike" which becomes unbounded in the singular limit. In a scaling which resolves the fast dynamics, the spike is approximated by the (unstable) homoclinic solution of the scalar Fisher equation. However, numerical experiments indicate that, contrary to intuition, the singular 1-pulse solutions will sometimes be stable.
The purpose of this lecture is twofold. First, a formal stability analysis is discussed which in which a reduced, scalar nonlocal eigenvalue problem (the NLEP equation) is derived from matched asymptotics which predicts that these waves will indeed be stable in certain ranges of parameters. The formal results are in excellent agrement with the numerical calculations.
The second part of the talk is concerned with a rigorous stability analysis of these solutions based upon the Evans function for the wave and on the stability index. The analysis provides a complete mathematical explanation for the mysterious disappearance of the unstable eigenvalue predicted by the fast reduced equation and it validates the predictions of the formal NLEP equation.
Another aspect of the analysis is the role of the continuous spectrum, which approaches the origin in the singular limit. This is a commonly encountered problem in stability calcuations near a singular limit. A novel method for handling this difficulty is introduced, which involves the analytic continuation of the Evans function and the stability index into the continuous spectrum. This requires that the spectral plane be lifted to a certain Riemann surface, since the Evans function has a branch point near the origin. The branch point plays a crucial role in the calculation of the index and therefore, in obtaining the correct eigenvalue count near the origin of the spectral plane.
This is joint work with Arjen Doelman and Tasso Kaper.