Slowly Modulated Two-Pulse Solutions in the Gray--Scott Model II: Geometric Theory, Bifurcations, and Splitting Dynamics

Arjen Doelman, Wiktor Eckhaus, Tasso J. Kaper
2001 SIAM Journal on Applied Mathematics  
In this second paper, we develop a geometrical method to systematically study the singular perturbed problem associated to slowly modulated two-pulse solutions. It enables one to see that the characteristics of these solutions are strongly determined by the flow on a slow manifold and, hence, also to identify the saddle-node bifurcations and bifurcations to classical traveling waves in which the solutions constructed in part I are created and annihilated. Moreover, we determine the geometric
more » ... gin of the critical maximum wave speeds discovered in part I. In this paper, we also study the central role of the slowly varying inhibitor component of the two-pulse solutions in the pulse-splitting bifurcations. Finally, the validity of the quasi-stationary approximation is established here, and we relate the results of both parts of this work to the literature on self-replication.
doi:10.1137/s0036139900372429 fatcat:gmywur5labejpbtuenomj75bpm