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 » ... ne the geometric origin 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