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Slowly modulated two-pulse solutions in the Gray–Scott model. II: Geometric theory, bifurcations, and splitting dynamics. (English) Zbl 0989.35073

The presented paper continues the study of A. Doelman, W. Eckhaus and T. J. Kaper regarding the slowly modulated two pulse solutions in the one-dimensional Gray-Scott model.
The introduction is followed by Section 2, “Geometry of governing equations”. Divided in three sub-sections, “Dynamics on the slow manifold”, “The fast subsystem when \(\varepsilon=0\)” and “Persistent fast connections”, Section 2 presents the fundamental geometric properties of the studied system. The third section, made by “The right-moving pulse with slowly changing \(c(t)\): Hooking up the slow and fast segments”, “The ODE for \(c(t)\) in case Ia”, “Bounded domains and \(N\)-pulse solutions \((N \neq 2)\)” and “The validity of the quasi-stationary approach” contains the construction of the basic slowly modulated two-pulse solution (case Ia), the consideration of the bounded interval and asymmetric cases and the establishment of the validity of the quasi-stationary approach. Section 4, “Geometric constructions of two-pulse solutions: cases Ib and IIa”, treats the cases Ib in “case Ib: \(\varepsilon\Delta/ \delta=O(1)\), the bifurcation of traveling waves” and IIa in “case IIa: \(\delta/ \varepsilon=0(1)\), a saddle-node bifurcation of two-pulse solutions”, respectively. The authors discuss the implications of the analytical results for the understanding of the self-replicating process in the section “The self-replication process” and they finish their study by relating it to the existing literature on self-replication.

MSC:

35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
92E20 Classical flows, reactions, etc. in chemistry
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