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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.

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.

Reviewer: Iulius I.Grosu (Iaşi)

### 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 |