Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach
type: Item
instance of: scholarly article
title: Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach
zbMATH Open document ID: 0943.34039
DOI: 10.1016/S0167-2789(98)00180-8
author: Arjen Doelman
author: Robert Gardner
author: Tasso J. Kaper
published in: Physica D
publication date: 1999-12-06T00:00:00Z
review text: The authors develop a stability analysis in order to facilitate the understanding of the recently discovered phenomena of self replicating pulses. They analyze the stability of singular homoclinic stationary solutions and spatially periodic stationary ones in the 1D Gray-Scott model. The paper is organized as follows. The introduction is followed by Section 2, Single pulse and spatially periodic stationary patterns , divided into two subsections: Single pulse homoclinic solutions and multiple pulse, spatially periodic stationary states. The paper continues with Section 3, The main stability results and with Section 4, Reduction of the singularly perturbed eigenvalue problem to a nonlocal eigenvalue problem, divided into The fast system , The slow system and Determination of \(c\) subsections. The 5th Section, The nonlocal eigenvalue problem is made of Explicit eigenvalue formulae and Hopf bifurcations and is followed by Section 6, Disappearance of one pulse homoclinic stationary states, divided into Asymptotically small \(d\) \((\beta<1)\), Asymptotically large \(d\) \((\beta>1)\) and \(d=O\) (1) but small \((\beta\sim 1)\). The study contains Numerical simulations , formed by The homoclinic one pulse pattern \((m = 1)\) and Spatially periodic \(N\) pulse patterns \((m > 1)\) and followed by conclusions in Section 8.
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 1374373
zbMATH Keywords: self replicating pulses
zbMATH Keywords: activator inhibitor system
zbMATH Keywords: autocatalysis
zbMATH Keywords: pattern formation
zbMATH Keywords: stability
zbMATH Keywords: Hopf bifurcations
zbMATH Keywords: nonlocal eigenvalue problem
zbMATH Keywords: 1D Gray-Scott model
describes a project that uses: Algorithm 731
describes a project that uses: CWRESU
describes a project that uses: CWRESX
MaRDI profile type: Publication
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full work available at URL: s0167-2789(98)00180-8
OpenAlex ID: W2140202856
description: scientific article; zbMATH DE number 1374373