Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model
type: Item
instance of: scholarly article
title: Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model
zbMATH Open document ID: 1105.35132
DOI: 10.4171/IFB/140
author: Theodore Kolokolnikov
author: Michael J. Ward
author: Juncheng Wei
published in: Interfaces and Free Boundaries
publication date: 2006-11-21T00:00:00Z
review text: The authors are concerned with the translational stability of equilibrium spike solutions to a one-dimensional Gray-Scott (GS) model in particular parameter regimes. The dimensionless GS model is described by the system of equations \[ \begin{aligned} v_{t} & =v_{xx}-v+Auv^{2},\qquad-1<x<1,\quad t>0,\\ \tau u_{t} & =Du_{xx}+1-u-uv^{2},\qquad-1<x<1,\quad t>0, \end{aligned} \] with boundary conditions \[ v_{x}\left( \pm1,t\right) =0,\qquad u_{x}\left( \pm1,t\right) =0. \] First, stability of symmetric \(k\)-spike solutions with respect to small eigenvalues of order \(O\left( \varepsilon^{2}\right) \) that govern translational instabilities of the spike profile is studied. Then, a formal asymptotic analysis is employed to investigate the existence of asymmetric \(k\)-spike patterns where the spikes have different heights. Finally, translational instabilities for the GS model in the intermediate regime when \(\tau\) is asymptotically large as \(\varepsilon\rightarrow0\) are analyzed. Numerical experiments are performed to validate the asymptotic theory. The paper concludes with a number of remarks and several open problems for further study.
Mathematics Subject Classification ID: 35Q80
Mathematics Subject Classification ID: 92C15
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 5074418
zbMATH Keywords: Gray-Scott model
zbMATH Keywords: translational instability
zbMATH Keywords: spike pattern
zbMATH Keywords: Hopf bifurcation
reviewed by: Yuriy V. Rogovchenko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5074418