Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model
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.
scientific article; zbMATH DE number 5074418