Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension
The paper deals with the coupled reaction-diffusion systems \[ u_t=\epsilon^2 u_{xx}+f(u,w), \] \[ \tau w_t = D w_{xx} + g(u, w) \] with homogeneous Neumann boundary conditions on a one dimensional interval. Here \(\tau\), \(\epsilon\) and \(D\) are positive parameters. This system includes such well known models as the model of Belousov-Shabotinskij in water-in-oil microemulsion, the Brusselator model and many others. Under some generic conditions on the nonlinearities \(f\), \(g\) and in the singular limit \(\epsilon\to + 0\), such a system admits a steady state for which \(u\) consists of sharp back-to-back interfaces. For a sufficiently large \(D\) and for sufficiently small \(\tau\), the mentioned steady state is known to be stable in time. On the other hand, it is also known that in the so-called shadow limit \(D\to\infty\), patterns having more than one interface are unstable. The authors analyse in detail the transition between the stable patterns when \(D = O(1)\) and the shadow system when \(D\to\infty\). Direct numerical computations of stability and dynamics are performed, and these computations are in excellent agreement with corresponding results as predicted by the theoretical results derived in the paper.
scientific article; zbMATH DE number 6015380