Mathematical modeling of spatiotemporal patterns formed at a traveling reaction front

In some chemical systems, the reaction proceeds in the form of a propagating wave. An example is the propagation of a combustion wave. At the front of such a wave, different oscillatory regimes and the appearance of spatiotemporal structures can be observed. We propose a qualitative mechanism for the formation of patterns at the front of the reaction. It is assumed that the reason is the interaction of two subsystems, one corresponding to the propagating front and the other describing the emerging patterns. The appropriate mathematical model contains two blocks: for the travelling front, we use a model of the Fisher-Kolmogorov-Petrovsky-Piskunov type, while patterns at the front are described by the FitzHugh-Nagumo type model. Earlier, we applied this approach to explain the occurrence of autowaves-target waves and spirals-at the front of the reaction. In the present paper, we demonstrate in numerical simulations that this approach also works effectively to explain stationary relative to the front patterns, the so-called Turing or cellular structures, that are observed experimentally, in particular, at the front of a combustion wave. We also investigate the dependence of these patterns on the thickness of the front and its speed, as well as on the degree of diffusion instability achieved within the front layer.