Pattern Formation and Localized Structures in Reaction-Diffusion Systems with Non-Fickian Transport

Abstract

We study the robust dynamical behaviors of reaction-diffusion systems where the transport gives rise to non-Fickian diffusion. A prototype model describing the deposition of molecules in a surface is used to show the generic appearance of Turing structures which can coexist with homogeneous states giving rise to localized structures through the pinning mechanism. The characteristic lengths of these structures are in the nanometer region in agreement with recent experimental observations.

Statistical properties of nanosized clusters on a surface in overdamped stochastic reaction-Cattaneo systems

“…[26,28,44,45]. It was shown that in such systems nano-localized patterns representing islands of adsorbate or vacancy islands can be controlled by adsorption and desorption rates.…”

Pattern formation and coexistence domains for a nonlocal population dynamics

“…On the other hand, the overwhelming majority of the studies have shown that Fickiantype diffusion [13][14][15] is unable to describe spreading of species in population equations formulated through reaction-diffusion models. Moreover, this is not the only possible criticism that one can find at the ordinary nonlinear reaction-diffusion approach.…”

Noise induced patterning in reaction-diffusion systems with non-Fickian diffusion

“…Obtained results can be applied to study patterns in adsorption/desorption processes in metals [26] deposition of a monolayer of molecules [32] and in processes of microstructure transformations of materials subject to intensive irradiation. One can await that formation of point defect clusters and clusters of particles in the system kind of ''vacanies + particles'' where a phase decompositions and ''chemical transformations'' caused by irradiation can be observed.…”

“…From the formal viewpoint one can consider two different models: in the case of lateral interactions (non-Fickian diffusion) one can use the local potential in the form φ(x) = − κ 2 x 2 , κ > 0 [26]; the ordinary phase separation scenario corresponds to the x 4 -potential, given by Eq. (16).…”