The goal of the proposed study is to develop a theory of pattern formation and nonlinear dynamics in reaction-diffusion systems with anomalous diffusion. A characteristic feature of most of the reaction-diffusion systems that have been studied to date is that diffusion is normal, i.e., the dependence of the mean square displacement of a randomly walking particle on time is linear. In some cases, however, a more general dependence is observed, in which the mean square displacement behaves as a power function of time, and the diffusion of the reactants is anomalous. If the exponent is less than one the diffusion process is slower than normal diffusion and is called subdiffusion. In contrast, if the exponent is greater than one it is faster than normal and is called superdiffusion. Both types of anomalous diffusion have been recognized to play important roles in various physical, chemical, biological, geological, and other processes. For anomalous diffusion the reaction-diffusion system of partial differential equations is replaced by a system of fractional partial differential equations which have been derived from appropriate continuous time random walk models at the molecular level. The fractional equations are equivalent to systems of integro-differential equations which are the objects of our investigation. The proposed studies will involve both general reaction-diffusion systems with anomalous diffusion, and specific applications. Investigations of general reaction-diffusion systems will help elucidate the universal mechanisms of pattern formation and nonlinear dynamics under anomalous diffusion conditions. The specific models will include the famous chemical dynamics models such as the Brusselator, Oregonator, Gray-Scott, Gierer-Meinhardt, Fitzhugh-Nagumo, and other models, as well as models for combustion, polymerization, and controlled drug delivery. The effects of both sub- and superdiffusion will be considered.
Reaction-diffusion systems are ubiquitous in many branches of science and have been attracting the attention of scientists, engineers and mathematicians for decades. Since the ground-breaking discoveries of Turing who showed that diffusion in a mixture of chemically reacting species could cause instability of a spatially uniform state leading to the formation of spatio-temporal patterns, and Belousov and Zhabotinskii who discovered oscillating chemical reactions, reaction-diffusion systems have become one of the paradigms for the formation of spatio-temporal patterns in systems far from thermodynamic equilibrium. The formation of such fascinating structures as spiral waves, spatially-regular, stationary patterns with different symmetries (hexagonal, stripe, etc.) as well as chemical turbulence have made reaction-diffusion systems the subject of numerous ongoing investigations. In many systems, however, the transport process is not via normal diffusion but rather via anomalous diffusion which can be slower (subdiffusion) or faster (superdiffusion) than normal diffusion. Subdiffusion often occurs in biogels, porous media and polymers, while superdiffusion is typical of some processes in plasmas, semiconductors, surface reactions and many others. Although many aspects of anomalous diffusion have been studied, nonlinear dynamic and pattern formation aspects were the subject of only a very limited number of works. These preliminary studies demonstrate the significance of anomalous diffusion and the necessity for its systematic investigation. The PIs will conduct systematic studies of the effect of anomalous diffusion on the formation of Turing patterns and spatially nonuniform oscillating patterns as well as spiral waves in reaction-diffusion systems, with specific applications to combustion waves, isothermal frontal polymerization waves, and controlled drug delivery.
