Joint Math/CMSS Colloquium

Abstract: Reaction-diffusion systems, which describe numerous phenomena in nature, exhibit different types of instabilities leading to pattern formation and spatio-temporal chaos. It has been recently understood that many random processes need for their description models of anomalous diffusion, that explain the observation of anomalously slow (subdiffusion) or fast (superdiffusion) growth of displacement moments of corresponding random walks. These models include a temporal memory and a spatial non-locality and involve integral operators in addition to differential operators. The interplay between the anomalous diffusion and the reactions is still hardly investigated. In the present talk, we analyze the development of instabilities in the systems with reactions and anomalous diffusion. In the case of superdiffusion (Levy flights), we use a model that involves a Riesz fractional spatial derivative. For oscillatory unstable reactions, we derive superdiffusive (nonlocal) complex Ginzburg-Landau and Kuramoto-Sivashinsky equations, and investigate their general properties and dynamics. In the case of subdiffusion, we construct a model that takes into account the change of the chemical composition between the jumps of molecules. The model includes a partial differential equation describing the reaction and "aging" of molecules, which is coupled with an integro-differential equation describing the random walk. We investigate the development of short-wave monotonic (Turing) and long-wave oscillatory (Hopf) instability and derive an amplitude (Ginzburg-Landau) equation for weakly nonlinear patterns which is free from non-localities and memory effects.