|Ph.D Thesis||Department of Applied Mathematics|
|Supervisor:||Prof. Nepomniashchy Alexander|
Formation of patterns in reaction - diffusion processes was studied in media subject to sub-diffusion or Levy flights. Sub-diffusive systems possess memory, modelled mathematically by integro-differential operators instead of the regular time derivative. The operators are not unique and are adjusted according to the physical properties of specific systems. A linear stability analysis was performed for two general models, and converse changes of stability properties were revealed. The first, more traditional model, introduced memory via a fractional derivative and resulted in the system's destabilization relatively to the normal case. The second model used a more complicated memory kernel and stabilised the system. In addition, a specific model was considered, stemming from the application of tumour cell invasion .
It was shown to bear stability properties compatible with the analysis of the first two general models .Amplitude equations near Turing and Hopf bifurcation points were obtained by a weakly non-linear analysis of the second model. It was shown that the universal behaviour near these points was preserved and governed by Ginzburg-Landau equations as long as the temporal scale of the memory process and the modulation evolution were set apart. The equation coefficients changed due to the presence of anomaly .
Super-diffusion of Levy flight type allows for an enhanced transport and is modelled by a fractional Laplacian. A reaction - diffusion system of this type possesses a Hopf bifurcation point similarly to a normal one, but the corresponding amplitude and non-linear phase diffusion equations are of a fractional order. In the extended parameter space the anomaly yields shifts of the phase diagram boundaries .
The real version of complex Ginzburg-Landau equation of a fractional order was used to study properties of front propagation in the context of phase transition in super-diffusive media. It was shown that the anomalous fronts are characterised by algebraically decaying tails instead of the normal exponential decay. The motion of domain walls and in particular coarsening was shown to occur on anomaly dependent scales, generalising the normal formulae. The normal properties of bidimensional fronts were extended for the anomalous case as well .
Anomalous diffusion bears intriguing and often unintuitive results in the context of stability and pattern formation. Careful adjustment of the model to the system is crucial for correct prediction of unstable modes and behaviour in close vicinity of bifurcations.