Samenvatting
A very fascinating phenomenon in nature is the spontaneous occurrence of patterns and other kinds of spatial structures. This can be found in a wide variety of disciplines although the structures itself are often very similar. For a long time, thes phenomena were studied separately in many of these different disciplines. It has become clear though, that the underlying principles and hence the description of such phenomena are in fact very universal.In more recent times, there has been an increased interest in so-called "nonlocal effects". Simply put, this means that the evolution in time of a point in space is not only determined by the physical situation at that particular point, but also by the situation in its vicinity. The existence of such nonlocal effects has been reported in many scientific disciplines.
In this master thesis, the effects of such a nonlocal interaction on the existence and stability of spatial structures in dissipative systems will be studied. Because the physical principles are so universal, there exist very generic model equations. This work will be done within the framework of Ginzburg-Landau equations. These are widely applicable models that allow to describe the formation of certain spatial structures in dissipative systems.
In the first chapter, the necessary concepts for this work are introduced and the outline and goals of this master thesis are explained in more detail.
The second chapter will be a thorough study of the consequences of a nonlocal interaction on the stability and existence of the spatial structures that are considered. This stability analysis will show that a nonlocal interaction will result in the existence of localized structures in the Real Ginzburg-Landau equation. This is a model where localized structures do not exist in its local form. For a more general model that already has localized structures in its local form, the influence of a nonlocal interaction is also analyzed.
In the third chapter, some results of the stability analysis will be verified. By relying on numerical techniques, we will gain more insight on the influence of a nonlocal interaction on the spatial dynamics.
The last chapter will consist of the conclusions of this work and some future perspectives.
| Datum prijs | 2009 |
|---|---|
| Originele taal | English |
| Begeleider | Jan Danckaert (Promotor), Lendert Gelens (Jury), Irina Veretennicoff (Jury), Guy Van Der Sande (Jury), Marc Sioen (Jury) & Tlidi Mustapha (Jury) |