Pattern formation under the influence of confinement and conservation

Abstract

This thesis consists of two parts dealing with different so far unsolved problems in the field of pattern formation theory. The first part studies the effects of restricting pattern formation to a finite domain - a scenario that is omnipresent in nature. In the second part we identify and investigate a new phase separation phenomenon in active systems with a conservation law - the so-called active phase separation. In a first publication we show that physical boundaries generically lead to a reflection effect for nonlinear traveling waves. This reflection forces systems that show traveling waves in large extended systems into a standing wave pattern if the system becomes sufficiently short. We also identify bands of stable standing waves with different numbers of nodes, allowing for transitions between different standing wave patterns. This generic result is especially relevant for the Min protein system that plays a crucial role in the cell division process of the bacterium E. coli. Thereby the Min proteins show a traveling wave pattern on large extended membranes in in vitro experiments, while inside a cell a standing wave-like pattern is observed. Finite domains for patterns can also be generated without hard physical boundaries. Instead the control parameter that switches the system between a patterned state and a homogeneous state can be varied spatially in a way that it suppresses the pattern in one region and allows it in another. A possible experimental realization for this scenario are light-sensitive chemical reactions where the pattern formation process can be enhanced or inhibited using an illumination mask. We figure out that the steepness of the variation from a sub- to a supercritical control parameter influences the orientation of stripe patterns in two spatial dimensions. For steep step-like control parameter drops, the stripes favor a orientation parallel to the control parameter variation. For smooth ramp-like drops on the other hand, they favor a perpendicular orientation. This also implies that the orientation of stripes will switch from parallel to perpendicular when decreasing the steepness of the drop. This transition can be understood with the decreasing importance of local resonance effects induced by the control parameter drop. In another way, a control parameter drop also influences traveling wave pattern in one dimension. While again local resonance effects are important, the control parameter drop there leads to four different wave patterns depending on the group velocity. For small group velocities, the traveling wave pattern thereby fills the whole supercritical domain forming a filled state. Increasing the group velocity will confine the pattern to one side of the supercritical domain. Even higher group velocities induce a state with a time-dependent amplitude of the wave pattern - a so-called blinking state. Thereby both left- and right-traveling waves occur whose amplitudes change periodically in time. Increasing the group velocity further leads to a return of a wave…