Join Dr Priya Subramanian from the University of Leeds for this week's seminar.
The dynamics of many physical systems such as transitional fluid flows, soft matter and even biological systems often evolve to asymptotic states that exhibit spatial and temporal variations in their properties such as density, temperature, etc. Such patterns arise due to spontaneous symmetry breaking instabilities which commonly include loss of stability for the trivial (uniform) state along with either the promotion of a single preferred wavelength or two non-zero wavelengths. Nonlinear interactions in the system then determine the final stable pattern observed in experiments from among a group of possible patterned states.
Starting with a prototypical model for pattern formation, we will look at methods such as linear stability analysis, weakly nonlinear analysis and numerical continuation methods which can help predict results from numerical simulations of the model. We will also note that in systems in which the loss of stability is accompanied with the promotion of two different non-zero wavelengths, more complicated patterns such as superlattice patterns and quasipatterns can be stable configurations. Finally, 2D and 3D results of quasipatterns in a model for soft matter crystallization are presented.