Patterns, defects, and phase singularities
I will discuss a few open questions related to defect formation in pattern-forming PDEs. I will start with numerical simulations of the complex Ginzburg-Landau (CGL) and the Swift-Hohenberg (SH) equations, and emphasize an interesting connection between defect formation and phase singularities. In the case of CGL, I will describe a formal approach that links phase gradients to defect creation. For SH, I will focus on a special category of defects: grain boundaries. My goal will be to discuss a well-known instability leading to the formation of dislocations at the core of these defects and to connect these observations with properties of the associated phase diffusion equation, the regularized Cross-Newell equation. This will be accomplished by combining analytical considerations with numerical results on the phase structure of grain boundaries, each approach being informed by and also guiding the other. I will finish with a list of open problems, some of which are currently being investigated. Parts of this work are in collaboration with Nick Ercolani (University of Arizona) and Nikola Kamburov (Pontificia Universidad Catolica de Chile).