Front motion and the growth of localized patterns
AbstractStationary localized patterns in bistable dissipative systems are confined by fronts connecting the pattern to a background homogeneous state. Such patterns are stationary whenever the fronts are pinned to the pattern state. When parameters are changed the fronts may unpin leading to the growth of the pattern as the pattern invades the stable homogeneous state. I will describe analytical techniques for computing the front speed focusing on the location and frequency of the phase slips that are necessary to grow the pattern at a rate that is consistent with the front motion. The process will be illustrated using numerical simulations of the Swift-Hohenberg equation and the forced complex Ginzburg-Landau equation in both one and two spatial dimensions.
This is joint work with R Krechetnikov (University of California at Santa Barbara) and Yi-ping Ma (University of Chicago).