"Localized states in one and two spatial dimensions" by Yi-Ping Ma
April 7, 2011 - 7:00am
The 1:1 forced complex Ginzburg-Landau equation (FCGL) is a non-variational system that exhibits bistability between equilibria and thus admits traveling front solutions. A localized state consisting of an inner equilibrium embedded in an outer equilibrium can be formed by assembling two identical fronts back-to-back. In this talk, I will first describe the bifurcation structure of 1D steady localized states that takes the form of collapsed snaking (CS) if the inner equilibrium is temporally stable, and defect-mediated snaking (DMS) if the inner equilibrium is modulationally unstable. Outside their existence ranges, the steady localized states undergo time evolutions collectively referred to as depinning dynamics. Moving on to 2D, I will first introduce the temporal dynamics of quasi-1D periodic stripes leading to planar localized hexagons. In exploring fully 2D steady solutions, the bifurcation structure of radially symmetric localized states again depends on whether the inner equilibrium is temporally stable or modulationally unstable. The time evolution of these fully 2D localized states leads to either radially expanding or contracting fronts or to localized hexagons bounded by an axisymmetric front. At the end, I will describe the case when the inner equilibrium becomes Hopf unstable in time, which in turn yields localized spatiotemporal chaos that bears some resemblance to turbulent spots in shear flows.