报告时间：10:00-11:00,2016.8.8

报告地点：D223

 Abstract:

Spatially localized states in pattern forming PDEs often result from bistability between a spatially homogeneous state and a spatially periodic state, and can be understood from the dynamical systems perspective. This talk focuses on the 1:1 forced complex Ginzburg-Landau equation, which is the normal form PDE for harmonically forced Hopf bifurcations in spatially extended systems.

In this PDE, a new type of localized patterns exist in parameter regimes where two spatially homogeneous equilibria exist on the upper and lower segments of an S-shaped bifurcation curve, and a supercritical Turing bifurcation occurs on one of these equilibria. Numerical continuation reveals the existence of steady localized wavetrains in 1D and localized target patterns in 2D. The bifurcation structures of these localized patterns differ significantly from classical pattern forming PDEs such as the Swift-Hohenberg equation. Spectral stability of these steady localized patterns also exhibits new features intimately related to the underlying bifurcation structures. Direct numerical simulation reveals the depinning dynamics of these steady localized patterns, as well as the existence of fully 2D localized patterns which are not necessarily steady, including localized hexagonal patterns bounded by circular and planar fronts.