Pattern Formation in the wake of growth mechanisms
Externally mediated, or “triggered,” spatial patterns have become a topic of recent interest in many fields, where researchers wish to harness natural pattern forming processes to form novel and functional materials. Such mechanisms arise in many different types of natural and man-made physical systems such as quenching interfaces in alloy melts, moving masks in ion bombardment, growing interfaces in biological systems, and traveling reaction fronts. These mechanisms can be encoded mathematically in a step-like parameter dependence that allows patterns in a half plane, and suppresses them in the complement, while the boundary of the pattern-forming region propagates with fixed normal velocity. We use techniques from dynamical systems and functional analysis, as well as numerical continuation algorithms, to explore how these mechanisms affect the pattern left in the wake, finding for example how the wavenumbers depend on speed of the parameter interface. In this talk we will discuss recent results in both one and two spatial dimensions in a variety of different pattern forming systems such as the Cahn-Hilliard, Complex Ginzburg-Landau, Swift-Hohenberg, and reaction-diffusion systems.