Growing Stripes, with and without Wrinkles

Abstract

We present results on stripe formation in the Swift--Hohenberg equation with a directional quenching term. Stripes are “grown” in the wake of a moving parameter step line, and we analyze how the orientation of stripes changes depending on the speed of the quenching line and on a lateral aspect ratio. We observe stripes perpendicular to the quenching line, but also stripes created at oblique angles, as well as periodic wrinkles created in an otherwise oblique stripe pattern. Technically, we study stripe formation as traveling-wave solutions in the Swift--Hohenberg equation and in reduced Cahn--Hilliard and Newell--Whitehead--Segel models, analytically, through numerical continuation, and in direct simulations.