Growing stripes, with and without wrinkles

TL;DR

This paper investigates stripe formation in the Swift-Hohenberg equation with a moving quenching line, analyzing how stripe orientation and wrinkles depend on quenching speed and aspect ratio using analytical, numerical, and simulation methods.

Contribution

It introduces a comprehensive analysis of stripe and wrinkle patterns formed by directional quenching in the Swift-Hohenberg and related models, combining multiple approaches.

Findings

Stripes form perpendicular and oblique to the quenching line.

Wrinkles occur in oblique stripe patterns.

Stripe orientation depends on quenching speed and aspect ratio.

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.