Wavenumber selection via spatial parameter jump

TL;DR

This paper investigates how a spatial parameter jump affects pattern formation in the Swift-Hohenberg equation, revealing a narrow wavenumber band that expands linearly with the jump size, contrasting with the usual square root growth.

Contribution

It introduces a novel analysis of spatial inhomogeneity effects on pattern selection, demonstrating a linear growth of the wavenumber band with the jump size.

Findings

Wavenumber band width grows linearly with jump size

Existence of steady-state solutions transitioning from homogeneous to patterned states

Numerical simulations confirm theoretical predictions

Abstract

The Swift-Hohenberg equation describes an instability which forms finite-wavenumber patterns near onset. We study this equation posed with a spatial inhomogeneity; a jump-type parameter that renders the zero solution stable for $x<0$ and unstable for $x>0$. Using normal forms and spatial dynamics, we prove the existence of a family of steady-state solutions that represent a transition in space from a homogeneous state to a striped pattern state. The wavenumbers of these stripes are contained in a narrow band whose width grows linearly with the size of the jump. This represents a severe restriction from the usual constant-parameter case, where the allowed band grows with the square root of the parameter. We corroborate our predictions using numerical continuation and illustrate implications on stability of growing patterns in direct simulations.