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. This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.
|Original language||English (US)|
|Journal||Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|
|State||Published - Apr 13 2018|
Bibliographical noteFunding Information:
The authors gratefully acknowledge support through NSF DMS-1612441.
Data accessibility. This article has no additional data. Competing interests. We declare we have no competing interests. Funding. The authors gratefully acknowledge support through NSF DMS-1612441.
- Inhomogeneous media
- Normal forms
- Spatial dynamics
- Turing instability