On the band-width of stable nonlinear stripe patterns in finite size systems

TL;DR

This paper investigates how the stability range of nonlinear stripe patterns varies with system size and boundary conditions, using analytical and numerical methods on the Swift-Hohenberg model and related equations.

Contribution

It provides a comprehensive analysis of the influence of system size and boundary conditions on stripe pattern stability in finite systems, including new insights into pattern behavior in small domains.

Findings

Stable wavenumber range increases as system size decreases.

Boundary conditions significantly affect pattern stability and evolution.

Small systems can exhibit simultaneous stability of multiple patterns.

Abstract

Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geological scales as cloud patterns. They all share the universal property of being stable at different wavenumbers $q$, i.e., they are multistable. The stable wavenumber range of the stripe patterns, which is limited by the Eckhaus- and zigzag instabilities even in finite systems for several boundary conditions, increases with decreasing system size. This enlargement comes about because suppressing degrees of freedom from the two instabilities goes along with the system reduction, and the enlargement depends on the boundary conditions, as we show analytically and numerically with the generic Swift-Hohenberg (SH) model and the universal Newell-Whitehead-Segel equation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basic state is simultaneously stable in certain parameter ranges, which is especially important for Turing pattern in cells. In addition, we explain why below a certain system width stripe pattern behave quasi-one-dimensional in two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in medium-sized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.