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

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 mediumsized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.Nonlinear stripe patterns are ubiquitous in nature, and their driving mechanisms are as diverse as the systems themselves in which they occur [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] . Stripe patterns have the universal property of being stable at different values of the wavenumber, and these stable wavenumber regions are the so-called Busse balloons after their pioneer 1,3,11,16,17 . To the instabilities bounding the stable wavenumber range of stripe patterns count the generic Eckhaus instability, a longwavelength longitudinal (compressional) instability, and the zigzag instability, a long-wavelength transverse instability 1,18 .Stable wavenumber ranges are restricted even in large systems by pattern suppressing boundary conditions at the domain sides 19,20 or are even selected by spatial inhomogeneities, e.g. via so-called ramps [21][22][23] . In contrast, in short systems the stable wavenumber range can be enlarged, as in Ref. 24 for quasione-dimensional systems predicted and experimentally confirmed in Ref. 25,26 . Such a range extension depends on the boundary conditions and the second spatial dimension as we explain in this work analytically and numerically by investigating the generic Swift-Hohenberg model and the universal Newell-Whitehead-Segel equation in rectangular domains. Finite size effects on patterns are also highly relevant for Turing patterns in small systems, as for instance in cells [27][28][29][30] .