Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

TL;DR

This paper investigates how nonuniform forcing gradients influence stripe pattern orientation in the Swift-Hohenberg equation, revealing stability conditions and competition with other effects through analysis and numerical simulations.

Contribution

It introduces a detailed analysis of stripe pattern orientation under nonuniform forcing in the Swift-Hohenberg model, highlighting the competition between gradient effects and other influences.

Findings

Stripes aligned with the gradient are stable to small perturbations.

Perpendicular stripes become unstable when aligned against the gradient.

Forcing gradients can reorient existing stripe patterns.

Abstract

Spatio-temporal pattern formation in complex systems presents rich nonlinear dynamics which leads to the emergence of periodic nonequilibrium structures. One of the most prominent equations for the theoretical and numerical study of the evolution of these textures is the Swift-Hohenberg (SH) equation, which presents a bifurcation parameter (forcing) that controls the dynamics by changing the energy landscape of the system, and has been largely employed in phase-field models. Though a large part of the literature on pattern formation addresses uniformly forced systems, nonuniform forcings are also observed in several natural systems, for instance, in developmental biology and in soft matter applications. In these cases, an orientation effect due to forcing gradients is a new factor playing a role in the development of patterns, particularly in the class of stripe patterns, which we investigate through the nonuniformly forced SH dynamics. The present work addresses amplitude instability of stripe textures induced by forcing gradients, and the competition between the orientation effect of the gradient and other bulk, boundary, and geometric effects taking part in the selection of the emerging patterns. A weakly nonlinear analysis suggests that stripes are stable with respect to small amplitude perturbations when aligned with the gradient, and become unstable to such perturbations when when aligned perpendicularly to the gradient. This analysis is vastly complemented by a numerical work that accounts for other effects, confirming that forcing gradients drive stripe alignment, or even reorient them from preexisting conditions. However, we observe that the orientation effect does not always prevail in the face of competing effects, whose hierarchy is suggested to depend on the magnitude of the forcing gradient.