Pattern formation in the damped Nikolaevskiy equation

TL;DR

This paper investigates how weak damping affects pattern formation and stability in the Nikolaevskiy equation, revealing a critical damping threshold that leads to instability of all periodic solutions.

Contribution

It introduces an analysis of damping effects on the Nikolaevskiy equation, combining numerical methods and weakly nonlinear analysis to identify stability thresholds.

Findings

All periodic solutions are unstable at zero damping.

A critical damping value exists where all solutions become unstable.

The transition to Eckhaus instability is characterized by a cusp near the marginal stability curve.

Abstract

The Nikolaevskiy equation has been proposed as a model for seismic waves, electroconvection and weak turbulence; we show that it can also be used to model transverse instabilities of fronts. This equation possesses a large-scale "Goldstone" mode that significantly influences the stability of spatially periodic steady solutions; indeed, all such solutions are unstable at onset, and the equation exhibits so-called soft-mode turbulence. In many applications, a weak damping of this neutral mode will be present, and we study the influence of this damping on solutions to the Nikolaevskiy equation. We examine the transition to the usual Eckhaus instability as the damping of the large-scale mode is increased, through numerical calculation and weakly nonlinear analysis. The latter is accomplished using asymptotically consistent systems of coupled amplitude equations. We find that there is a critical value of the damping below which (for a given value of the supercriticality parameter) all periodic steady states are unstable. The last solutions to lose stability lie in a cusp close to the left-hand side of the marginal stability curve.