Nonlinear stability at the Eckhaus boundary

TL;DR

This paper investigates the diffusive stability of spatially periodic equilibria at the Eckhaus boundary in the real Ginzburg-Landau equation, revealing that nonlinear effects are crucial for understanding stability at this critical point.

Contribution

It establishes the diffusive stability of equilibria exactly at the Eckhaus boundary, accounting for nonlinear effects that are marginal in the linearized dynamics.

Findings

Equilibria at the Eckhaus boundary are diffusively stable.

Nonlinear terms are marginal and influence the stability analysis.

The limit profile is governed by a nonlinear equation.

Abstract

The real Ginzburg-Landau equation possesses a family of spatially periodic equilibria. If the wave number of an equilibrium is strictly below the so called Eckhaus boundary the equilibrium is known to be spectrally and diffusively stable, i.e., stable w.r.t. small spatially localized perturbations. If the wave number is above the Eckhaus boundary the equilibrium is unstable. Exactly at the boundary spectral stability holds. The purpose of the present paper is to establish the diffusive stability of these equilibria. The limit profile is determined by a nonlinear equation since a nonlinear term turns out to be marginal w.r.t. the linearized dynamics.