Diffusive stability of spatially periodic solutions of the Brusselator model

TL;DR

This paper rigorously analyzes the stability of Turing patterns in the Brusselator reaction-diffusion model, confirming that Ginzburg Landau equations accurately predict stability in the small-amplitude limit.

Contribution

It introduces a rigorous small-amplitude stability analysis of Turing patterns in the Brusselator model using Lyapunov-Schmidt reduction, validating the Ginzburg Landau approximation.

Findings

Stability predictions match Ginzburg Landau equations in the small-amplitude limit.

Validation of the Eckhaus criterion for pattern stability.

Rigorous confirmation of weakly unstable approximation.

Abstract

Applying the Lyapunov-Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift-Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg Landau amplitude equations, rigorously validating the standard weakly unstable approximation and Eckhaus criterion.