Diffusive stability of Turing patterns via normal forms

TL;DR

This paper develops a normal form approach to analyze the stability of Turing patterns in reaction-diffusion systems on the real line, revealing an approximate conservation law and proving nonlinear stability for small perturbations.

Contribution

It introduces a novel normal form coordinate system that captures the dynamics near Turing patterns and establishes their nonlinear stability with sharp decay rates.

Findings

Normal form construction reveals an approximate discrete conservation law.

Proved nonlinear stability of Turing patterns for small perturbations.

Established sharp decay rates for perturbations in $L^1\cap L^\infty$.

Abstract

We investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a "normal form" coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction-diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove nonlinear stability of Turing patterns with respect to perturbations that are small in $L^1\cap L^\infty$, with sharp rates.