On the Ginzburg-Landau approximation for quasilinear pattern forming reaction-diffusion-advection systems

TL;DR

This paper proves that the Ginzburg-Landau equation accurately predicts the dynamics of quasilinear pattern-forming reaction-diffusion-advection systems near the first instability, using a simple, broadly applicable theorem.

Contribution

It introduces a straightforward theorem based on maximal regularity results that validates the Ginzburg-Landau approximation for these complex systems.

Findings

The theorem is successfully applied to the Gray-Scott-Klausmeier model.

The approach is general and applicable to various reaction-diffusion-advection systems.

The Ginzburg-Landau equation is confirmed as a reliable predictor near the first instability.

Abstract

We prove that the Ginzburg-Landau equation correctly predicts the dynamics of quasilinear pattern-forming reaction-diffusion-advection systems, close to the first instability. We present a simple theorem which is easily applicable for such systems and relies on key maximal regularity results. The theorem is applied to the Gray-Scott-Klausmeier vegetation-water interaction model and its application to general reaction-diffusion-advection systems is discussed.