Ginzburg-Landau amplitude equation for nonlinear nonlocal models

TL;DR

This paper demonstrates that the Ginzburg-Landau equation universally describes large-scale pattern modulations near instability in nonlinear nonlocal models, and introduces a new framework to derive its coefficients from diverse systems.

Contribution

It establishes the broad applicability of the Ginzburg-Landau equation for nonlocal models and presents a novel method to analytically determine its coefficients from model parameters.

Findings

Ginzburg-Landau equation describes pattern modulations at instability onset.

The new framework derives analytical coefficients for the equation.

Applicability extends to models with higher order nonlocal interactions.

Abstract

Regular spatial structures emerge in a wide range of different dynamics characterized by local and/or nonlocal coupling terms. In several research fields this has spurred the study of many models, which can explain pattern formation. The modulations of patterns, occurring on long spatial and temporal scales, can not be captured by linear approximation analysis. Here, we show that, starting from a general model with long range couplings displaying patterns, the spatio-temporal evolution of large scale modulations at the onset of instability is ruled by the well-known Ginzburg-Landau equation, independently of the details of the dynamics. Hence, we demonstrate the validity of such equation in the description of the behavior of a wide class of systems. We introduce a novel mathematical framework that is also able to retrieve the analytical expressions of the coefficients appearing in the Ginzburg-Landau equation as functions of the model parameters. Such framework can include higher order nonlocal interactions and has much larger applicability than the model considered here, possibly including pattern formation in models with very different physical features.