Formation of localized structures in bistable systems through nonlocal spatial coupling II: The nonlocal Ginzburg Landau Equation

TL;DR

This paper investigates how nonlocal spatial coupling affects front interactions in the Ginzburg-Landau equation, revealing conditions under which localized structures can form due to oscillatory front behavior.

Contribution

It demonstrates that nonlocal terms can induce spatial oscillations in fronts, enabling localized structures, with analysis across different kernel types including positive-definite and Mexican-hat kernels.

Findings

Nonlocal coupling induces spatial oscillations in fronts.

Localized structures can emerge from front pinning.

Different kernels have distinct effects on front dynamics.

Abstract

We study the influence of a linear nonlocal spatial coupling on the interaction of fronts connecting two equivalent stable states in the prototypical 1-D real Ginzburg-Landau equation. While for local coupling the fronts are always monotonic and therefore the dynamical behavior leads to coarsening and the annihilation of pairs of fronts, nonlocal terms can induce spatial oscillations in the front, allowing for the creation of localized structures, emerging from pinning between two fronts. We show this for three different nonlocal influence kernels. The first two, mod-exponential and Gaussian, are positive-definite and decay exponentially or faster, while the third one, a Mexican-hat kernel, is not positive definite.