Weakly non-linear dynamics in reaction -- diffusion systems with L\'{e}vy flights

TL;DR

This paper derives a fractional Ginzburg-Landau equation from reaction-diffusion systems with Lévy flights near a Hopf bifurcation, extending known properties of the classical equation to fractional cases.

Contribution

It introduces a fractional Laplacian into the amplitude equation of reaction-diffusion systems, generalizing the complex Ginzburg-Landau and Kuramoto-Sivashinsky equations.

Findings

Derived fractional Ginzburg-Landau equation near Hopf bifurcation.

Extended properties of classical equations to fractional Laplacian cases.

Established a fractional analogue of the Kuramoto-Sivashinsky equation.

Abstract

Reaction--diffusion equations with a fractional Laplacian are reduced near a long wave Hopf bifurcation. The obtained amplitude equation is shown to be the complex Ginzburg-Landau equation with a fractional Laplacian. Some of the properties of the normal complex Ginzburg-Landau equation are generalised for the fractional analogue. In particular, an analogue of Kuramoto-Sivashinsky equation is derived.