Instability in Reaction-Superdiffusion Systems

TL;DR

This paper investigates how superdiffusion influences instabilities in reaction-diffusion systems, revealing new stability criteria, deriving amplitude equations, and demonstrating effects on pattern formation through numerical simulations.

Contribution

It introduces a generalized free energy for reaction-superdiffusion systems, derives a fractional Ginzburg-Landau equation near Hopf bifurcations, and explores superdiffusion's impact on the Brusselator model.

Findings

Superdiffusion modifies the stability and dynamics near Turing bifurcations.

A fractional complex Ginzburg-Landau equation describes amplitude near Hopf bifurcations.

Numerical simulations show superdiffusion affects spatio-temporal pattern formation.

Abstract

We study the effect of superdiffusion on the instability in reaction-diffusion systems. It is shown that reaction-superdiffusion systems close to a Turing instability are equivalent to a time-dependent Ginzburg-Landau model and the corresponding free energy is introduced. This generalized free energy which depends on the superdiffusion exponent governs the stability, dynamics and the fluctuations of reaction-superdiffusion systems near the Turing bifurcation. In addition, we show that for a general n-component reaction-superdiffusion system, a fractional complex Ginzburg- Landau equation emerges as the amplitude equation near a Hopf instability. Numerical simulations of this equation are carried out to illustrate the effect of superdiffusion on spatio-temporal patterns. Finally the effect of superdiffusion on the instability in Brusselator model, as a special case of reaction-diffusion systems, is studied. In general superdiffusion introduces a new parameter that changes the behavior of the system near the instability.