title: Instability in reaction-superdiffusion systems.
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 spatiotemporal patterns. Finally, the effect of superdiffusion on the instability in the 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.
authors.affiliation: Department of Physics, Tafresh University, Tafresh 39518 79611, Iran.
authors.forename: Reza
authors.initials: R
authors.lastname: Torabi
authors.affiliation: Department of Physics, Tafresh University, Tafresh 39518 79611, Iran.
authors.forename: Zahra
authors.initials: Z
authors.lastname: Rezaei