Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensional
superdiffusive Brusselator model are analyzed. The superdiffusive Brusselator model
differs from its regular counterpart in that the Laplacian operator of the regular model
is replaced by ∂α/∂|ξ|α, 1 < α
< 2, an integro-differential operator that reflects the nonlocal behavior of
superdiffusion. The order of the operator, α, is a measure of the rate of
superdiffusion, which, in general, can be different for each of the two components. A
weakly nonlinear analysis is used to derive two coupled amplitude equations describing the
slow time evolution of the Turing and Hopf modes. We seek special solutions of the
amplitude equations, namely a pure Turing solution, a pure Hopf solution, and a mixed mode
solution, and analyze their stability to long-wave perturbations. We find that the
stability criteria of all three solutions depend greatly on the rates of superdiffusion of
the two components. In addition, the stability properties of the solutions to the
anomalous diffusion model are different from those of the regular diffusion model.
Numerical computations in a large spatial domain, using Fourier spectral methods in space
and second order Runge-Kutta in time are used to confirm the analysis and also to find
solutions not predicted by the weakly nonlinear analysis, in the fully nonlinear regime.
Specifically, we find a large number of steady state patterns consisting of a localized
region or regions of stationary stripes in a background of time periodic cellular motion,
as well as patterns with a localized region or regions of time periodic cells in a
background of stationary stripes. Each such pattern lies on a branch of such solutions, is
stable and corresponds to a different initial condition. The patterns correspond to the
phenomenon of pinning of the front between the stripes and the time periodic cellular
motion. While in some cases it is difficult to isolate the effect of the diffusion
exponents, we find characteristics in the spatiotemporal patterns for anomalous diffusion
that we have not found for regular (Fickian) diffusion.