Instabilities Two- and Three-Dimensional Patterns

There are a number of important differences between quadratic and cubic fronts however, the most striking is found in their behavior in two- and three-dimensional configurations. We shall focus on the two-dimensional case in which the diffusivities of A and B may take on significantly different values. (Similar behavior is found in the three-dimensional case.) now use a two-dimensional Laplacian defined by -t- dVdV with  

The behavior found for the quadratic system in the one-dimensional case is directly applicable to the two-dimensional configuration planar fronts are exhibited with velocities given by Eq. [88] for the case of equal diffusivities of A and B. When the diffusivities significantly differ, planar fronts are still observed however, now the velocity scales with the diffusion coefficient D = Dr according to Eq. [89]. The one-dimensional solution for the cubic system is also valid for the two-dimensional configuration however, the cubic front may exhibit lateral instabilities that are not observed in the quadratic system. will now consider the stability of cubic autocatalysis fronts. 

The opposite is true along the concave segments, where the wave front is retarded relative to the planar front. Here, there is a diffusive focusing of B into the region ahead of the front, leading to a local increase in wave velocity. These local increases and decreases in propagation velocity tend to eliminate die local curvature. Thus we may say that the diffusion of the autocatalyst has a stabiliz-ing effect on the planar wave. 

Based on these qualitative arguments, we may predict that for 8 1, in which the diffusion coefficient of the autocatalyst is greater than that of the reactant, the overall tendency will be a stabilization of the planar front. For 8 1, however, we anticipate that the planar front will lose stability as the destabilizing influence of the reactant diffusion becomes dominant. To test this prediction, Eqs. [91] were numerically integrated with 8 = 1 and 8 = 5. The initial conditions correspond to a discontinuity in the concentrations of A and B a = 0, p = 1 for X s Xq and a = 1, p = 0 for x Xg at some convenient Xq for all y. To perturb the planar front, the middle third in the y direction was displaced sli tly forward. For 8 = 1, the perturbation eventually completely decays away to yield a planar front. For 8 = 5, the perturbation evolves to produce a distinctly nonplanar front in which four spatial oscillations in the y direction are displayed. The concentration profiles of a and P for the front are 

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