Symmetric patterns in the Dirichlet problem for a two-cell cubic autocatalytor reaction model

摘要

By using an approach developed by one of the authors, approximate solutions of the soft periodic boundary conditions for a two-cell reaction diffusion model have been obtained. The system is considered with reactant A and autocatalyst B. The reaction is taken cubic in the autocatalyst in the two-cell with linear exchange through A. The formal exact solution is obtained which is symmetric with respect to the mid-point of the container. Approximate solutions are found through the Picard iterative sequence of solutions constructed after the exact one. It is found that the solution obtained is not unique. When the initial conditions are periodic, the most dominant modes initiate to traveling waves in systems with moderate size. Symmetric configurations forming a parabolic one for large time are observed. In systems of large size, spatially symmetric chaos are produced which are stationary in time. Furthermore, it is found the symmetric pattern formation hold irrespective of the condition of linear instability against small spatial disturbance.