Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2007-01-26
Nonlinear Sciences
Pattern Formation and Solitons
To appear, Physica D
Scientific paper
10.1016/j.physd.2007.07.014
Certain two-component reaction-diffusion systems on a finite interval are known to possess mesa (box-like) steadystate patterns in the singularly perturbed limit of small diffusivity for one of the two solution components. As the diffusivity D of the second component is decreased below some critical value Dc, with Dc = O(1), the existence of a steady-state mesa pattern is lost, triggering the onset of a mesa self-replication event that ultimately leads to the creation of additional mesas. The initiation of this phenomena is studied in detail for a particular scaling limit of the Brusselator model. Near the existence threshold Dc of a single steady-state mesa, it is shown that an internal layer forms in the center of the mesa. The structure of the solution within this internal layer is shown to be governed by a certain core problem, comprised of a single non-autonomous second-order ODE. By analyzing this core problem using rigorous and formal asymptotic methods, and by using the Singular Limit Eigenvalue Problem (SLEP) method to asymptotically calculate small eigenvalues, an analytical verification of the conditions of Nishiura and Ueyema [Physica D, 130, No. 1, (1999), pp. 73-104], believed to be responsible for self-replication, is given. These conditions include: (1) The existence of a saddle-node threshold at which the steady-state mesa pattern disappears; (2) the dimple-shaped eigenfunction at the threshold, believed to be responsible for the initiation of the replication process; and (3) the stability of the mesa pattern above the existence threshold. Finally, we show that the core problem is universal in the sense that it pertains to a class of reaction-diffusion systems, including the Gierer-Meinhardt model with saturation, where mesa self-replication also occurs.
Kolokolnikov Theodore
Ward Martin J.
Wei Janyan
No associations
LandOfFree
Self-Replication of Mesa Patterns in Reaction-Diffusion Systems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Self-Replication of Mesa Patterns in Reaction-Diffusion Systems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Self-Replication of Mesa Patterns in Reaction-Diffusion Systems will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-568518