论文标题
对奇异扰动反应扩散系统局部斑点花生形变形的弱非线性分析
Weakly Nonlinear Analysis of Peanut-Shaped Deformations for Localized Spots of Singularly Perturbed Reaction-Diffusion Systems
论文作者
论文摘要
在较大的扩散比的奇异极限中，在空间定位的2-D斑点上发生了多种多样的两个组件反应扩散系统。众所周知，这种局部，远程平衡的模式可以表现出各种不同的不稳定性，例如呼吸振荡，斑点歼灭和斑点自我复制行为。先前对Schnakenberg和Brusselator系统的数值模拟表明，局部斑点的局部花生形线性不稳定性是启动完全非线性的点自我复制事件的机制。 从弱非线性理论的开发和实施以实现局部斑点的形状变形，可以通过正常形式振幅方程来表明，稳态斑点的花生形线性不稳定性对于Schnakenberg和Brusselator反应 - 反应扩散系统始终是亚临界值。通过使用全局分叉软件{\ em pde2path} [H. 观点。
Spatially localized 2-D spot patterns occur for a wide variety of two component reaction-diffusion systems in the singular limit of a large diffusivity ratio. Such localized, far-from-equilibrium, patterns are known to exhibit a wide range of different instabilities such as breathing oscillations, spot annihilation, and spot self-replication behavior. Prior numerical simulations of the Schnakenberg and Brusselator systems have suggested that a localized peanut-shaped linear instability of a localized spot is the mechanism initiating a fully nonlinear spot self-replication event. From a development and implementation of a weakly nonlinear theory for shape deformations of a localized spot, it is shown through a normal form amplitude equation that a peanut-shaped linear instability of a steady-state spot solution is always subcritical for both the Schnakenberg and Brusselator reaction-diffusion systems. The weakly nonlinear theory is validated by using the global bifurcation software {\em pde2path} [H.~Uecker et al., Numerical Mathematics: Theory, Methods and Applications, {\bf 7}(1), (2014)] to numerically compute an unstable, non-radially symmetric, steady-state spot solution branch that originates from a symmetry-breaking bifurcation point.
