论文标题
一圈钉子
A ring of spikes
论文作者
论文摘要
对于Schnakenberg模型，我们考虑了n个尖峰的高度对称配置，其位置位于单位磁盘或环内的常规n gon的顶点。我们将这种配置称为钉环。环半径以修饰的绿色功能为特征。对于磁盘，我们发现相对于小特征值，一个9或更多的尖峰环总是不稳定的。相反，只要馈电率$ a $就足够大，磁盘内的8个或更少的尖峰环在磁盘内稳定。更普遍地，对于足够高的进料率，只要环足足够薄，就可以稳定$ n $尖峰的环。随着$ a $的减少，我们表明该环首先是由于小特征值而稳定的，然后是由于较大的特征值，尽管这两个阈值都被渐近少量的距离分开。对于磁盘内8个尖峰的环，不稳定似乎是超临界的，并将环变形为类似正方形的配置。对于不到8个尖峰，这种不稳定性是亚临界的，导致峰值死亡。
For the Schnakenberg model, we consider a highly symmetric configuration of N spikes whose locations are located at the vertices of a regular N-gon inside either a unit disk or an annulus. We call such configuration a ring of spikes. The ring radius is characterized in terms of the modified Green's function. For a disk, we find that a ring of 9 or more spikes is always unstable with respect to small eigenvalues. Conversely, a ring of 8 or less spikes is stable inside a disk provided that the feed-rate $A$ is sufficiently large. More generally, for sufficiently high feed-rate, a ring of $N$ spikes can be stabilized provided that the annulus is thin enough. As $A$ is decreased, we show that the ring is destabilized due to small eigenvalues first, and then due to large eigenvalues, although both of these thresholds are separated by an asymptotically small amount. For a ring of 8 spikes inside a disk, the instability appears to be supercritical, and deforms the ring into a square-like configuration. For less than 8 spikes, this instability is subcritical and results in spike death.
