双稳态离散晶格中局域态的蛇形和孤立点
Snaking and isolas of localized states in bistable discrete lattices
摘要：
我们考虑离散双稳态Allen-Cahn方程中的局域态。该模型方程以最简单的方式结合了双稳态和局部细胞间耦合。稳定的局域态的存在是可能的钉扎到底层晶格;他们不存在于等效的连续方程。特别是，我们解决的存在性“隔离”：封闭曲线的分歧图中的解决方案。孤立点出现在一些非周期性的边界条件在一个空间维度，但似乎一般出现在两个维度。我们指出如何在1D的分叉图的功能有助于解释一些（不直观的）功能的分叉图在2D。
英文摘要：
We consider localized states in a discrete bistable Allen-Cahn equation. This model equation combines bistability and local cell-to-cell coupling in the simplest possible way. The existence of stable localized states is made possible by pinning to the underlying lattice; they do not exist in the equivalent continuum equation. In particular we address the existence of 'isolas': closed curves of solutions in the bifurcation diagram. Isolas appear for some non-periodic boundary conditions in one spatial dimension but seem to appear generically in two dimensions. We point out how features of the bifurcation diagram in 1D help to explain some (unintuitive) features of the bifurcation diagram in 2D.
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