论文标题
二嵌段共聚物模型中的循环对称诱导的干草叉分叉
Cyclic symmetry induced pitchfork bifurcations in the diblock copolymer model
论文作者
论文摘要
用于二嵌段共聚物的Ohta-Kawasaki模型表现出丰富的平衡分叉结构。即使在一维基域上，分叉集的特征在于高水平的多稳定性和许多次要分叉点。这些分叉中的许多是干草叉类型的。在先前的工作中，作者表明，如果干草叉分叉是由简单的$ \ mathbb {z} _2 $ symetry-Breaking引起的，那么可以使用计算机辅助的证明技术来使用扩展系统来严格验证它们。但是，许多二嵌段共聚物干草叉分叉不能以这种方式进行治疗。在本文中，我们表明，在这些涉及的情况下，基于偶数的循环群，循环群体行动是为了使它们的存在负责。我们提出了建立此类分叉点的理论结果，并表明它们可以被描述为合适的非线性扩展非线性系统的非排定溶液。使用后者的表征，我们还证明了计算机辅助的证明技术可用于验证此类分叉。尽管本文提出的方法仅应用于二嵌段共聚物模型，但我们希望它们也适用于其他抛物线偏微分方程。
The Ohta-Kawasaki model for diblock copolymers exhibits a rich equilibrium bifurcation structure. Even on one-dimensional base domains the bifurcation set is characterized by high levels of multi-stability and numerous secondary bifurcation points. Many of these bifurcations are of pitchfork type. In previous work, the authors showed that if pitchfork bifurcations are induced by a simple $\mathbb{Z}_2$ symmetry-breaking, then computer-assisted proof techniques can be used to rigorously validate them using extended systems. However, many diblock copolymer pitchfork bifurcations cannot be treated in this way. In the present paper, we show that in these more involved cases, a cyclic group action is responsible for their existence, based on cyclic groups of even order. We present theoretical results establishing such bifurcation points and show that they can be characterized as nondegenerate solutions of a suitable extended nonlinear system. Using the latter characterization, we also demonstrate that computer-assisted proof techniques can be used to validate such bifurcations. While the methods proposed in this paper are only applied to the diblock copolymer model, we expect that they will also apply to other parabolic partial differential equations.
