论文标题
六边形模式形成的动力过渡理论
Dynamical Transition Theory of Hexagonal Pattern Formations
论文作者
论文摘要
本文的主要目标是从动态过渡论的角度了解六角形模式的形成。我们考虑从抽象非线性耗散系统的稳态过渡。为了阐明混合模式的形成，例如六角形模式，我们考虑了系统的线性化操作员具有两个关键的真实特征值，以临界值$λ_c$的控制参数$λ$，而相关的eigenmodes具有滚动和矩形模式的相关特征。通过使用中心歧管减少，我们获得了临界过渡值$λ_c$的系统的简化方程。通过对这些方程式的分析，我们充分表征了所有可能的过渡场景，而当还原方程的二次部分的系数不会消失。我们考虑三个问题，这是2D Swift-Hohenberg方程的两个变体和3D表面张力驱动的对流，以证明我们在这里获得的所有主要理论结果确实可以实现。
The main goal of this paper is to understand the formation of hexagonal patterns from the dynamical transition theory point of view. We consider the transitions from a steady state of an abstract nonlinear dissipative system. To shed light on the formation of mixed mode patterns such as the hexagonal pattern, we consider the case where the linearized operator of the system has two critical real eigenvalues, at a critical value $λ_c$ of a control parameter $λ$ with associated eigenmodes having a roll and rectangular pattern. By using center manifold reduction, we obtain the reduced equations of the system near the critical transition value $λ_c$. By a through analysis of these equations, we fully characterize all possible transition scenarios when the coefficients of the quadratic part of the reduced equations do not vanish. We consider three problems, two variants of the 2D Swift-Hohenberg equation and the 3D surface tension driven convection, to demonstrate that all the main theoretical results we obtain here are indeed realizable.
