半球形斑图形成的动力学跃迁理论
Dynamical Transition Theory of Hexagonal Pattern Formations
摘要：
本文的主要目的是从动力学跃迁理论的角度来理解六边形图案的形成。我们考虑了一个抽象的非线性耗散系统从定态到稳定态的转变。为了阐明混合模式的形成，如六边形模式，我们考虑的情况下，系统的线性化运营商有两个关键的实本征值，在一个临界值$λ_c$的控制参数$λ$与相关的本征模式具有滚动和矩形图案。通过中心流形约化，得到了系统在临界转变值$λ_c$附近的约化方程。通过对这些方程的分析，我们充分刻画了当约化方程的二次部分的系数不为零时所有可能的过渡情形。我们考虑三个问题，两个变种的2D Swift-Hohenberg方程和3D表面张力驱动的对流，以证明我们在这里得到的所有主要的理论结果确实是可实现的。
英文摘要：
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.
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