Cahn-Hilliard Equations on Lattices: Dynamic Transitions and Pattern Formations

TL;DR

The paper develops a lattice-based analysis of the Cahn–Hilliard equation using center-manifold reductions and Ma–Wang dynamic transition theory to classify dynamic transitions as continuous or catastrophic. It treats multiplicities 2, 4, and 6 arising from lattice geometry, derives reduced cubic (and in resonant cases quadratic) dynamics, and identifies the resulting transition states and patterns, including rolls, squares, and hexagons (the latter enabled by lattice resonance). It also extends the analysis to long-range interactions, showing qualitatively similar transition scenarios with hexagonal patterns arising under appropriate conditions. The work provides explicit examples on square and parallelogram lattices and yields criteria, via parameters $\gamma_2,\gamma_3$ and the geometry-determined coefficients, for the emergence of specific patterns and the nature of the transitions. Overall, it offers a rigorous framework for predicting pattern selection and transition structure in lattice CH dynamics with practical implications for material science and pattern formation.

Abstract

This article examines the dynamic phase transitions and pattern formations attributed to binary systems modeled by the Cahn-Hilliard equation. In particular, we consider a two-dimensional lattice structure and determine how different choices of the spanning vectors influence the resulting dynamical tramsitions and pattern formations. As the basic steady-state loses its linear stability, the binary system undergoes a dynamic transition which is shown to be characterized by both the geometry of the domain and the choice of physical parameters of the model. Unlike rectangular domains, we are able to observe the emergence of hexagonally-packed circles, as well as the familiar rolls and square structures. We begin with the decomposition of our function space into a stable and unstable eigenspace before calculating the center manifold that maps the former to the later. In analyzing the resulting reduced equations, we consider the different multiplicities that the critical eigenvalue can have, which is shown to be geometry-dependent. We briefly consider the long-range interaction model and determine that it produces similar results to the original model.

Figures (9)

• Figure 3.1: Examples of various lattice structures in which the critical eigenvalue has multiplicity; (a) four, (b) two, and (c) six.
• Figure 5.1: Straight line orbits for: (a) $\xi+\eta<0$ and $\xi<0$; (b) $\xi+\eta>0$ and $\xi>0$; (c) $\xi+\eta>0$ and $\xi<0$; (d) $\xi+\eta<0$ and $\xi >0$.
• Figure 5.2: Horizontal rolls exhibited by the stationary solution $u(x,t)=\sqrt{2}\cos(50x_2)-\sqrt{2}\sin(50x_2)$.
• Figure 5.3: Vertical rolls exhibited by the stationary solution $u(x,t)=\sqrt{2}\cos(50x_1)-\sqrt{2}\sin(50x_1)$.
• Figure 5.4: Square-packed circles exhibited by stationary solution $u(x,t)=\sqrt{2}[ \cos(50x_1)-\sin(50x_1)+\cos(50x_2)-\sin(50x_2)]$.

Theorems & Definitions (15)

• Theorem 3.1: Existence of Transition
• Remark 3.1
• Theorem 4.1: Transition Types of Multiplicity Four Case
• Theorem 5.1: Structure of $\Sigma_\lambda$
• Theorem 6.1: Transition Types for Multiplicity Two
• Theorem 7.1: Transition Types for Multiplicity Six Case
• Theorem 7.2: Structure of $\Sigma_\lambda$
• Theorem 8.1: Transition Types with $k_3^c = k_1^c + k_2^c$
• proof
• Theorem 8.2: Stability of Roll Solutions