Bifurcation Analysis Reveals Solution Structures of Phase Field Models

TL;DR

This paper introduces a bifurcation analysis approach to explore the solution structures of phase field models, aiding understanding of morphological evolution in materials and biological systems.

Contribution

It develops a new analytical method based on bifurcation analysis to study phase field models, verified through numerical methods on key equations.

Findings

Revealed complex solution structures of phase field equations.

Provided insights into morphological evolution phenomena.

Validated analytical results with numerical continuation methods.

Abstract

Phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems. Here, we develop a new analytical approach based on bifurcation analysis to explore the mathematical solution structure of phase field models. Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions. To elucidate the idea, we apply this analytical approach to three representative phase field equations: Allen-Cahn equation, Cahn-Hilliard equation, and Allen-Cahn-Ohta-Kawasaki system. The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.