Bifurcation Analysis Reveals Solution Structures of Phase Field Models

Xinyue Evelyn Zhao, Long Qing Chen, Wenrui Hao, Yanxiang Zhao

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


The 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 the 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: the Allen-Cahn equation, the Cahn-Hilliard equation, and the Allen-Cahn-Ohta-Kawasaki system. The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.

Original languageEnglish (US)
Pages (from-to)64-89
Number of pages26
JournalCommunications on Applied Mathematics and Computation
Issue number1
StatePublished - Mar 2024

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics


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