Bifurcation Analysis Reveals Solution Structures of Phase Field Models

Xinyue Evelyn Zhao; Long-Qing Chen; Wenrui Hao; Yanxiang Zhao

doi:10.1007/s42967-022-00221-1

Communications on Applied Mathematics and Computation ›› 2022, Vol. 6 ›› Issue (1) : 64-89. DOI: 10.1007/s42967-022-00221-1

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Bifurcation Analysis Reveals Solution Structures of Phase Field Models

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Abstract

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.

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Xinyue Evelyn Zhao, Long-Qing Chen, Wenrui Hao, Yanxiang Zhao. Bifurcation Analysis Reveals Solution Structures of Phase Field Models. Communications on Applied Mathematics and Computation, 2022, 6(1): 64‒89 https://doi.org/10.1007/s42967-022-00221-1

Funding

Office of Science(DESC0020145); Directorate for Mathematical and Physical Sciences(DMS-2052685); Directorate for Mathematical and Physical Sciences(2142500); Simons Foundation(357963)