Bifurcation Analysis Reveals Solution Structures of Phase Field Models

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