  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.
