Stability and Bifurcations in a Model of Phase Transitions With Order Parameter

A one-dimensional model of phase transitions with convex strain energy is investigated within the limits of nonlinear bar theory. The model is a special case of a coupled field theory that has been developed by Fried and Gurtin to study the nucleation and propagation of phase boundaries. The system of governing equations studied here consists of a wave equation coupled to a nonlinear reaction-diffusion equation. Using phase plane methods, the equilibria of the system have been constructed in order to obtain the macroscopic response of the bar and the bifurcation diagram. It is demonstrated that a large number of coexisting spatially periodic, inhomogeneous solutions can occur, with the number of these solutions being inversely proportional to the diffusion coefficient in the reaction-diffusion subsystem. A stability analysis of the equilibria is presented.