Numerical simulation of unstable processes in phase decomposition problem

We carry out numerical simulation of phase decomposition processes in binary alloys for a conservative phase field system. For the case $x\in\mathbb R^1$ explicit and implicit difference schemes are proposed. Numerical results illustrate the main stages of the process dynamics: the interface motion, the evolution and bifurcation of a “soliton”, and the motion and bifurcation of an oscillating singularity.