  In this article, we discuss the stability of soft quasicrystalline phases in
a coupled-mode Swift-Hohenberg model for three-component systems, where the
characteristic length scales are governed by the positive-definite gradient
terms. Classic two-mode approximation method and direct numerical minimization
are applied to the model. In the latter approach, we apply the projection
method to deal with the potentially quasiperiodic ground states. A variable
cell method of optimizing the shape and size of higher-dimensional periodic
cell is developed to minimize the free energy with respect to the order
parameters. Based on the developed numerical methods, we rediscover decagonal
and dodecagonal quasicrystalline phases, and find diverse periodic phases and
complex modulated phases. Furthermore, phase diagrams are obtained in various
phase spaces by comparing the free energies of different candidate structures.
It does show not only the important roles of system parameters, but also the
effect of optimizing computational domain. In particular, the optimization of
computational cell allows us to capture the ground states and phase behavior
with higher fidelity. We also make some discussions on our results and show the
potential of applying our numerical methods to a larger class of mean-field
free energy functionals.
