  In our recent work we found a surprising breakdown of symmetry conservation:
using standard numerical discretization with very high precision the computed
numerical solutions corresponding to very nice initial data may converge to
completely incorrect steady states due to the gradual accumulation of machine
round-off error. We solved this issue by introducing a new Fourier filter
technique for solutions with certain band gap properties. To further
investigate the attracting basin of steady states we classify in this work all
possible bounded nontrivial steady states for the Allen-Cahn equation. We
characterize sharp dependence of nontrivial steady states on the diffusion
coefficient and prove strict monotonicity of the associated energy. In
particular, we establish a certain self-replicating property amongst the
hierarchy of steady states and give a full classification of their energies and
profiles. We develop a new modulation theory and prove sharp convergence to the
steady state with explicit rates and profiles.
