  Formal asymptotic expansions have long been used to study the singularly
perturbed Allen-Cahn type equations and reaction-diffusion systems, including
in particular the FitzHugh-Nagumo system. Despite their successful role, it has
been largely unclear whether or not such expansions really represent the actual
profile of solutions with rather general initial data. By combining our earlier
result and known properties of eternal solutions of the Allen-Cahn equation, we
prove validity of the principal term of the formal expansions for a large class
of solutions.
