In this talk, I will discuss the quadruple junction solutions in
the entire three dimensional space to a vector-valued Allen-Cahn equation
which models multiple phase separation. The solution is the basic profile
of the local structure near a quadruple junction in three dimensional
crystalline material using the generalized Allen-Cahn model, and is the
three dimensional counterpart of two dimensional triple junction solutions.
I will start with one dimensional heteroclinic solutions, and describe
how higher dimensional solutions can be constructed from the lower dimensional ones,
and explain the complications and difficulties in constructing such a solution
in three dimensions.