Symmetric quadruple phase transitions Changfeng GuiM. Schatzman 35J1535J2035J6035J65Allen-Cahn equationquadruple well potentialquadruple junctionphase transitionphase separationmultiple well potentialtriple junctionheteroclinic solutionvariational methodsymmetryspectrumlinearized operator A quadruple junction solution from $\mathbb{R}^3$ to $\mathbb{R}^3$ is constructed for a generalized Allen-Cahn equation with symmetric quadruple well potential. This solution is a three dimensional counterpart of the two dimensional triple junction solution, yet displays more complicated structure and new features. The model may be used to understand the quadruple junctions in phase separations and grain formations of crystalline materials. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3089 10.1512/iumj.2008.57.3089 en Indiana Univ. Math. J. 57 (2008) 781 - 836 state-of-the-art mathematics http://iumj.org/access/