Defects are weak and self-dual solutions of the Cross-Newell phase diffusion equation for natural patterns

We show that defects are weak solutions of the phase diffusion equation for the macroscopic order parameter for natural patterns. Further, by exploring a new class of nontrivial solutions for which the graph of the phase function has vanishing Gaussian curvature (in 3D, all sectional curvatures) except at points, we are able to derive explicit expressions which capture the anatomies of point and line (and surface) defects in two and three dimensional patterns, together with their topological characters and energetic constraints.