Abstract

Many important partial differential equations used in engineering or theoretical physics describe the motion of lines or surfaces. Locations of particular interest are those where the motion becomes (nearly) singular, and the line or surface forms a cusp or a tip with a very large curvature. Physical examples occur when viscous fluid is poured into a beaker, or when caustics form at the bottom of a coffee cup.

Recent work suggests the structure of many such singularities can be understood using a local geometrical description. The local shape corresponds exactly to a solution of the partial differential equation. Building on this insight, we propose a preliminary classification of such singular points, including higher dimensions of both the surface and the space it lives in. We also investigate how the observed singularity can be understood from the point of view of the dynamical equation, and how the geometrical description could be used in more complex situations.