  Some of the most impressive singular wave fronts seen in Nature are the
transbasin oceanic internal waves, which may be observed from the Space Shuttle
as they propagate and interact with each other, for example, in the South China
Sea. The characteristic feature of these strongly nonlinear wavefronts is that
they reconnect when two of them collide transversely. We derive the EPDiff
equation, and use it to model this phenomenon as elastic collisions between
singular wave fronts (solitons) whose momentum is distributed along curves
moving in the plane. Numerical methods for EPDiff based on compatible
differencing algorithms (CDAs) are used for simulating these collisions among
curves. The numerical results show the same nonlinear behavior of wavefront
reconnections as that observed for internal waves in the South China Sea. We
generalize the singular solutions of EPDiff for other applications, in
computational anatomy and in imaging science, where the singular wavefronts are
evolving image outlines, whose momentum may be distributed on surfaces moving
though space in three dimensions. The key idea is always momentum exchange
during collisions of the wavefronts. A suite of 2d and 3d numerical simulations
provide collision rules for the wavefront reconnection phenomenon in a variety
of scenarios.
