Complete integrability of shock clustering

The emergence of structure from disorder is interesting in several physical models such as vortex coalescence in 2d flows, and domain growth in materials science. One mathematical model for such phenomena is to study the equations of continuum physics with random initial data.

We show that this problem has a very rich structure even for the simplest nonlinear equations. Our main result is that the evolution of shock statistics for scalar conservation laws with convex flux and suitable random data is completely integrable. These results sit at an interesting junction of kinetic theory, integrable systems, and probability theory. Little background will be presumed in any of these areas.