Burgers turbulence is the study of the statistics of shocks in Burgers equation with random initial data or forcing. More generally, we consider the statistics of shock collisions for a scalar conservation law with random initial data. We derive exact kinetic models for the evolution of the statistics of the entropy solution under natural assumptions on the initial data. Quite remarkably, these kinetic models have an integrable structure, in the form of a Lax pair.
Our work provides a new approch to a remarkable exact solution of Groeneboom for the statistics of shocks in Burgers equation with white noise initial data. It also suggests tantalizing connections with the Tracy-Widom limit laws in the theory of random matrices.
This is joint work with Ravi Srinivasan.