An Exact Discretization of a Lax Equation for Shock Clustering and Burgers Turbulence I: Dynamical Aspects and Exact Solvability

We consider a finite dimensional system which arises as an exact discretization of the Lax equation for shock clustering, which describes the evolution of the generator of a Markov process with a finite number of states. The spectral curve of the system is a nodal curve which is fully reducible. In this work, we will show that the flow is conjugate to a straight line motion, and we will also show that the equation is exactly solvable. En route, we will establish a dictionary between an open, dense set of the lower triangular infinitesimal generator matrices and an associated set of algebro-geometric data.