4 - Nodal Curves and a Class of Solutions of the Lax Equation for Shock Clustering and Burgers Turbulence

Summary

In this paper, we present an expository account of the work done in the last few years in understanding a matrix Lax equation which arises in the study of scalar hyperbolic conservation laws with spectrally negative pure-jump Markov initial data. We begin with its extension to general N x N matrices, which is Liouville integrable on generic coadjoint orbits of a matrix Lie group. In the probabilistically interesting case in which the Lax operator is the generator of a pure-jump Markov process, the spectral curve is generically a fully reducible nodal curve. In this case, the equation is not Liouville integrable, but we can show that the flow is still conjugate to a straight line motion, and the equation is exactly solvable. En route, we establish a dictionary between an open, dense set of lower triangular generator matrices and algebro-geometric data which plays an important role in our analysis.