Exact statistical properties of the Burgers equation

The one-dimensional Burgers equation in the inviscid limit with white noise initial condition is revisited. The one- and two-point distributions of the Burgers field as well as the related distributions of shocks are obtained in closed analytical forms. In particular, the large distance behaviour of spatial correlations of the field is determined. Since higher-order distributions factorize in terms of the one- and two- point functions, our analysis provides an explicit and complete statistical description of this problem.