Exact statistical properties of the Burgers equation

TL;DR

This paper provides a comprehensive analytical characterization of the statistical properties of the one-dimensional Burgers equation with white noise initial conditions, including distributions of the field and shocks, and spatial correlations.

Contribution

It derives closed-form expressions for one- and two-point distributions and analyzes large-distance correlations, offering a complete statistical description of the inviscid Burgers equation with white noise.

Findings

Explicit formulas for one- and two-point distributions.

Determination of large-distance spatial correlation behavior.

Complete statistical description of the Burgers field and shocks.

Abstract

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 behavior of spatial correlations of the field is determined. Since higher order distributions factorize in terms of the one and two points functions, our analysis provides an explicit and complete statistical description of this problem.