Instantons in Burgers Equation

TL;DR

This paper derives the instanton solution for the forced Burgers equation, explaining the exponential tail of velocity difference distributions and generalizing results to arbitrary dimensions, with implications for the KPZ universality class.

Contribution

It presents an explicit instanton solution for the forced Burgers equation and extends it to arbitrary dimensions, connecting to the operator product conjecture and KPZ equation.

Findings

Derived the instanton solution for the velocity difference PDF tail.

Confirmed the asymptotic exactness of the WKB expansion under the conjecture.

Extended the solution to arbitrary dimensions, revealing angular dependence.

Abstract

The instanton solution for the forced Burgers equation is found. This solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent. The results agree with the one found recently by Polyakov, who used the operator product conjecture. If this conjecture is true, then our WKB asymptotics of the Wyld functional integral is exact to all orders of the perturbative expansion around the instanton solution. We explicitly checked this in the first order. We also generalized our solution for the arbitrary dimension of Burgers (=KPZ) equation. As a result we found the angular dependence of the velocity difference PDF.