The Instanton Solution of Forced Burgers Equation in Polyakov's Approach

TL;DR

This paper derives an instanton (shock) solution for the forced Burgers equation using an extended operator product expansion in Polyakov's turbulence approach, revealing new tail behaviors in the velocity PDF.

Contribution

It introduces a generalized OPE that accounts for instanton solutions in Burgers turbulence, providing new insights into shock dynamics and probability distribution tails.

Findings

Left-tail of PDF behaves as u^{-7/2}

Extended OPE captures shock solutions

Asymptotic behavior of N-point generating function calculated

Abstract

We calculate the coefficients of the operator product expansion (OPE), in Polyakov's approach for Burgers turbulence. We show that the OPE has to be generalized and it is shown that the extra term gives us the instanton solution (shock solution) of Burgers equation. We consider the effect of the new-term in the OPE, on the right and left-tail of probability distribution function (PDF). It is shown that the left-tail of PDF, where is dominated by the well-separated shocks behaves as $ W(u)\sim u^{-7/2}$. Finally we calculate the assymptotic behaviour of the N-point generating function of the velocity field, using the new OPE.