Instantons in Burgers Equation

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 perturbation expansion around the instanton solution. We explicitly checked this in the first order. We also generalized our solution for the arbitrary dimention of Burgers (=KPZ) equation. As a result we find the angular dependence of the velocity difference PDF. There are two complementary views of the turbulence problem. One could regard it as kinetics, in which case the time dependence of velocity probability distribution function (PDF) must be studied. The Wyld functional integral describes the correlations functions of the velocity field in this picture. Another view is the Hopf (or Fokker-Planck) approach, where the equal time PDF is studied. For the random force distributed as white noise in time, the closed functional equations (Fokker-Planck equation) can be derived. In the case of thermal noise the Boltzmann distribution can be derived as an asymptotic solution of this equation. One of the authors [1] reduced the Hopf equations for the full Navier-Stokes equation to the one dimensional functional equation (loop equation). The WKB solutions of this equation were studied, leading to the area law for the velocity circulation PDF. In the recent paper by Polyakov [2] a similar method of solving the randomly driven Burgers equation was proposed. It reduced the problem of computations of their correlation functions to the solution of a certain partial differential equation. This equation for the velocity difference PDF can be explicitly solved. The derivation of the Polyakov equation was based on the conjecture of the existence of the operator product expansion.