Velocity-difference probability density functions for Burgers turbulence

In this paper the Polyakov equation [Phys. Rev. E 52, 6183 (1995)] for the velocity-difference probability density functions, with the random Gaussian external force, with the correlation function κ(y)∼1-${\mathrm{y}}^{\mathrm{\alpha }}$, is analyzed. Solutions for the cases α={2,1/2,1} are found, which agree very well with available numerical results. It is also argued that the stationary regime of Burgers turbulence can depend not only on the distribution of the external force, but also on the dissipative regularization.