Energy Decay in Burgers Turbulence and Interface Growth. the Problem of Random Initial Conditions - II

We present a study of the Burgers equation in one and two dimensions $d=1,2$ following the analytic approach indicated in the previous paper I. For the problem of the initial conditions decay we consider two classes of initial condition distributions $Q_{1,2} \sim \exp[-(1/4D)\int({\nabla}h)^2$d{\bf x}] where $h$-field is unbounded ($Q_1$) or bounded ($Q_2, |h|\leq H$). Avoiding the replica trick and using an integral representation of the logarithm we study the tractable field theory which has $d=2$ as a critical dimension. It is shown that the degenerate one-dimensional case has three stages of decay, when the kinetic energy density diminishes with time as $t^{-2/3}$, $t^{-2}$, $t^{-3/2}$ contrary to the predictions of the similarity hypothesis based on the second-order correlator of the distribution. In two dimensions we find the kinetic energy decay which is proportional to $t^{-1}\ln^{-1/2}(t)$. It is shown that the pure diffusion equation with the $Q_2$-type initial condition has non-trivial energy decay exponents indicating connection with the $O(2)$ non-linear $\sigma$-model.