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 a previous paper [S. E. Esipov and T. J. Newman, Phys. Rev. E 48, 1046 (1993)]. For the problem of initial-condition decay we consider two classes of initial-condition distributions ${\mathit{Q}}_{1,2}$∼exp[-(1/4D)F(∇h${)}^2$dx], where the h field is unbounded (${\mathit{Q}}_1$) or bounded (${\mathit{Q}}_2$,‖h‖≤H). In one dimension these distributions give examples of nondegenerate and degenerate Burgers models of turbulence, respectively. Avoiding the replica trick and using an integral representation of the logarithm we study the exact analytically 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 ${\mathit{t}}^{\mathrm{-}2/3}$, ${\mathit{t}}^{\mathrm{-}2}$, and ${\mathit{t}}^{\mathrm{-}3/2}$, 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 ${\mathit{t}}^{\mathrm{-}1}$ ${\ln }^{\mathrm{-}1/2}$(t). It is shown that the pure diffusion equation with the ${\mathit{Q}}_2$-type initial condition has nontrivial energy decay exponents indicating connection with the O(2) nonlinear σ model.