Shocks and Structure Functions of Burgers and KPZ Equations

In this paper we calculate the structure functions of one dimensional Burgers and KPZ equations with and without noise, and we interpret our findings using shocks. Most of our results are based on direct numerical simulation of the KPZ equation. Using the solution of noiseless Burgers equation we show that the exponent cq (< |h(x+r)−h(x)| q >∝ rqcq ) of the structure function of noiseless KPZ is 1. For KPZ equation with white noise, using the well known asymptotic solution and using lattice gas model of KPZ equation, we show that cq is 1/2. The exponent for the corresponding Burgers equation is zero. Regarding the Burgers equation with colored noise with ρ > 3/2, where ρ is the exponent of noise f of the corresponding KPZ equation, 〈|f(k)|2〉 ∝ k−2ρ, we find that < |u(x + r) − u(x)|q >∝ r, indicating presence of large shocks. The corresponding KPZ equation has cq = 1. For small ρ(ρ < 1/2), the structure function exponent for Burgers equation is close to zero and cq is close to 1/2. However, in the range 1/2 < ρ < 3/2, cq increases from 1/2 to 1 as we increase ρ. Our numercal results show that for ρ 6= 0, c2 differs from α, where < h(x + r)h(x) >∝ rα. Our α is close to that of Medina et al. for 0 < ρ < 1/2, but differs significantly from Medina et al.’s in the