Structure functions in the stochastic Burgers equation

We study analytically and numerically structure functions $S_q\mathrm{}\left(r\right)\mathrm{}$ in the one-dimensional Burgers equation, driven by noise with variance $\propto |k{|}^{\beta }$ in Fourier space, (a) when the noise is cut off at some length $l_c,$ and (b) when it is not. We present exact relations satisfied by $S_3\mathrm{}\left(r\right)\mathrm{}$ (the von Karman–Howarth relation) and $S_4\mathrm{}\left(r\right)\mathrm{}$ that form the basis of our analysis. When there is a cutoff length, shocks occur and $S_q\mathrm{}\left(r\right)\mathrm{}\propto r$ for $q>~2$ for $\delta <r<\mathrm{}{l}_c$ where δ is the shock thickness for all β between -1 and 2. We deduce this behavior from the exact relations along with an ansatz that is verified numerically. When there is no cutoff length, multifractal behavior is known to occur only when β<0. Through a study of exact expression $S_3$ we highlight the difference between multifractality in this case as compared to the case with a cutoff.