On applying Lie-group symmetry analysis to the functional Hopf-Burgers equation in spectral space

The quest to find new statistical symmetries in the theory of turbulence is an ongoing research endeavor which is still in its beginning and exploratory stage. In our comment we show that the recently performed study of Waclawczyk and Oberlack [J. Math. Phys. 54, 072901 (2013)] failed to present such new statistical symmetries. Despite their existence within a functional Fourier space of the statistical Burgers equation, also referred to as the spectral Hopf-Burgers equation, they all can be reduced to the classical and well-known symmetries of the underlying deterministic Burgers equation itself, except for one symmetry, but which, as we will demonstrate, is only a mathematical artefact without any physical meaning. Moreover, we show that the proposed connection between the translation invariance of the multi-point moments and a symmetry transformation associated to a certain invariant solution of the inviscid Hopf-Burgers equation is invalid. In general, their study constructs and discusses new particular solutions of this functional equation without referring them to the well-established general solution. Finally, we also see a shortcoming in the presented methodology as being too restricted to construct a complete set of Lie point symmetries for functional equations. In particular, for the considered spectral Hopf-Burgers equation essential symmetries are not captured.