Comment on “Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation” [J. Math. Phys. 54, 072901 (2013)]

Abstract

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 Wacławczyk 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, 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 functional Burgers equation is invalid. In general, their study constructs and discusses new particular solutions of the functional Burgers 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 Burgers equation essential symmetries are not captured.

Comments on Janocha et al. Lie Symmetry Analysis of the Hopf Functional-Differential Equation. Symmetry 2015, 7, 1536–1566

“…Hence, for an overall consistent analysis, Equation (71) must already apply in Equation (70); but, even this consequence still has no effect on the incompatibility feature of Equation [14] in [1] with the transformation of Equation (64)). Hence, when strictly following the arguments in [1], no partial integration in the last term of Equation [15] can be executed, which means that the appearance of the last term in Equation [16] is incorrect, in particular as it misleadingly suggests a non-zero contribution to this equation. The same problem one faces a page later when the expressions for the infinitesimals ζ ;y(x) and ζ ;y(x)y(x) get constructed, which all are incorrect, as they always involve too many terms.…”

“…This confusion in the dependencies also brings us to the next issue. E.3: The partial integration in Equation [15] to obtain Equation [16] is not justified. Since y(x )dx is identified or treated in [1] as an own independent variable next to x (see the arguments, e.g., on p. 1545 and p. 1549), the relative variation of both variables in the untransformed, as well as in the transformed domain must be zero:…”

“…This correct ansatz should then replace the inconsistent one in [1]. (Remark: First attempts to extend the classical Lie-point symmetry analysis from partial differential and integro-differential to functional equations can be found in [14,15]; a shortcoming of these attempts is discussed in [16]). …”

“…As was shown in [16,18], this transformation violates the classical principle of cause and effect, and henceforth, the transformation ) is admitted as a symmetry by the considered Hopf-Burgers Equation (A29), it nevertheless is unphysical in the same sense as X phys 6 in that it also violates the classical causality principle and, hence, has to be discarded, as well; for more details, see [19,20] and the review [21].…”