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

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## Abstract

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## 1. Introduction: The Problem of Combining Implicit and Explicit Functional Dependence

- (i)
- Caution has to be exercised when using the notation of Hopf [13] for the functional derivative (shown here only for the 1D case):$$\frac{\delta}{\delta y\left(x\right)}=\frac{\partial}{\partial y\left(x\right)dx}$$$$\left[\frac{\delta}{\delta y\left(x\right)}\right]=\frac{1}{\left[y\right]\xb7L}$$
- (ii)
- When considering a transformation, e.g., on the 2D set of variables $(y\left({x}^{\prime}\right),x)\mapsto (\overline{y}\left({\overline{x}}^{\prime}\right),\overline{x})$, it is necessary to realize that there is only one physical space, i.e., only one set $G\subseteq {\mathbb{R}}^{1}$ where both spatial variables ${x}^{\prime}$ and x belong to: ${x}^{\prime},x\in G$. Hence, when treating ${x}^{\prime}$ and x as true variables, i.e., as quantities that can vary between all values in G, the transformation rule for ${x}^{\prime}\mapsto {\overline{x}}^{\prime}$ must obviously be the same as for $x\mapsto \overline{x}$. Consequently, the transformation on the above set of variables can also be formally written as $(y\left(x\right),x)\mapsto (\overline{y}\left(\overline{x}\right),\overline{x})$. To simplify formal expressions, we will make use of this notation, as long as no ambiguity arises.

- (a)
- In order to perform a consistent Lie-group symmetry analysis for a functional differential equation (FDE), such as Equation (3), it is crucial to correctly identify and separate the independent from the dependent variables. Looking at Equation (3), it is clear that t and x are to be identified as independent variables and Φ as a dependent one. But, how does one identify the variable $y=y\left(x\right)$? Is it a dependent or an independent variable? Therefore, in order to arrive at unique results, we must specify the order of the mathematical operations in Equation (3). Since by construction the functional derivatives have priority over the usual differentiation and integration processes (note that, due to this priority of the functional derivatives over the partial derivatives, they do not commute, that means $\frac{\partial}{\partial x}\frac{\delta}{\delta y\left(x\right)}\ne \frac{\delta}{\delta y\left(x\right)}\frac{\partial}{\partial x}$), the variable y has to be identified as an independent variable, which additionally can be differentiated relative to x, i.e., ${\partial}_{x}y={y}^{\prime}\left(x\right)$. This property of having different dependencies of y, namely the hierarchy of being an independent functional variable relative to x, which then can be differentiated to it, has to be carefully monitored in Equation (3) when transforming this equation, otherwise one runs into a conflict of dependencies. But, this exact monitoring has not been done in [1], and thus, a conflict of variables takes place throughout that study.
- (b)
- The dependent variable Φ in Equation (3) only depends on the functional variable y and on time t:$$\Phi =\Phi \left(y\right(x),t)$$
- (c)
- When y acts as a dependent variable, it only depends on x, since the function $y=y\left(x\right)$ is defined as a time-independent function, i.e., ${\partial}_{t}y\left(x\right)=0$ (for more details, see [13]). Any symmetries thus found must be compatible with this condition, i.e., in the transformed domain, we must obtain this independency, as well: ${\partial}_{\overline{t}}\overline{y}\left(\overline{x}\right)=0$.

- (1)
- From the information that one directly gets from the new approach of Equation (8), it is unclear how the functional variable y should transform. The only way to retrieve this information is to enforce the consistent condition $\overline{\omega}=\overline{y}\xb7d\overline{x}$, which will lead to:$$\begin{array}{cc}\hfill \omega +{\xi}_{\omega}\phantom{\rule{0.166667em}{0ex}}\epsilon =\overline{\omega}=\overline{y\xb7dx}& =\overline{y}\xb7d\overline{x}\hfill \\ & =\left(y+{\xi}_{y}\phantom{\rule{0.166667em}{0ex}}\epsilon \right)\xb7\left(dx+{\partial}_{x}{\xi}_{x}\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}\epsilon \right)\hfill \\ & =y\xb7dx+({\xi}_{y}\phantom{\rule{0.166667em}{0ex}}dx+y\xb7{\partial}_{x}{\xi}_{x}\phantom{\rule{0.166667em}{0ex}}dx)\phantom{\rule{0.166667em}{0ex}}\epsilon +\mathcal{O}\left({\epsilon}^{2}\right)\hfill \end{array}$$$${\xi}_{\omega}=({\xi}_{y}+y\xb7{\partial}_{x}{\xi}_{x})\phantom{\rule{0.166667em}{0ex}}dx$$
- (2)
- The more severe problem, which essentially turns the proposed symmetry analysis in [1] into an incomplete and also inconsistent analysis, is the fact that the dependent variable Φ, as defined in Definition 5 [1] (p. 1542), only depends on y, x and t and not on the combined variable ω of Equation (7). Surely, instead of the original system of Equation (5), one can consider a more general system in rewriting the original system F equivalently into:$$F(\omega ,y,x,t,{\Phi}^{*},\underset{1}{{\Phi}^{*}},\underset{2}{{\Phi}^{*}},\underset{3}{{\Phi}^{*}})=0$$$$\left.\begin{array}{cc}\hfill \overline{\omega}& =\omega +{\xi}_{\omega}(\omega ,y,x,t,{\Phi}^{*})\phantom{\rule{0.166667em}{0ex}}\epsilon \hfill \\ \hfill \overline{y}& =y+{\xi}_{y}(\omega ,y,x,t,{\Phi}^{*})\phantom{\rule{0.166667em}{0ex}}\epsilon \hfill \\ \hfill \overline{x}& =x+{\xi}_{x}(\omega ,y,x,t,{\Phi}^{*})\phantom{\rule{0.166667em}{0ex}}\epsilon \hfill \\ \hfill \overline{t}& =t+{\xi}_{t}(\omega ,y,x,t,{\Phi}^{*})\phantom{\rule{0.166667em}{0ex}}\epsilon \hfill \\ \hfill {\overline{\Phi}}^{*}& ={\Phi}^{*}+{\eta}_{{\Phi}^{*}}(\omega ,y,x,t,{\Phi}^{*})\phantom{\rule{0.166667em}{0ex}}\epsilon \hfill \end{array}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}\right\}$$

## 2. A Proposal for a Consistent Treatment in Generating Lie-Point Symmetries for FDEs

#### 2.1. Construction Guideline for Prolongations in the Example of a PDE

#### 2.2. First Order Prolongations for the Functional Hopf–Burgers Equation

- D.1:
- Equation [18] in [1] and the subsequent equation below it, each contain two terms in the prolonged infinitesimals for the correspondingly-transformed first order variables ${\overline{\Phi}}_{,\overline{t}}$ and ${\overline{\Phi}}_{,\overline{y\left({z}_{1}\right)}}$ [1] (p. 1543), namely the last two terms in ${\zeta}_{;t}$ and ${\zeta}_{;y\left(x\right)}$ [1] (p. 1546), which are not present in our corresponding results of Equations (43) and (44). These terms being proportional to the variations of the infinitesimal ${\xi}_{x}$ appear to be due to a technical error performed in [1], which we will discuss in detail in the next section (see Discussions E.1 and E.2)
- D.2:
- The prolonged infinitesimal ${\zeta}_{;y\left(x\right)}$ resulting from Equation (44) also differs in another, independent respect from the result obtained in [1] (p. 1546). To see this difference, one first has to recognize that with Equation (44), we did not directly obtain a transformation rule for the differential variable ${\Phi}_{,y\left(x\right)}$; instead, we only obtained a rule for the combined product variable ${\Phi}_{,y\left(x\right)}\phantom{\rule{0.166667em}{0ex}}dx$. To retrieve the transformation rule just for the variable ${\Phi}_{,y\left(x\right)}$, we first have to transform the 1D volume element $dx$, which for the infinitesimal transformation $x\mapsto \overline{x}$ of Equation (33) up to first order is expressed through the 1D Jacobian:$$dx=(1-\epsilon \xb7{\xi}_{x,x})\phantom{\rule{0.166667em}{0ex}}d\overline{x}+\mathcal{O}\left({\epsilon}^{2}\right)$$$${\overline{\Phi}}_{,\overline{y}\left(\overline{x}\right)}={\Phi}_{,y\left(x\right)}+\epsilon \xb7{\zeta}_{;y\left(x\right)}+\mathcal{O}\left({\epsilon}^{2}\right)$$$$\begin{array}{cc}\hfill {\zeta}_{;y\left(x\right)}=-{\Phi}_{,y\left(x\right)}\phantom{\rule{0.166667em}{0ex}}{\xi}_{x,x}+{\eta}_{,y\left(x\right)}+{\Phi}_{,y\left(x\right)}& {\eta}_{,\Phi}-{\Phi}_{,t}\phantom{\rule{0.166667em}{0ex}}{\xi}_{t,y\left(x\right)}-{\Phi}_{,t}{\Phi}_{,y\left(x\right)}\phantom{\rule{0.166667em}{0ex}}{\xi}_{t,\Phi}\hfill \\ & -\int d{x}^{\prime}\phantom{\rule{0.166667em}{0ex}}{\Phi}_{,y\left({x}^{\prime}\right)}\phantom{\rule{0.166667em}{0ex}}{\xi}_{y\left({x}^{\prime}\right),y\left(x\right)}-\int d{x}^{\prime}\phantom{\rule{0.166667em}{0ex}}{\Phi}_{,y\left({x}^{\prime}\right)}{\Phi}_{,y\left(x\right)}\phantom{\rule{0.166667em}{0ex}}{\xi}_{y\left({x}^{\prime}\right),\Phi}\hfill \end{array}$$$$\overline{t}={e}^{2\epsilon}t=t+2t\phantom{\rule{0.166667em}{0ex}}\epsilon +\mathcal{O}\left({\epsilon}^{2}\right),\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\overline{x}={e}^{\epsilon}x=x+x\phantom{\rule{0.166667em}{0ex}}\epsilon +\mathcal{O}\left({\epsilon}^{2}\right),\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\overline{y}\left(\overline{x}\right)=y\left(x\right),\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\overline{\Phi}=\Phi $$$${\overline{\Phi}}_{,\overline{y}\left(\overline{x}\right)}={e}^{-\epsilon}{\Phi}_{,y\left(x\right)}={\Phi}_{,y\left(x\right)}-\epsilon \phantom{\rule{0.166667em}{0ex}}{\Phi}_{,y\left(x\right)}+\mathcal{O}\left({\epsilon}^{2}\right)$$$$\left[{\Phi}_{,y\left(x\right)}\right]=\left[\frac{\delta \Phi}{\delta y\left(x\right)}\right]=\frac{\left[\Phi \right]}{\left[y\right]\xb7L}$$$${\xi}_{t}=2t,\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\xi}_{x}=x,\phantom{\rule{1.em}{0ex}}\phantom{\rule{4pt}{0ex}}{\xi}_{y\left(x\right)}=0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\eta =0$$$${\zeta}_{;y\left(x\right)}=-{\Phi}_{,y\left(x\right)}$$
- D.3:
- The restricting Equation (45) for the generally assumed infinitesimals of the point variables of Equation (33) automatically emerges as a consistent by-product in our analysis when systematically determining their first order prolongations under the restriction ${\Phi}_{,x}=0$. This result which, of course, reduces to the set of restrictions:$${\eta}_{,x}=0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.222222em}{0ex}}{\xi}_{t,x}=0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.222222em}{0ex}}{\xi}_{y\left(x\right),x}=0$$The difference is that all three restrictions in Equation (55) tell us that in order to warrant consistency with ${\Phi}_{,x}=0$, the infinitesimals η, ${\xi}_{t}$ and ${\xi}_{y\left(x\right)}$ should not show any explicit dependence on the spatial variable x, i.e., when looking in particular at ${\xi}_{y\left(x\right)}$, then, instead of the general form given in Equation (33), it should only show the reduced dependence ${\xi}_{y\left(x\right)}={\xi}_{y\left(x\right)}(t,y\left(x\right),\Phi )$. However, Equation [37] in [1] does not induce this restriction, as can be readily seen, e.g., in the second term of Equation [47], which is treated as a non-zero term. Equation [37], therefore, does not induce the same restriction on the infinitesimal ${\xi}_{y\left(x\right)}$ as the corresponding and consistent restriction derived by us in Equation (55).The problem is that the authors in [1] artificially distinguish between the terms ${\xi}_{y\left(z\right),x}$ and ${\xi}_{y\left(x\right),x}$ as given in Equations [37] and [47], respectively, if $z\ne x$. The reason is that in Definition 5 [1] (p. 1542), the infinitesimal ${\xi}_{y\left(x\right)}$ is misleadingly defined as a multipoint function depending explicitly on two spatial points z and x. However, such a definition is inappropriate and, as we have shown in this study, is not even required in order to perform a consistent analysis (see also Discussion (ii) in Section 1). Hence, due to this multipoint definition in Definition 5 [1] (p. 1542), we face in Equation [37] the misleading consequence that ${\xi}_{y\left(z\right)}$ must be explicitly independent of the spatial variable x while still depending explicitly on the spatial variable z. This can be clearly seen, e.g., in the proposed ansatz function for ${\xi}_{y\left(z\right)}$ in Equation [50], which is not compatible to the consistent constraint ${\xi}_{y\left(z\right),z}=0$ in Equation (55) when written relative to z.Regarding the first restriction ${\eta}_{,x}=0$ in Equation (55), it is not clear from the analysis in [1] whether this consistent restriction has been initially assumed or not. Only in the end of their analysis, when solving the overdetermined system, a consistent ansatz function in Equation [45] is proposed in an ad hoc manner.

## 3. Points for Correction in [1]

- E.1:
- As written in [1], the variation of the transformed Hopf functional $\overline{\Phi}$ with respect to t given through Equation [12] is incorrect. Their argument of why the temporal variation in the last term of Equation [12] is only acting on the transformed expansion coefficients ${\overline{y}}_{n}$ and not also on the orthogonal functions ${h}_{n}$ and the volume element $d{\overline{x}}^{\prime}$ is not convincing. Because, when following their preceding argument in decomposing $y\left(z\right)dz$ into a complete and transformable set of orthogonal basis functions, the correct variation up to first order will not lead to Equation [12], but instead, it will lead to:$$\frac{\mathcal{D}\overline{\Phi}}{\mathcal{D}t}=\frac{\mathcal{D}\overline{\Phi}}{\mathcal{D}\overline{t}}\frac{\mathcal{D}\overline{t}}{\mathcal{D}t}+\frac{\mathcal{D}\overline{\Phi}}{\mathcal{D}\overline{x}}\frac{\mathcal{D}\overline{x}}{\mathcal{D}t}+{\int}_{\overline{G}}\frac{\mathcal{D}\overline{\Phi}}{\mathcal{D}\overline{y\left({x}^{\prime}\right)d{x}^{\prime}}}\sum _{n=1}^{\infty}\frac{\mathcal{D}\phantom{\rule{0.166667em}{0ex}}{\overline{y}}_{n}{h}_{n}({\overline{x}}^{\prime}-\epsilon {\xi}_{{x}^{\prime}})d{\overline{x}}^{\prime}}{\mathcal{D}t}$$$$\begin{array}{cc}\hfill 0& =y\left(z\right)dz-\sum _{n=1}^{\infty}{y}_{n}{h}_{n}\left(z\right)dz=\overline{y\left(z\right)dz}-\epsilon {\xi}_{y\left(z\right)}dz-\sum _{n=1}^{\infty}{y}_{n}{h}_{n}\left(z\right)dz+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \\ & =\overline{y\left(z\right)dz}-\epsilon {\int}_{G}d{z}^{\prime}\delta ({z}^{\prime}-z){\xi}_{y\left({z}^{\prime}\right)}dz-\sum _{n=1}^{\infty}{y}_{n}{h}_{n}\left(z\right)dz+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \\ & =\overline{y\left(z\right)dz}-\epsilon {\int}_{G}d{z}^{\prime}{\xi}_{y\left({z}^{\prime}\right)}dz\sum _{n=1}^{\infty}{h}_{n}\left(z\right){h}_{n}\left({z}^{\prime}\right)-\sum _{n=1}^{\infty}{y}_{n}{h}_{n}\left(z\right)dz+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \\ & =\overline{y\left(z\right)dz}-\sum _{n=1}^{\infty}\left({y}_{n}+\epsilon {\int}_{G}d{z}^{\prime}{\xi}_{y\left({z}^{\prime}\right)}{h}_{n}\left({z}^{\prime}\right)\right){h}_{n}\left(z\right)dz+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \\ & =\overline{y\left(z\right)dz}-\sum _{n=1}^{\infty}\left({y}_{n}+\epsilon {\int}_{G}d{z}^{\prime}{\xi}_{y\left({z}^{\prime}\right)}{h}_{n}\left({z}^{\prime}\right)\right){h}_{n}\left(z\right)\left(1-\epsilon {\partial}_{\overline{z}}{\xi}_{z}\right)d\overline{z}+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \\ & =\overline{y\left(z\right)dz}-\sum _{n=1}^{\infty}{\overline{y}}_{n}{\overline{h}}_{n}\left(\overline{z}\right)d\overline{z}+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \end{array}$$$$\begin{array}{c}{\overline{h}}_{n}\left(\overline{z}\right)={h}_{n}\left(z\right)={h}_{n}(\overline{z}-\epsilon {\xi}_{z})\ne {h}_{n}\left(\overline{z}\right)\end{array}$$$$\begin{array}{c}{\overline{y}}_{n}={y}_{n}+\epsilon \left(\phantom{\rule{0.166667em}{0ex}}{\int}_{G}d{z}^{\prime}{\xi}_{y\left({z}^{\prime}\right)}{h}_{n}\left({z}^{\prime}\right)-{y}_{n}{\partial}_{z}{\xi}_{z}\right)\end{array}$$Comparing now the above result of Equation (58) to Equation [11] up to first order, we see that the basis functions do not transform as $h\left(z\right)=h\left(\overline{z}\right)$, as claimed in [1] (p. 1543), but instead as $h\left(z\right)=\overline{h}\left(\overline{z}\right)$ as in Equation (59). Moreover, the expansion coefficients ${\overline{y}}_{n}$ do not transform independently, but are induced by the transformations of the variables $y\left(z\right)dz$ and z according to Equation (60). However, note that the transformation of Equation (60) is not complete. It still must be supplemented by a consistency constraint, because when decomposing the time-independent 1D scalar (differential) variable $y\left(z\right)dz$ into a set of orthogonal functions $\left\{{h}_{n}\left(z\right)\right\}$:$$y\left(z\right)dz=\sum _{n=1}^{\infty}{y}_{n}{h}_{n}\left(z\right)dz$$$${y}_{n}={\int}_{G}{h}_{n}\left(z\right)y\left(z\right)dz$$
- E.2:
- In going from Equation [13] to Equation [14], the authors in [1] assumed that the transformed expansion coefficients ${\overline{y}}_{n}$ should satisfy the condition:$$\frac{\mathcal{D}{\overline{y}}_{n}}{\mathcal{D}\overline{x}}=0$$$$\left.\begin{array}{c}\hfill \overline{t}=t+\epsilon \xb7{t}^{2},\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\overline{x}=x+\epsilon \xb7tx,\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\overline{y\left(x\right)dx}=y\left(x\right)dx+\epsilon \xb7ty\left(x\right)dx\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\\ \hfill \overline{\Phi}=\Phi +\epsilon \xb7{\textstyle \frac{i}{2}\Phi \int xy\left(x\right)dx}\phantom{\rule{113.81102pt}{0ex}}\end{array}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\right\}$$$${\overline{y}}_{n}={y}_{n}+\epsilon \xb7t\left(\phantom{\rule{0.166667em}{0ex}}{\int}_{G}{h}_{n}\left(x\right)y\left(x\right)dx-{y}_{n}\right)={y}_{n}$$$$d\overline{x}=dx+\epsilon \xb7tdx$$$$\begin{array}{cc}\hfill {\text{LHS}}_{\text{Eq.[13]}}:& \phantom{\rule{1.em}{0ex}}\frac{\mathcal{D}\overline{y\left(x\right)dx}}{\mathcal{D}t}=\frac{\partial \overline{y\left(x\right)dx}}{\partial t}=\epsilon \xb7y\left(x\right)dx+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \\ \hfill {\text{RHS}}_{\text{Eq.[13]}}:& \phantom{\rule{1.em}{0ex}}\sum _{n=1}^{\infty}\frac{\mathcal{D}{\overline{y}}_{n}}{\mathcal{D}t}{\overline{h}}_{n}\left(\overline{x}\right)d\overline{x}+\sum _{n=1}^{\infty}{\overline{y}}_{n}\frac{\mathcal{D}{\overline{h}}_{n}\left(\overline{x}\right)d\overline{x}}{\mathcal{D}t}=\sum _{n=1}^{\infty}{\overline{y}}_{n}\frac{\partial {\overline{h}}_{n}\left(\overline{x}\right)d\overline{x}}{\partial t}\hfill \end{array}$$$$\begin{array}{cc}& \phantom{\rule{14.22636pt}{0ex}}=\sum _{n=1}^{\infty}\epsilon {\overline{y}}_{n}{h}_{n}\left(x\right)dx+\mathcal{O}\left({\epsilon}^{2}\right)=\epsilon \xb7y\left(x\right)dx+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {\text{LHS}}_{\text{Eq.[14]}}:& \phantom{\rule{1.em}{0ex}}\sum _{n=1}^{\infty}\frac{\mathcal{D}{\overline{y}}_{n}}{\mathcal{D}t}{\overline{h}}_{n}\left(\overline{x}\right)d\overline{x}=0\hfill \\ \hfill {\text{RHS}}_{\text{Eq.[14]}}:& \phantom{\rule{1.em}{0ex}}\frac{\mathcal{D}\overline{y\left(x\right)dx}}{\mathcal{D}t}-\frac{\mathcal{D}\overline{y\left(x\right)dx}}{\mathcal{D}\overline{x}}\frac{\mathcal{D}\overline{x}}{\mathcal{D}t}=\epsilon \xb7y\left(x\right)dx-\epsilon \xb7x\frac{\partial \overline{y\left(x\right)dx}}{\partial \overline{x}}+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \end{array}$$$$\begin{array}{cc}& \phantom{\rule{14.22636pt}{0ex}}=\epsilon \xb7\left(y\left(x\right)-x{\partial}_{x}y\left(x\right)\right)dx+\mathcal{O}\left({\epsilon}^{2}\right)\hfill \end{array}$$To nonetheless allow for Equation [14], additional constraints for the infinitesimals ${\xi}_{y\left(x\right)}dx$ and ${\xi}_{x}$ must be placed. However, this has not been done in [1]; instead, Equation [14] is imposed as a generally valid relation without any restrictions, representing thus a technical error, which runs through the entire study. Additionally, to simply declare “that t, x and the infinite set $\left\{{y}_{n}\right\}$ are the independent variables” [1] (p. 1545), such that $\mathcal{D}{\overline{y}}_{n}/\mathcal{D}\overline{x}=0$ of Equation (63) imposes no restrictions, is incorrect, because, when following their initial argument, the variables t, x and $y\left(x\right)dx$ form the independent variables and not t, x and ${y}_{n}$; see the above Discussion E.1, where we explicitly show that the expansion coefficients ${y}_{n}$ do not transform independently from the other variables. 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 $\overline{y\left({x}^{\prime}\right)d{x}^{\prime}}$ is identified or treated in [1] as an own independent variable next to ${x}^{\prime}$ (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:$$\frac{\mathcal{D}y\left({x}^{\prime}\right)d{x}^{\prime}}{\mathcal{D}{x}^{\prime}}=\frac{\mathcal{D}\overline{y\left({x}^{\prime}\right)d{x}^{\prime}}}{\mathcal{D}\overline{{x}^{\prime}}}=0$$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 ${\zeta}_{;y\left(x\right)}$ and ${\zeta}_{;y\left(x\right)y\left(x\right)}$ get constructed, which all are incorrect, as they always involve too many terms. Note that the condition of Equation (71) is at the heart of every Lie-point symmetry analysis, in that if α and β are independent variables, then their relative variation must be zero; analogously for the corresponding transformed variables $\overline{\alpha}$ and $\overline{\beta}$:$$\frac{\mathcal{D}\alpha}{\mathcal{D}\beta}=\frac{\mathcal{D}\beta}{\mathcal{D}\alpha}=\frac{\mathcal{D}\overline{\alpha}}{\mathcal{D}\overline{\beta}}=\frac{\mathcal{D}\overline{\beta}}{\mathcal{D}\overline{\alpha}}=0$$
- E.4:
- The conclusion made in [1] (p. 1551), namely that if the multipoint velocity correlation functions vanish at infinity for an unbounded domain, then all functional derivatives of Φ vanish at infinity, too, is generally incorrect. In particular, their argument that if Equation [24] is imposed, i.e.,$${U}^{t}(x=\pm \infty )=0$$$${\Phi}_{,y\left(x\right)}{|}_{x=\pm \infty}=i\phantom{\rule{0.166667em}{0ex}}\u2329{U}^{t}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{e}^{i({U}^{t},y)}\u232a{|}_{x=\pm \infty}=0$$$$\u2329{e}^{i({U}^{t},y)}\u232a=\int {e}^{i\left(v\right(x),y)}\phantom{\rule{0.166667em}{0ex}}{f}^{t}\left(v\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathfrak{D}v\left(x\right)$$$$\u2329{U}^{t}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{e}^{i({U}^{t},y)}\u232a{|}_{x=\pm \infty}={\left[\phantom{\rule{0.166667em}{0ex}}\int {e}^{i\left(v\right(x),y)}\phantom{\rule{0.166667em}{0ex}}v\left(x\right)\phantom{\rule{0.166667em}{0ex}}{f}^{t}\left(v\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathfrak{D}v\left(x\right)\right]}_{x=\pm \infty}$$$$\Phi \left(t,y\left({x}^{\prime}\right)\right)=exp\left(\phantom{\rule{0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}f\left(t\right)\int d{x}^{\prime}\phantom{\rule{0.166667em}{0ex}}y\left({x}^{\prime}\right)\left({e}^{-\lambda {x}^{\prime 2}}+\int d{x}^{\u2033}d{x}^{\u2034}g({x}^{\u2033},{x}^{\u2034})\phantom{\rule{0.166667em}{0ex}}y\left({x}^{\u2033}\right)y\left({x}^{\u2034}\right)\right)\right)$$$${\Phi}^{*}\left(t,y\left({x}^{\prime}\right)\right)=\Phi \left(t,-y\left({x}^{\prime}\right)\right),\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\Phi (t,0)=1,\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\Phi \left(t,y\left({x}^{\prime}\right)\right)|\le 1$$$${\Phi}_{,y\left(x\right)}=\frac{\delta \Phi}{\delta y\left(x\right)}=if\left(t\right)\xb7\left(\phantom{\rule{0.166667em}{0ex}}{e}^{-\lambda {x}^{2}}+\int d{x}^{\prime}d{x}^{\u2033}\left(g({x}^{\prime},{x}^{\u2033})+2g(x,{x}^{\prime})\right)\phantom{\rule{0.166667em}{0ex}}y\left({x}^{\prime}\right)y\left({x}^{\u2033}\right)\phantom{\rule{0.166667em}{0ex}}\right)\xb7\Phi $$$$\u2329{U}^{t}\left(x\right)\u232a=\frac{1}{i}{\Phi}_{,y\left(x\right)}{|}_{y=0}=f\left(t\right)\xb7{e}^{-\lambda {x}^{2}}$$$$\u2329{U}^{t}(x=\pm \infty )\u232a={\left[\phantom{\rule{0.166667em}{0ex}}f\left(t\right)\xb7{e}^{-\lambda {x}^{2}}\phantom{\rule{0.166667em}{0ex}}\right]}_{x=\pm \infty}=0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\lambda >0$$$$\u2329{U}^{t}\left({x}_{1}\right)\cdots {U}^{t}\left({x}_{n}\right)\u232a{|}_{\parallel \mathit{x}\parallel \to \infty}=\frac{1}{{i}^{n}}{\Phi}_{,y\left({x}_{1}\right)\cdots y\left({x}_{n}\right)}{|}_{y=0,\parallel \mathit{x}\parallel \to \infty}=0$$$${\Phi}_{,y\left(x\right)}{|}_{x=\pm \infty}={\left[if\left(t\right)\left(\phantom{\rule{0.166667em}{0ex}}{e}^{-\lambda {x}^{2}}+\int d{x}^{\prime}d{x}^{\u2033}\left(g({x}^{\prime},{x}^{\u2033})+2g(x,{x}^{\prime})\right)\phantom{\rule{0.166667em}{0ex}}y\left({x}^{\prime}\right)y\left({x}^{\u2033}\right)\phantom{\rule{0.166667em}{0ex}}\right)\Phi \right]}_{x=\pm \infty}$$$${\Phi}_{,y\left(x\right)}{|}_{x=\pm \infty}=if\left(t\right)\Phi \int d{x}^{\prime}d{x}^{\u2033}g({x}^{\prime},{x}^{\u2033})\phantom{\rule{0.166667em}{0ex}}y\left({x}^{\prime}\right)y\left({x}^{\u2033}\right)\ne 0$$Worthwhile to note here is that when looking at the structure of the Hopf–Burgers Equation (3) more closely, one can see that the functional derivatives do not even have to converge for $\left|x\right|\to \infty $. Since by definition, $y\left(x\right)$ must be an asymptotically-decaying function, cf. [13], the functional derivatives appearing in Equation (3) are thus allowed to diverge within a certain prescribed order. For example, if $y\left(x\right)$ decays as $1/{x}^{n}$ for $\left|x\right|\to \infty $, then the (viscous) first-order functional derivative may diverge, but not faster than ${x}^{n}$ and the (inertial) second-order one not faster than ${x}^{n-1}$. This can be easily verified by power counting as within these assumptions, the integral operator over x in Equation (3) is well-conditioned.

## Author Contributions

## Conflicts of Interest

## Appendix A. Proof that ${\mathit{X}}_{\mathbf{4}}^{\text{phys}}$-${\mathit{X}}_{\mathbf{6}}^{\text{phys}}$in [1] Are Not Admitted as Symmetry Transformations

#### A.1. Transformation ${\mathsf{T}}_{4}$

#### A.2. Transformation ${\mathsf{T}}_{5}$

#### A.3. Transformation ${\mathsf{T}}_{6}$

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Comments on Janocha *et al*. Lie Symmetry Analysis of the Hopf Functional-Differential Equation. *Symmetry* 2015, 7, 1536–1566. *Symmetry* **2016**, *8*, 23.
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Comments on Janocha *et al*. Lie Symmetry Analysis of the Hopf Functional-Differential Equation. *Symmetry* 2015, 7, 1536–1566. *Symmetry*. 2016; 8(4):23.
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