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

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

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

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:

$$\frac{\mathcal{D}\overline{\Phi }}{\mathcal{D}t}=\frac{\mathcal{D}\overline{\Phi }}{\mathcal{D}\overline{t}}\frac{\mathcal{D}\overline{t}}{\mathcal{D}t}+\frac{\mathcal{D}\overline{\Phi }}{\mathcal{D}\overline{x}}\frac{\mathcal{D}\overline{x}}{\mathcal{D}t}+{\int }_{\overline{G}}\frac{\mathcal{D}\overline{\Phi }}{\mathcal{D}\overline{y\left(x^{\prime }\right)d{x}^{\prime }}}\sum _{n=1}^{\infty }\frac{\mathcal{D}{\overline{y}}_n{h}_n({\overline{x}}^{\prime }-\epsilon {\xi }_{x^{\prime }})d{\overline{x}}^{\prime }}{\mathcal{D}t}$$

(57)

This considerably influences all further results, e.g., the final result in Equation [15] is not given by three, but only by the first two terms on its right-hand side, which then coincides again with the above (correct) variation of Equation (57), knowing that $\mathcal{D}\overline{\Phi }/\mathcal{D}\overline{x}=0$. Hereby, it should be pointed out that the formulation for the transformed decomposition of $y\left(z\right)dz$ in Equation [11] is misleading, since it suggests that the basis functions transform frame-indifferently, i.e., as $h_n\left(z\right)=h_n\left(\overline{z}\right)$, but which is not the case, as can be easily seen when directly transforming the initial decomposition for $y\left(z\right)dz$ in Equation [10] according to Definition 5 [1] (p. 1542) up to first order:

$$\begin{array}{cc}\hfill 0& =y\left(z\right)dz-\sum _{n=1}^{\infty }{y}_n{h}_n\left(z\right)dz=\overline{y\left(z\right)dz}-\epsilon {\xi }_{y\left(z\right)}dz-\sum _{n=1}^{\infty }{y}_n{h}_n\left(z\right)dz+\mathcal{O}\left({\epsilon }^2\right)\hfill \\ & =\overline{y\left(z\right)dz}-\epsilon {\int }_{G}d{z}^{\prime }\delta (z^{\prime }-z){\xi }_{y\left(z^{\prime }\right)}dz-\sum _{n=1}^{\infty }{y}_n{h}_n\left(z\right)dz+\mathcal{O}\left({\epsilon }^2\right)\hfill \\ & =\overline{y\left(z\right)dz}-\epsilon {\int }_{G}d{z}^{\prime }{\xi }_{y\left(z^{\prime }\right)}dz\sum _{n=1}^{\infty }{h}_n\left(z\right)h_n\left(z^{\prime }\right)-\sum _{n=1}^{\infty }{y}_n{h}_n\left(z\right)dz+\mathcal{O}\left({\epsilon }^2\right)\hfill \\ & =\overline{y\left(z\right)dz}-\sum _{n=1}^{\infty }\left(y_n+\epsilon {\int }_{G}d{z}^{\prime }{\xi }_{y\left(z^{\prime }\right)}{h}_n\left(z^{\prime }\right)\right)h_n\left(z\right)dz+\mathcal{O}\left({\epsilon }^2\right)\hfill \\ & =\overline{y\left(z\right)dz}-\sum _{n=1}^{\infty }\left(y_n+\epsilon {\int }_{G}d{z}^{\prime }{\xi }_{y\left(z^{\prime }\right)}{h}_n\left(z^{\prime }\right)\right)h_n\left(z\right)\left(1-\epsilon {\partial }_{\overline{z}}{\xi }_z\right)d\overline{z}+\mathcal{O}\left({\epsilon }^2\right)\hfill \\ & =\overline{y\left(z\right)dz}-\sum _{n=1}^{\infty }{\overline{y}}_n{\overline{h}}_n\left(\overline{z}\right)d\overline{z}+\mathcal{O}\left({\epsilon }^2\right)\hfill \end{array}$$

(58)

where the transformation rules for ${\overline{y}}_n$ and ${\overline{h}}_n\left(\overline{z}\right)$ are thus given as:

$$\begin{array}{c}{\overline{h}}_n\left(\overline{z}\right)=h_n\left(z\right)=h_n(\overline{z}-\epsilon {\xi }_z)\ne h_n\left(\overline{z}\right)\end{array}$$

(59)

$$\begin{array}{c}{\overline{y}}_n=y_n+\epsilon \left({\int }_{G}d{z}^{\prime }{\xi }_{y\left(z^{\prime }\right)}{h}_n\left(z^{\prime }\right)-y_n{\partial }_z{\xi }_z\right)\end{array}$$

(60)

(Remark: We could not find any reasonable argument or see any analytical advantage for why the authors in [1] have made use of the formal expansion of Equation (58) at this stage of their analysis. In the way as their cited book by Klauder [17] already discusses in Section 3, in particular in Sections 3.5 and 3.7, this expansion only becomes relevant as soon as one specifies any functional for which one faces problems in practically evaluating its functional derivatives, where it sometimes is necessary to soften or to smear out the Dirac delta function. But, since all functionals in Section 2 of [1] are treated arbitrarily and no specifications are needed, such an expansion only unnecessarily complicates their analysis at this stage. Also, note that in the derivation of Equation (58) we assume that the set of basis functions $\left\{h_n\left(z\right)\right\}$ is complete and orthonormal, i.e., that they satisfy the relation ${\sum }_{n=1}^{\infty }{h}_n\left(z^{\prime }\right)h_n\left(z\right)=\delta (z^{\prime }-z)$).

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 $\left\{h_n\left(z\right)\right\}$:

$$y\left(z\right)dz=\sum _{n=1}^{\infty }{y}_n{h}_n\left(z\right)dz$$

(61)

any variable transformation must of course be compatible with the following constraint, namely that the expansion coefficients $y_n$ constitute simple constants, i.e., ${\partial }_t{y}_n={\partial }_x{y}_n=0$, which, if the set $\left\{h_n\left(z\right)\right\}$ is complete and orthonormal, are disclosed by:

$$y_n={\int }_G{h}_n\left(z\right)y\left(z\right)dz$$

(62)

Hence, when transforming the relation of Equation (62), it will inherently induce restrictions on the transformation rule for the expansion coefficients $y_n\mapsto {\overline{y}}_n$ of Equation (60) and, thus, on the infinitesimals ${\xi }_{y\left(z\right)}dz$ and ${\xi }_z$. However, again, this information was not revealed in [1].

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:

$$\frac{\mathcal{D}{\overline{y}}_n}{\mathcal{D}\overline{x}}=0$$

(63)

(Remark: Note that Equation [13] in [1] contains a typing error: in the second term on the right-hand side, “$y_n$” has to be replaced by “${\overline{y}}_n$”). However, a restriction such as Equation (63) would have severe consequences for the transformation rule of the variables $y\left(x\right)dx$ and x, and it is not advisable to enforce this unnecessary restriction; in particular, as this restriction is artificially generated in [1], namely by introducing a functional decomposition of $y\left(x\right)dx$ in Equation [10], which has, in the end, even no necessity in itself to successfully analyze the Hopf–Burgers Equation (3) for Lie-point symmetries (see e.g., our proposed approach in Section 2). To corroborate our claim, let us separately verify the relations of Equations [13] and [14] for the following transformation (up to first order), which was found as the symmetry solution $X_4^{\text{phys}}$ in [1] (p. 1561) for the unbounded domain $G=(-\infty ,+\infty )$:

$$\left.\begin{array}{c}\hfill \overline{t}=t+\epsilon ·t^2,\overline{x}=x+\epsilon ·tx,\overline{y\left(x\right)dx}=y\left(x\right)dx+\epsilon ·ty\left(x\right)dx\\ \hfill \overline{\Phi }=\Phi +\epsilon ·\frac{i}{2}\Phi \int xy\left(x\right)dx\end{array}\right\}$$

(64)

in order to see that this transformation is only compatible with Equation [13], but not to Equation [14]. The required transformation rule for the expansion coefficients $y_n$ is induced by Equation (64), which, for this particular configuration up to first order, all transform invariantly according to Equation (60) when using Equation (62):

$${\overline{y}}_n=y_n+\epsilon ·t\left({\int }_G{h}_n\left(x\right)y\left(x\right)dx-y_n\right)=y_n$$

(65)

while the spatial 1D volume element transforms as:

$$d\overline{x}=dx+\epsilon ·tdx$$

(66)

Now, since the left- and right-hand sides of Equation [13] evaluate to:

$$\begin{array}{cc}\hfill {\text{LHS}}_{\text{Eq.[13]}}:& \frac{\mathcal{D}\overline{y\left(x\right)dx}}{\mathcal{D}t}=\frac{\partial \overline{y\left(x\right)dx}}{\partial t}=\epsilon ·y\left(x\right)dx+\mathcal{O}\left({\epsilon }^2\right)\hfill \\ \hfill {\text{RHS}}_{\text{Eq.[13]}}:& \sum _{n=1}^{\infty }\frac{\mathcal{D}{\overline{y}}_n}{\mathcal{D}t}{\overline{h}}_n\left(\overline{x}\right)d\overline{x}+\sum _{n=1}^{\infty }{\overline{y}}_n\frac{\mathcal{D}{\overline{h}}_n\left(\overline{x}\right)d\overline{x}}{\mathcal{D}t}=\sum _{n=1}^{\infty }{\overline{y}}_n\frac{\partial {\overline{h}}_n\left(\overline{x}\right)d\overline{x}}{\partial t}\hfill \end{array}$$

(67)

$$\begin{array}{cc}& =\sum _{n=1}^{\infty }\epsilon {\overline{y}}_n{h}_n\left(x\right)dx+\mathcal{O}\left({\epsilon }^2\right)=\epsilon ·y\left(x\right)dx+\mathcal{O}\left({\epsilon }^2\right)\hfill \end{array}$$

(68)

and the corresponding sides of Equation [14] to:

$$\begin{array}{cc}\hfill {\text{LHS}}_{\text{Eq.[14]}}:& \sum _{n=1}^{\infty }\frac{\mathcal{D}{\overline{y}}_n}{\mathcal{D}t}{\overline{h}}_n\left(\overline{x}\right)d\overline{x}=0\hfill \\ \hfill {\text{RHS}}_{\text{Eq.[14]}}:& \frac{\mathcal{D}\overline{y\left(x\right)dx}}{\mathcal{D}t}-\frac{\mathcal{D}\overline{y\left(x\right)dx}}{\mathcal{D}\overline{x}}\frac{\mathcal{D}\overline{x}}{\mathcal{D}t}=\epsilon ·y\left(x\right)dx-\epsilon ·x\frac{\partial \overline{y\left(x\right)dx}}{\partial \overline{x}}+\mathcal{O}\left({\epsilon }^2\right)\hfill \end{array}$$

(69)

$$\begin{array}{cc}& =\epsilon ·\left(y\left(x\right)-x{\partial }_{x}y\left(x\right)\right)dx+\mathcal{O}\left({\epsilon }^2\right)\hfill \end{array}$$

(70)

we see that while Equation [13] is generally satisfied, Equation [14] is only satisfied for the special case when $y\left(x\right)=c·x$, where c is an arbitrary integration constant. But, since the considered transformation of Equation (64) is only defined in the unbounded domain $G=(-\infty ,+\infty )$, the function $y\left(x\right)$ must be an asymptotically decaying function, cf. [13], which thus forces the integration constant to be $c=0$. That means we have to demand $y\left(x\right)=0$, but which, obviously, does not make any sense. In other words, the transformation of Equation (64) is not compatible to Equation [14], simply because it is not generally valid. (Remark: See Discussion E.1, in particular the transformation of Equation (59), for why Equation (68) is in fact the only correct formulation for the right-hand side of Equation [13] and not its formulation as given in [1]).

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:

$$\frac{\mathcal{D}y\left(x^{\prime }\right)d{x}^{\prime }}{\mathcal{D}{x}^{\prime }}=\frac{\mathcal{D}\overline{y\left(x^{\prime }\right)d{x}^{\prime }}}{\mathcal{D}\overline{x^{\prime }}}=0$$

(71)

(Remark: Note that the total variation operator $\mathcal{D}$ only scans for explicit dependencies. 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 ${\zeta }_{;y\left(x\right)}$ 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 $\overline{\beta }$:

$$\frac{\mathcal{D}\alpha }{\mathcal{D}\beta }=\frac{\mathcal{D}\beta }{\mathcal{D}\alpha }=\frac{\mathcal{D}\overline{\alpha }}{\mathcal{D}\overline{\beta }}=\frac{\mathcal{D}\overline{\beta }}{\mathcal{D}\overline{\alpha }}=0$$

(72)

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.,

$$U^t(x=\pm \infty )=0$$

(73)

that then the evaluation, e.g., of the first order functional derivative at infinity will give the zero result:

$${\Phi }_{,y\left(x\right)}{|}_{x=\pm \infty }=i〈U^t\left(x\right)e^{i(U^t,y)}〉{|}_{x=\pm \infty }=0$$

(74)

cannot be correct in the general sense, simply because the above coarse-graining operator $\langle \rangle$ is not defined as a point-based ensemble operator, where the above reasoning would then be generally true, but as a function-based ensemble operator that is expressed as a path integral over all possible configuration states $U^t$ in the following way (see Equation [1] or Definition 2 in [1] (p. 1538)):

$$〈e^{i(U^t,y)}〉=\int e^{i\left(v\right(x),y)}{f}^t\left(v\left(x\right)\right)\mathfrak{D}v\left(x\right)$$

(75)

where $f^t$ is the probability density functional and $\mathfrak{D}v\left(x\right)$ the functional volume element over an ensemble of all possible spatial velocity fields $v\left(x\right)$ (for more details, see, e.g., [13]). Hence, a point-wise evaluation as proposed in Equation (74) is ill-conditioned, i.e., to evaluate the following mathematical object:

$$〈U^t\left(x\right)e^{i(U^t,y)}〉{|}_{x=\pm \infty }={\left[\int e^{i\left(v\right(x),y)}v\left(x\right)f^t\left(v\left(x\right)\right)\mathfrak{D}v\left(x\right)\right]}_{x=\pm \infty }$$

(76)

is in general not well defined, since a priori it is not clear how and in which way all functional elements within the path integral collectively behave when x tends to infinity. To substantiate our argument, let us give a concrete example where the evaluation ${\Phi }_{,y\left(x\right)}{|}_{x=\pm \infty }$ in Equation (74) is not zero. Consider, for example, the following well-defined specification of the functional Φ:

$$\Phi \left(t,y\left(x^{\prime }\right)\right)=exp\left(if\left(t\right)\int d{x}^{\prime }y\left(x^{\prime }\right)\left(e^{-\lambda x^{\prime 2}}+\int d{x}^{″}d{x}^{‴}g(x^{″},x^{‴})y\left(x^{″}\right)y\left(x^{‴}\right)\right)\right)$$

(77)

where f is an arbitrary real function, $\lambda >0$ some positive real-valued constant and g a two-dimensional real and symmetric scalar function decaying sufficiently fast at space infinity, i.e., $g\left(\mathit{x}\right):=g(x_1,x_2)=g(x_2,x_1)\to 0$ for $\parallel \mathit{x}\parallel \to \infty$. (Remark: Note that $y\left(x\right)$ is defined as a real function; see [13]). In the wording of [1], the specification of Equation (77) represents a physically-relevant functional, since it respects all the constraints of Definition 11 given in [1] (p. 1559):

$${\Phi }^{*}\left(t,y\left(x^{\prime }\right)\right)=\Phi \left(t,-y\left(x^{\prime }\right)\right),\Phi (t,0)=1,|\Phi \left(t,y\left(x^{\prime }\right)\right)|\le 1$$

(78)

Now, by taking the first order functional derivative of Equation (77), we get:

$${\Phi }_{,y\left(x\right)}=\frac{\delta \Phi }{\delta y\left(x\right)}=if\left(t\right)·\left(e^{-\lambda x^2}+\int d{x}^{\prime }d{x}^{″}\left(g(x^{\prime },x^{″})+2g(x,x^{\prime })\right)y\left(x^{\prime }\right)y\left(x^{″}\right)\right)·\Phi$$

(79)

When evaluating this result for $y\left(x\right)=0$, one yields the first order velocity correlation function (see Equation [5] in [1] (p. 1539))

$$〈U^t\left(x\right)〉=\frac{1}{i}{\Phi }_{,y\left(x\right)}{|}_{y=0}=f\left(t\right)·e^{-\lambda x^2}$$

(80)

which then, when evaluated at the infinite far point:

$$〈U^t(x=\pm \infty )〉={\left[f\left(t\right)·e^{-\lambda x^2}\right]}_{x=\pm \infty }=0,\lambda >0$$

(81)

satisfies the correct imposed condition of Equation (73). Similarly, for all higher (multipoint) moments, we also get:

$$〈U^t\left(x_1\right)\cdots U^t\left(x_n\right)〉{|}_{\parallel \mathit{x}\parallel \to \infty }=\frac{1}{i^n}{\Phi }_{,y\left(x_1\right)\cdots y\left(x_n\right)}{|}_{y=0,\parallel \mathit{x}\parallel \to \infty }=0$$

(82)

That means the specified functional of Equation (77) is compatible to the (instantaneous) velocity constraint of Equation (73) in that it induces the correctly-conditioned (statistical) multipoint moments. But, now, if we directly evaluate, e.g., the functional derivative of Equation (79) only at the infinite far point:

$${\Phi }_{,y\left(x\right)}{|}_{x=\pm \infty }={\left[if\left(t\right)\left(e^{-\lambda x^2}+\int d{x}^{\prime }d{x}^{″}\left(g(x^{\prime },x^{″})+2g(x,x^{\prime })\right)y\left(x^{\prime }\right)y\left(x^{″}\right)\right)\Phi \right]}_{x=\pm \infty }$$

(83)

then it does not necessarily evaluate to zero if $y\ne 0$; in particular, it will evaluate to:

$${\Phi }_{,y\left(x\right)}{|}_{x=\pm \infty }=if\left(t\right)\Phi \int d{x}^{\prime }d{x}^{″}g(x^{\prime },x^{″})y\left(x^{\prime }\right)y\left(x^{″}\right)\ne 0$$

(84)

Hence, in contrast to the general claim of Equation (74) made in [1] (p. 1551), it is possible to construct a functional Φ, e.g., Equation (77), which satisfies all physical constraints of Equation (78) and which induces the correct imposed condition of Equation (73), but which in general does not evaluate to zero when taking its functional derivative at infinity, as we can clearly see in Equation (84). This problem will then continue for all higher order functional derivatives. That means, the reasoning in [1] (p. 1551) “that all functional derivatives of Φ vanish for $x\to \pm \infty$” if the condition of Equation [24], i.e., $U^t(x=\pm \infty )=0$ is imposed, is incorrect in the sense that this argument may not be used, e.g., to perform partial integrations, as it was generally done throughout [1].

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.