# Reply to Frewer et al. 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

## 2. Relation to Other Works

## 3. Technical Errors

## 4. Choice of variables

## 5. Breaking of Symmetries

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A.

## Infinitesimals

## Determining the System of Equations for the Infinitesimals

- $\mathcal{A}=0$ reads:$$\frac{\partial {\eta}_{\varphi}}{\partial t}-{\int}_{G}y\left(x\right)\left(i\frac{{\partial}^{3}{\eta}_{\varphi}}{\partial x\partial {\left(y\left(x\right)\mathrm{d}x\right)}^{2}}+\nu \frac{{\partial}^{3}{\eta}_{\varphi}}{\partial {x}^{2}\partial y\left(x\right)\mathrm{d}x}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x=0$$
- $\mathcal{B}=0$ reads:$$\begin{array}{cc}\hfill 0& =\nu \frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(z\right)\frac{\partial {\eta}_{\varphi}}{\partial \varphi}\right)-\nu \frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(z\right)\frac{\partial {\xi}_{t}}{\partial t}\right)-\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial t}-\nu \frac{{\partial}^{2}{\xi}_{\gamma \left(z\right)}}{\partial {z}^{2}}\hfill \\ & \phantom{\rule{1.em}{0ex}}-2iy\left(z\right)\frac{{\partial}^{2}}{\partial z\partial \varphi}\frac{\delta {\eta}_{\varphi}}{\delta y\left(z\right)}+i\nu {\int}_{G}\frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(x\right)y\left(z\right)\frac{\partial}{\partial x}\frac{{\delta}^{2}{\xi}_{t}}{\delta y{\left(x\right)}^{2}}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x\hfill \\ & \phantom{\rule{1.em}{0ex}}+i{\int}_{G}y\left(x\right)\frac{\partial}{\partial x}\frac{{\delta}^{2}{\xi}_{\gamma \left(z\right)}}{\delta y{\left(x\right)}^{2}}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x-i\frac{\partial}{\partial z}\left(y\left(z\right)\frac{\partial}{\partial z}\frac{\delta {\xi}_{z}}{\delta y\left(z\right)}\right)+2i\frac{\partial}{\partial z}\left(y\left(z\right)\frac{\partial}{\partial \varphi}\frac{\delta {\eta}_{\varphi}}{\delta y\left(z\right)}\right)\hfill \\ & \phantom{\rule{1.em}{0ex}}-i{\int}_{G}\frac{\partial}{\partial x}\left(y\left(x\right)\frac{{\delta}^{2}{\xi}_{\gamma \left(z\right)}}{\delta y{\left(x\right)}^{2}}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x+i\frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(z\right)\frac{\delta {\xi}_{z}}{\delta y\left(z\right)}\right)\hfill \\ & \phantom{\rule{1.em}{0ex}}+{\nu}^{2}{\int}_{G}\frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(x\right)y\left(z\right)\frac{{\partial}^{2}}{\partial {x}^{2}}\frac{\delta {\xi}_{t}}{\delta y\left(x\right)}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x+\nu {\int}_{G}y\left(x\right)\frac{{\partial}^{2}}{\partial {x}^{2}}\frac{\delta {\xi}_{\gamma \left(z\right)}}{\delta y\left(x\right)}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x\hfill \\ & \phantom{\rule{1.em}{0ex}}-2\nu {\int}_{G}\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial}{\partial x}\frac{\delta {\xi}_{\gamma \left(z\right)}}{\delta y\left(x\right)}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x-\nu \frac{\partial}{\partial z}\left(y\left(z\right)\frac{{\partial}^{2}{\xi}_{z}}{\partial {z}^{2}}\right)-\nu \frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(z\right)\frac{\partial {\eta}_{\varphi}}{\partial \varphi}\right)\hfill \\ & \phantom{\rule{1.em}{0ex}}+\nu {\int}_{G}\frac{{\partial}^{2}}{\partial {x}^{2}}\left(y\left(x\right)\frac{\delta {\xi}_{\gamma \left(z\right)}}{\delta y\left(x\right)}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x+2\nu \frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(z\right)\frac{\partial {\xi}_{z}}{\partial z}\right)\hfill \end{array}$$
- $\mathcal{C}=0$ reads:$$\begin{array}{cc}\hfill 0& =-i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial {\eta}_{\varphi}}{\partial \varphi}\right)\delta (x-z)+i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial {\xi}_{t}}{\partial t}\right)\delta (x-z)+i\frac{\partial {\xi}_{\gamma \left(x\right)}}{\partial x}\delta (x-z)\hfill \\ & \phantom{\rule{1.em}{0ex}}+{\int}_{G}\frac{\partial}{\partial z}\left(y\left(a\right)y\left(z\right)\frac{\partial}{\partial a}\frac{{\delta}^{2}{\xi}_{t}}{\delta y{\left(a\right)}^{2}}\right)\delta (x-z)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}a+2iy\left(x\right)\frac{\partial}{\partial x}\frac{\delta {\xi}_{\gamma \left(z\right)}}{\delta y\left(x\right)}\hfill \\ & \phantom{\rule{1.em}{0ex}}-2i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\delta {\xi}_{\gamma \left(z\right)}}{\delta y\left(x\right)}\right)+i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial {\eta}_{\varphi}}{\partial \varphi}\right)\delta (x-z)-i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial {\xi}_{x}}{\partial x}\right)\delta (x-z)\hfill \\ & \phantom{\rule{1.em}{0ex}}-i\nu {\int}_{G}\frac{\partial}{\partial z}\left(y\left(a\right)y\left(z\right)\frac{{\partial}^{2}}{\partial {a}^{2}}\frac{\delta {\xi}_{t}}{\delta y\left(a\right)}\right)\delta (x-z)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}a+\nu y\left(x\right)\frac{{\partial}^{3}{\xi}_{\gamma \left(z\right)}}{\partial {x}^{2}\partial \varphi}+\nu y\left(x\right)\frac{{\partial}^{2}{\xi}_{\gamma \left(z\right)}}{\partial {x}^{2}}\hfill \\ & \phantom{\rule{1.em}{0ex}}-\nu \frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial x}\right)-\nu \frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial x}\right)\hfill \end{array}$$
- $\mathcal{D}=0$ reads:$$\begin{array}{cc}\hfill 0& =-{\nu}^{2}\frac{{\partial}^{4}}{\partial {z}^{2}\partial {x}^{2}}\left(y\left(x\right)y\left(z\right)\frac{\partial {\xi}_{t}}{\partial \varphi}\right)-\nu \frac{{\partial}^{2}}{\partial {x}^{2}}\left(y\left(x\right)\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial \varphi}\right)+2i\nu \frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(x\right)y\left(z\right)\frac{{\partial}^{2}}{\partial x\partial \varphi}\frac{\delta {\xi}_{t}}{\delta y\left(x\right)}\right)\hfill \\ & \phantom{\rule{1.em}{0ex}}+2iy\left(x\right)\frac{{\partial}^{2}}{\partial x\partial \varphi}\frac{\delta {\xi}_{\gamma \left(z\right)}}{\delta y\left(x\right)}-i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{{\partial}^{2}{\xi}_{x}}{\partial x\partial \varphi}\right)\delta (x-z)-2i\nu \frac{{\partial}^{3}}{\partial {z}^{2}\partial x}\left(y\left(x\right)y\left(z\right)\frac{\partial}{\partial \varphi}\frac{\delta {\xi}_{t}}{\partial y\left(x\right)}\right)\hfill \\ & \phantom{\rule{1.em}{0ex}}-2i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial}{\partial \varphi}\frac{\delta {\xi}_{\gamma \left(z\right)}}{\delta y\left(x\right)}\right)\delta (x-z)-2i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial}{\partial \varphi}\frac{\delta {\xi}_{\gamma \left(z\right)}}{\delta y\left(x\right)}\right)\hfill \\ & \phantom{\rule{1.em}{0ex}}+2i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{{\partial}^{2}{\eta}_{\varphi}}{\partial {\varphi}^{2}}\right)\delta (x-z)+2i\frac{{\partial}^{2}}{\partial {x}^{2}}\left(y\left(x\right)\frac{\partial {\xi}_{x}}{\partial \varphi}\right)\delta (x-z)\hfill \\ & \phantom{\rule{1.em}{0ex}}-\nu \frac{\partial}{\partial x}\left(y\left(x\right)\frac{{\partial}^{2}{\xi}_{\gamma \left(z\right)}}{\partial x\partial \varphi}\right)-2\nu \frac{\partial}{\partial x}\left(y\left(x\right)\frac{{\partial}^{2}{\xi}_{\gamma \left(z\right)}}{\partial x\partial \varphi}\right)-\nu \frac{\partial}{\partial x}\left(y\left(x\right)\frac{{\partial}^{2}{\xi}_{\gamma \left(z\right)}}{\partial x\partial \varphi}\right)\hfill \\ & \phantom{\rule{1.em}{0ex}}+2\nu \frac{{\partial}^{2}}{\partial {x}^{2}}\left(y\left(x\right)\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial \varphi}\right)+{\nu}^{2}\frac{{\partial}^{4}}{\partial {z}^{2}\partial {x}^{2}}\left(y\left(x\right)y\left(z\right)\frac{\partial {\xi}_{t}}{\partial \varphi}\right)+\nu \frac{{\partial}^{2}}{\partial {x}^{2}}\left(y\left(x\right)\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial \varphi}\right)\hfill \\ & \phantom{\rule{1.em}{0ex}}+\nu \frac{{\partial}^{2}}{\partial {x}^{2}}\left(y\left(x\right)\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial \varphi}\right)-2{\nu}^{2}\frac{{\partial}^{3}}{\partial {z}^{2}\partial x}\left(y\left(x\right)y\left(z\right)\frac{{\partial}^{2}{\xi}_{t}}{\partial x\partial \varphi}\right)+{\nu}^{2}\frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(x\right)y\left(z\right)\frac{{\partial}^{3}{\xi}_{t}}{\partial {x}^{2}\partial \varphi}\right)\hfill \end{array}$$
- $\mathcal{E}=0$ reads:$$\begin{array}{cc}\hfill 0& =2i\nu \frac{{\partial}^{3}}{\partial {z}^{2}\partial x}\left(y\left(x\right)y\left(z\right)\frac{\partial {\xi}_{t}}{\partial \varphi}\right)\delta (x-a)+i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial \varphi}\right)\delta (x-a)\hfill \\ & \phantom{\rule{1.em}{0ex}}+2\frac{\partial}{\partial x}\left(y\left(x\right)y\left(z\right)\frac{{\partial}^{2}}{\partial z\partial \varphi}\frac{\delta {\xi}_{t}}{\delta y\left(z\right)}\right)\delta (x-a)+2iy\left(z\right)\frac{{\partial}^{2}{\xi}_{\gamma \left(x\right)}}{\partial z\partial \varphi}\delta (a-z)+iy\left(x\right)\frac{{\partial}^{2}{\xi}_{\gamma \left(z\right)}}{\partial x\partial \varphi}\delta (x-a)\hfill \\ & \phantom{\rule{1.em}{0ex}}-2\frac{{\partial}^{2}}{\partial z\partial x}\left(y\left(x\right)y\left(z\right)\frac{\partial}{\partial \varphi}\frac{\delta {\xi}_{t}}{\delta y\left(z\right)}\right)\delta (x-a)-2i\frac{\partial}{\partial z}\left(y\left(z\right)\frac{\partial {\xi}_{\gamma \left(x\right)}}{\partial \varphi}\right)\delta (a-z)\hfill \\ & \phantom{\rule{1.em}{0ex}}-i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial \varphi}\right)\delta (x-a)-2i\frac{\partial}{\partial z}\left(y\left(z\right)\frac{\partial {\xi}_{\gamma \left(x\right)}}{\partial \varphi}\right)\delta (a-z)\hfill \\ & \phantom{\rule{1.em}{0ex}}-i\nu \frac{{\partial}^{3}}{\partial {z}^{2}\partial x}\left(y\left(x\right)y\left(z\right)\frac{\partial {\xi}_{t}}{\partial \varphi}\right)\delta (x-a)-i\frac{\partial}{\partial x}\left(y\left(x\right)\frac{\partial {\xi}_{\gamma \left(z\right)}}{\partial \varphi}\right)\delta (x-a)\hfill \\ & \phantom{\rule{1.em}{0ex}}-i\nu \frac{{\partial}^{3}}{\partial {z}^{2}\partial x}\left(y\left(x\right)y\left(z\right)\frac{\partial {\xi}_{t}}{\partial \varphi}\right)\delta (x-a)+i\nu \frac{{\partial}^{2}}{\partial {z}^{2}}\left(y\left(x\right)y\left(z\right)\frac{{\partial}^{2}{\xi}_{t}}{\partial x\partial \varphi}\right)\delta (x-a)\hfill \\ & \phantom{\rule{1.em}{0ex}}+2i\nu \frac{{\partial}^{2}}{\partial z\partial x}\left(y\left(x\right)y\left(z\right)\frac{{\partial}^{2}{\xi}_{t}}{\partial z\partial \varphi}\right)\delta (x-a)-2i\nu \frac{\partial}{\partial x}\left(y\left(x\right)y\left(z\right)\frac{{\partial}^{3}{\xi}_{t}}{\partial {z}^{2}\partial \varphi}\right)\delta (x-a)\hfill \end{array}$$
- Equations $\mathcal{F}=0$, $\mathcal{G}=0$, $\mathcal{H}=0$, $\mathcal{I}=0$, $\mathcal{J}=0$, $\mathcal{K}=0$ and $\mathcal{L}=0$ are the same as in [1].

## Solution of the Determining System of Equations for the Infinitesimals

## References

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**MDPI and ACS Style**

Wacławczyk, M.; Janocha, D.D.; Oberlack, M.
Reply to Frewer *et al.* Comments on Janocha *et al.* Lie Symmetry Analysis of the Hopf Functional-Differential Equation. *Symmetry* 2015, *7*, 1536–1566. *Symmetry* **2016**, *8*, 24.
https://doi.org/10.3390/sym8040024

**AMA Style**

Wacławczyk M, Janocha DD, Oberlack M.
Reply to Frewer *et al.* Comments on Janocha *et al.* Lie Symmetry Analysis of the Hopf Functional-Differential Equation. *Symmetry* 2015, *7*, 1536–1566. *Symmetry*. 2016; 8(4):24.
https://doi.org/10.3390/sym8040024

**Chicago/Turabian Style**

Wacławczyk, Marta, Daniel D. Janocha, and Martin Oberlack.
2016. "Reply to Frewer *et al.* Comments on Janocha *et al.* Lie Symmetry Analysis of the Hopf Functional-Differential Equation. *Symmetry* 2015, *7*, 1536–1566" *Symmetry* 8, no. 4: 24.
https://doi.org/10.3390/sym8040024