# Second-Order Conditional Lie–Bäcklund Symmetries and Differential Constraints of Nonlinear Reaction–Diffusion Equations with Gradient-Dependent Diffusivity

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## Abstract

**:**

## 1. Introduction

## 2. Basic Notations and Theorems

**Definition 1**

**([45,46,47]).**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Equation (3) Admitting CLBS (1) and H–J SI (2)

- $\left(i\right)$
- For $k\ne -2,\phantom{\rule{4pt}{0ex}}0$,$$\begin{array}{c}\hfill \begin{array}{c}\hfill G(r,u)={g}_{1}\left(r\right){u}^{-\frac{k}{2}}+{g}_{2}\left(r\right)u+\frac{n-1}{2r}.\end{array}\end{array}$$
- $\left(ii\right)$
- For $k=-2$,$$\begin{array}{c}\hfill \begin{array}{c}\hfill G(r,u)={g}_{1}\left(r\right)u+{g}_{2}\left(r\right)ulnu+\frac{n-1}{2r}.\end{array}\end{array}$$
- $\left(iii\right)$
- For $k=0$,$$\begin{array}{c}\hfill \begin{array}{c}\hfill G(r,u)={g}_{1}\left(r\right)u+{g}_{2}\left(r\right).\end{array}\end{array}$$

## 4. Symmetry Reductions of Equation (3)

**Example**

**1.**

- For $b>\frac{27}{2}$,$$\begin{array}{c}\hfill u(r,t)=\frac{4(2b-27)}{{\left[\alpha \left(t\right)sinh\left({d}_{1}lnr\right)+\beta \left(t\right)cosh\left({d}_{1}lnr\right)\right]}^{2}{r}^{3}},\end{array}$$$$\begin{array}{c}\hfill \begin{array}{c}{\alpha}^{\prime}=\frac{3}{8}{\alpha}^{3}+\frac{1}{4}{d}_{1}{\alpha}^{2}\beta -\frac{3}{8}\alpha {\beta}^{2}-\frac{1}{4}{d}_{1}{\beta}^{3}-\frac{1}{2}a\alpha ,\hfill \\ {\beta}^{\prime}=-\frac{3}{8}{\beta}^{3}-\frac{1}{4}{d}_{1}\alpha {\beta}^{2}+\frac{3}{8}{\alpha}^{2}\beta +\frac{1}{4}{d}_{1}{\alpha}^{3}-\frac{1}{2}a\beta .\hfill \end{array}\end{array}$$
- For $b=\frac{27}{2}$,$$\begin{array}{c}\hfill u(r,t)=\frac{4}{{\left[\alpha \left(t\right)+\beta \left(t\right)lnr\right]}^{2}{r}^{3}},\end{array}$$$$\begin{array}{c}\hfill \begin{array}{c}{\alpha}^{\prime}=\frac{3}{2}\alpha {\beta}^{2}-\frac{1}{2}a\alpha +{\beta}^{3},\hfill \\ {\beta}^{\prime}=\frac{3}{2}{\beta}^{3}-\frac{1}{2}a\beta .\hfill \end{array}\end{array}$$The solutions of this system of ODEs with $a=0$ are listed below:$$\begin{array}{c}\hfill \begin{array}{c}\alpha \left(t\right)=\frac{{c}_{2}\mp ln({c}_{1}-3t)}{3\sqrt{{c}_{1}-3t}},\hfill \\ \beta \left(t\right)=\pm \frac{1}{\sqrt{{c}_{1}-3t}}.\hfill \end{array}\end{array}$$
- For $b<\frac{27}{2}$,$$\begin{array}{c}\hfill u(r,t)=\frac{4(2b-27)}{{\left[\alpha \left(t\right)sin\left({d}_{2}lnr\right)+\beta \left(t\right)cos\left({d}_{2}lnr\right)\right]}^{2}{r}^{3}},\end{array}$$$$\begin{array}{c}\hfill \begin{array}{c}{\alpha}^{\prime}=-\frac{3}{8}{\alpha}^{3}+\frac{1}{4}{d}_{2}{\alpha}^{2}\beta -\frac{3}{8}\alpha {\beta}^{2}+\frac{1}{4}{d}_{2}{\beta}^{3}-\frac{1}{2}a\alpha ,\hfill \\ {\beta}^{\prime}=-\frac{3}{8}{\beta}^{3}-\frac{1}{4}{d}_{2}\alpha {\beta}^{2}-\frac{3}{8}{\alpha}^{2}\beta -\frac{1}{4}{d}_{2}{\alpha}^{3}-\frac{1}{2}a\beta .\hfill \end{array}\end{array}$$

**Example**

**2.**

- For $a\ne 0$,$$\begin{array}{c}\hfill u(r,t)={\left[-\frac{a}{n}\left(\alpha \left(t\right)lnr+\beta \left(t\right)\right)\right]}^{-\frac{n}{a}},\end{array}$$$$\begin{array}{c}{\alpha}^{\prime}=\frac{\left[a-(c+1)n\right]a}{{n}^{2}}\alpha +\frac{1}{n}{a}^{2}\beta +b,\hfill \\ {\beta}^{\prime}=\frac{\left[a-(c+1)n\right]a}{{n}^{2}}\beta +\frac{1}{n}{a}^{2}{\alpha}^{-1}{\beta}^{2}+d.\hfill \end{array}$$The solutions blow up along the curves $r=exp\left[-\beta \left(t\right)/\alpha \left(t\right)\right]$ when $a>0$ and extinguish along the curves when $a<0$.
- For $a=0$,$$\begin{array}{c}\hfill u(r,t)={r}^{\alpha \left(t\right)}\beta \left(t\right),\end{array}$$
**(i)**- For $b\ne 0$ and $b\ne n$,$$\begin{array}{c}\hfill \alpha \left(t\right)=bt+{c}_{1},\phantom{\rule{4pt}{0ex}}\beta \left(t\right)={c}_{2}{(bt+{c}_{1})}^{\frac{n}{b}}-\frac{c+d+1}{b-n}(bt+{c}_{1}).\end{array}$$
**(ii)**- For $b=0$,$$\begin{array}{c}\hfill \alpha \left(t\right)={c}_{1},\phantom{\rule{4pt}{0ex}}\beta \left(t\right)={c}_{2}exp\left(\frac{n}{{c}_{1}}t\right)-\frac{(c+d+1){c}_{1}}{n}.\end{array}$$
**(iii)**- For $b=n$,$$\begin{array}{c}\hfill \alpha \left(t\right)=nt+{c}_{1},\phantom{\rule{4pt}{0ex}}\beta \left(t\right)=\left[{c}_{2}+\frac{c+d+1}{n}ln(nt+{c}_{1})\right](nt+{c}_{1}).\end{array}$$

**Example**

**3.**

- For $k\ne 3$,$$\begin{array}{c}\hfill u(r,t)={\left[(3-k)\left(2\alpha \left(t\right)\sqrt{r}+\beta \left(t\right)\right)\right]}^{\frac{1}{3-k}},\end{array}$$$$\begin{array}{c}{\alpha}^{\prime}=\frac{1}{2}(k-3)(4nk+k-12n-4){\alpha}^{-1}\beta +(3-k)a\alpha +\frac{b}{2},\hfill \\ {\beta}^{\prime}=n{(k-3)}^{2}{\alpha}^{-2}{\beta}^{2}+(3-k)a\beta +c.\hfill \end{array}$$The solutions blow up along the curves $r={\left[-\beta \left(t\right)/\left(2\alpha \left(t\right)\right)\right]}^{2}$ when $k>3$ and extinguish along the curves when $k<3$.
- For $k=3$,$$\begin{array}{c}\hfill u(r,t)=exp[\alpha \left(t\right)\sqrt{r}+\beta \left(t\right)],\end{array}$$$$\begin{array}{c}\hfill {\alpha}^{\prime}=2{\alpha}^{-1}+b,\phantom{\rule{4pt}{0ex}}{\beta}^{\prime}=4n{\alpha}^{-2}+a+c.\end{array}$$

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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No. | m | n | $\mathit{D}\left(\mathit{u}\right)$ | $\mathit{Q}(\mathit{r},\mathit{u})$ | $\mathit{H}\left(\mathit{u}\right)$ | $\mathit{G}(\mathit{r},\mathit{u})$ | $\mathit{F}(\mathit{r},\mathit{u})$ |
---|---|---|---|---|---|---|---|

1 | 2 | 1 | $2(n+3)a$ | $-\frac{1}{2u}$ | $-\frac{1}{2r}$ | $\frac{ar}{u}$ | |

2 | 2 | 3 | $\frac{1}{{u}^{3}}$ | $\left(-\frac{2}{9}bc{r}^{3}+\frac{2}{3}a{r}^{\frac{3}{2}}+d\right){u}^{2}+bu-6c$ | $-\frac{2}{u}$ | $-\frac{1}{2r}$ | $cr{u}^{2}$ |

3 | 2 | 3 | $\frac{1}{{u}^{2}}$ | $au$ | $-\frac{3}{2u}$ | $-\frac{2}{r}$ | $\frac{(b-18)u}{{r}^{2}}$ |

4 | 2 | $\frac{1}{u}$ | $\left[a{r}^{\frac{3}{2}}-\frac{(n-6){(2n-3)}^{2}}{27{r}^{3}}+b\right]u$ | $-\frac{1}{u}$ | $-\frac{1}{2r}$ | $-\frac{(3-2n)u}{2{r}^{2}}$ | |

5 | 2 | 6 | $\frac{1}{u}$ | 0 | $-\frac{1}{u}$ | $\frac{1}{2r}$ | $-\frac{9u-2a}{2{r}^{2}}$ |

6 | $-1$ | ${u}^{k}$ | $au+(b+clnr){u}^{\frac{k}{2}}-\frac{k}{2}{u}^{k-1}$ | $-\frac{k}{2u}$ | $\frac{1}{r}$ | 0 | |

7 | $-1$ | ${u}^{2}$ | $(a+b{u}^{\frac{a}{n}})ulnr+(c+d{u}^{\frac{a}{n}})u$ | $-\frac{n+a}{nu}$ | $\frac{1}{r}$ | 0 | |

8 | $-1$ | 2 | ${u}^{2}$ | $\left[-2(h+1)lnr+a\right]u+(b+clnr){u}^{-h}$ | $\frac{h}{u}$ | $\frac{1}{r}$ | 0 |

9 | $-2$ | ${u}^{k}$ | $[b\sqrt{r}-2(k-3)(2kn+k-6n-4)r+c]{u}^{k-2}+au$ | $\frac{2-k}{u}$ | $\frac{1}{2r}$ | 0 | |

10 | $-2$ | ${u}^{3}$ | $\left[\frac{nr}{{(2n+1)}^{2}}+b\right]u+(b+c\sqrt{r}){u}^{\frac{4n+3}{2(2n+1)}}$ | $-\frac{4n+3}{2(2n+1)u}$ | $\frac{1}{2r}$ | 0 | |

11 | $-2$ | ${u}^{\frac{2(9n+5)}{3(2n+1)}}$ | $\left[b{r}^{\frac{1}{3}}+\frac{(3n+1)r}{3{(2n+1)}^{2}}+c\right]{u}^{\frac{2(3n+2)}{3(2n+1)}}+au$ | $-\frac{2(3n+2)}{3(2n+1)u}$ | $\frac{2}{3r}$ | 0 |

No. | $\mathit{D}\left(\mathit{u}\right)$ | $\mathit{Q}(\mathit{r},\mathit{u})$ | $\mathit{H}\left(\mathit{u}\right)$ | $\mathit{G}(\mathit{r},\mathit{u})$ |
---|---|---|---|---|

1 | ${u}^{k}$ | $au+\left(b+c{r}^{\frac{m+1}{m}}\right){u}^{\frac{1-k}{m}}$ | $\frac{k-1}{mu}$ | $-\frac{1}{mr}$ |

$k\ne -m+1$ | ||||

2 | ${u}^{-m+1}$ | $\left(alnu+b+c{r}^{\frac{m+1}{m}}\right)u$ | $-\frac{1}{u}$ | $-\frac{1}{mr}$ |

3 | ${u}^{-\frac{{(m-1)}^{2}(n+2)+5m-3}{mn+m-n+1}}$ | $\left(b+c{r}^{\frac{m+1}{m-1}}\right){u}^{\frac{mn+2m-n+2}{mn+m-n+1}}+au$ | $-\frac{mn+2m-n+2}{(mn+m-n+1)u}$ | $-\frac{2}{(m-1)r}$ |

4 | ${u}^{-\frac{{m}^{2}n-mn+2{m}^{2}+2m}{mn+m+1}}$ | $\left(b+c{r}^{\frac{m+1}{m}}\right){u}^{\frac{mn+2m+2}{mn+m+1}}+au$ | $-\frac{mn+2m+2}{(mn+m+1)u}$ | $-\frac{1}{mr}$ |

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## Share and Cite

**MDPI and ACS Style**

Ji, L.; Feng, W.
Second-Order Conditional Lie–Bäcklund Symmetries and Differential Constraints of Nonlinear Reaction–Diffusion Equations with Gradient-Dependent Diffusivity. *Symmetry* **2018**, *10*, 267.
https://doi.org/10.3390/sym10070267

**AMA Style**

Ji L, Feng W.
Second-Order Conditional Lie–Bäcklund Symmetries and Differential Constraints of Nonlinear Reaction–Diffusion Equations with Gradient-Dependent Diffusivity. *Symmetry*. 2018; 10(7):267.
https://doi.org/10.3390/sym10070267

**Chicago/Turabian Style**

Ji, Lina, and Wei Feng.
2018. "Second-Order Conditional Lie–Bäcklund Symmetries and Differential Constraints of Nonlinear Reaction–Diffusion Equations with Gradient-Dependent Diffusivity" *Symmetry* 10, no. 7: 267.
https://doi.org/10.3390/sym10070267