# Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

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## Abstract

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## 1. Introduction

## 2. Lie Symmetries

## 3. Conservation Laws for DD Equation in (1+1) Dimension

**Case 1**: ${Q}_{1}=x$. This case gives the following corresponding conserved density and flux:$$\begin{array}{ccc}\hfill {T}_{1}& =& b{u}_{xt}+x{u}_{t},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\displaystyle {X}_{1}}& =& {f}^{\prime}\left(u\right)x{u}_{x}-f\left(u\right)+(a{u}_{xxx}-b{u}_{ttx}-c{u}_{x})x-a{u}_{xx}+u.\hfill \end{array}$$**Case 2**: ${Q}_{2}=t$. For this case, we obtain conserved density and flux as$$\begin{array}{ccc}\hfill {T}_{2}& =& t{u}_{t}-u,\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\displaystyle {X}_{2}}& =& t({u}_{x}{f}^{\prime}\left(u\right)+a{u}_{xxx}-b{u}_{ttx}-c{u}_{x}).\hfill \end{array}$$**Case 3**: ${Q}_{3}=1$. We get conserved density and flux as$$\begin{array}{ccc}\hfill {T}_{3}& =& {u}_{t},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\displaystyle {X}_{3}}& =& {u}_{x}(-c+{f}^{\prime}\left(u\right))+a{u}_{xxx}-b{u}_{ttx}.\hfill \end{array}$$**Case 4**: ${Q}_{4}=tx$. For this last case, the corresponding conserved density and flux are$$\begin{array}{ccc}\hfill {T}_{4}& =& b(t{u}_{xt}-{u}_{x})+x(t{u}_{t}-u),\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\displaystyle {X}_{4}}& =& t({f}^{\prime}\left(u\right)x{u}_{x}-f\left(u\right))+(a{u}_{xxx}-b{u}_{ttx}-{u}_{x})x-a{u}_{xx}+cu).\hfill \end{array}$$

## 4. Conservation Laws for DD Equation in (2+1) Dimension

**Case 1**: For ${Q}_{1}=1$,$$\begin{array}{c}{T}_{1}={u}_{t},\hfill \\ {X}_{1}=b{u}_{xxx}-a{u}_{ttx}+a{u}_{tt}+({f}^{\prime}\left(u\right)-1){u}_{x},\hfill \\ {Y}_{1}=b{u}_{yyy}.\hfill \end{array}$$**Case 2**: For ${Q}_{2}=x$,$$\begin{array}{c}{T}_{2}=x{u}_{t},\hfill \\ {X}_{2}=xb{u}_{xxx}-b{u}_{xx}-xa{u}_{ttx}+a{u}_{tt}+(x{f}^{\prime}\left(u\right)-x){u}_{x}+u-f\left(u\right),\hfill \\ {Y}_{2}=xb{u}_{yyy}-xa{u}_{tty}+(x({f}^{\prime}\left(u\right)-x){u}_{y}.\hfill \end{array}$$**Case 3**: For ${Q}_{3}=y$,$$\begin{array}{c}{T}_{3}=y{u}_{t},\hfill \\ {X}_{3}=yb{u}_{xxx}-ya{u}_{ttx}+(y{f}^{\prime}\left(u\right)-y){u}_{x},\hfill \\ {Y}_{3}=yb{u}_{yyy}-b{u}_{yy}-ya{u}_{tty}+a{u}_{tt}+(y{f}^{\prime}\left(u\right)-y){u}_{y}+u-f\left(u\right).\hfill \end{array}$$**Case 4**: For ${Q}_{4}=t$,$$\begin{array}{c}{T}_{4}=t{u}_{t}-u,\hfill \\ {X}_{4}=tb{u}_{xxx}-at{u}_{ttx}+(t{f}^{\prime}\left(u\right)-t){u}_{x},\hfill \\ {Y}_{4}=tb{u}_{yyy}-at{u}_{tty}+(t{f}^{\prime}\left(u\right)-t){u}_{y}.\hfill \end{array}$$**Case 5**: For ${Q}_{5}=xy$,$$\begin{array}{c}{T}_{5}=xy{u}_{t},\hfill \\ {X}_{5}=xyb{u}_{xxx}-yb{u}_{xx}-xya{u}_{ttx}+ya{u}_{tt}+(xy({f}^{\prime}\left(u\right)-xy){u}_{x}+yu-yf\left(u\right),\hfill \\ {Y}_{5}=xyb{u}_{yyy}-xya{u}_{tty}-xb{u}_{yy}+xa{u}_{tt}+(xy{f}^{\prime}\left(u\right)-xy){u}_{y}+xu-xf\left(u\right).\hfill \end{array}$$**Case 6**: For ${Q}_{6}=tx$,$$\begin{array}{c}{T}_{6}=tx{u}_{t}-xu,\hfill \\ {X}_{6}=txb{u}_{xxx}-tb{u}_{xx}-txa{u}_{ttx}+ta{u}_{tt}+(tx({f}^{\prime}\left(u\right)-tx){u}_{x}+tu-tf\left(u\right),\hfill \\ {Y}_{6}=txb{u}_{yyy}-txa{u}_{tty}+(tx({f}^{\prime}\left(u\right)-tx){u}_{y}.\hfill \end{array}$$**Case 7**: For ${Q}_{7}=ty$,$$\begin{array}{c}{T}_{7}=ty{u}_{t}-yu,\hfill \\ {X}_{7}=tyb{u}_{xxx}-tya{u}_{ttx}+(ty({f}^{\prime}\left(u\right)-ty){u}_{x},\hfill \\ {Y}_{7}=txb{u}_{yyy}-tya{u}_{tty}+tb{u}_{yy}+ta{u}_{tt}+(ty({f}^{\prime}\left(u\right)-ty){u}_{y}+tu-tf\left(u\right).\hfill \end{array}$$**Case 8**: For ${Q}_{8}=txy$,$$\begin{array}{c}{T}_{8}=txy{u}_{t}-xyu,\hfill \\ {X}_{8}=txyb{u}_{xxx}-tyb{u}_{xx}-txya{u}_{ttx}+tya{u}_{tt}+(txy({f}^{\prime}\left(u\right)-txy){u}_{x}+ytu-ytf\left(u\right),\hfill \\ Y8=txyb{u}_{yyy}-txya{u}_{tty}-txb{u}_{yy}+txa{u}_{tt}+(txy({f}^{\prime}\left(u\right)-txy){u}_{y}+txu-txf\left(u\right).\hfill \end{array}$$

## 5. Travelling Waves for Equations (2) and (3)

#### 5.1. The (1+1)-DD Travelling Waves

#### 5.2. The (2+1)-DD Line Travelling Waves

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Plot of solution for Equation (46).

i | a | b | f(u) | ${\mathbf{v}}_{\mathit{k}}$ |
---|---|---|---|---|

1 | $\forall $ | $\forall $ | $\forall $ | ${\mathbf{v}}_{1}={\mathit{\partial}}_{\mathit{x}},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{2}={\partial}_{y},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{3}={\partial}_{t}$ |

2 | $\forall $ | 0 | $\forall $ | ${\mathbf{v}}_{1},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{2},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{3},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{4}=y{\partial}_{x}-x{\partial}_{y}$ |

3 | 0 | $\forall $ | ${k}_{1}{({k}_{2}-u)}^{n}+u-{k}_{3}$ | ${\mathbf{v}}_{1},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{2},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{3},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{5}=x{\partial}_{x}+y{\partial}_{y}+2t{\partial}_{t}+{\displaystyle \frac{2{k}_{2}-2u}{n-1}}{\partial}_{u}$ |

$(\mathrm{with}\phantom{\rule{0.166667em}{0ex}}n\ne 1,\phantom{\rule{0.166667em}{0ex}}{k}_{1}\ne 0)$ | ||||

4 | 0 | $\forall $ | ${k}_{1}{e}^{nu}+u+{k}_{2}$ | ${\mathbf{v}}_{1},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{2},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{3},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{6}=x{\partial}_{x}+y{\partial}_{y}+2t{\partial}_{t}-{\displaystyle \frac{2}{n}}{\partial}_{u}$ |

$(\mathrm{with}\phantom{\rule{0.166667em}{0ex}}n\ne 0)$ | ||||

5 | 0 | $\forall $ | ${k}_{1}ln({k}_{2}-u)+u-{k}_{3}$ | ${\mathbf{v}}_{1},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{2},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{3},\phantom{\rule{0.277778em}{0ex}}{\mathbf{v}}_{7}=x{\partial}_{x}+y{\partial}_{y}+2t{\partial}_{t}-(2{k}_{2}-2u){\partial}_{u}$ |

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**MDPI and ACS Style**

Gandarias, M.L.; Durán, M.R.; Khalique, C.M.
Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions. *Symmetry* **2020**, *12*, 950.
https://doi.org/10.3390/sym12060950

**AMA Style**

Gandarias ML, Durán MR, Khalique CM.
Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions. *Symmetry*. 2020; 12(6):950.
https://doi.org/10.3390/sym12060950

**Chicago/Turabian Style**

Gandarias, María Luz, María Rosa Durán, and Chaudry Masood Khalique.
2020. "Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions" *Symmetry* 12, no. 6: 950.
https://doi.org/10.3390/sym12060950