# Solitary and Periodic Wave Solutions of Fractional Zoomeron Equation

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

- (i)
- $\phantom{\rule{1.em}{0ex}}\mathrm{If}\phantom{\rule{1.em}{0ex}}\mu <0,\phantom{\rule{1.em}{0ex}}\mathrm{then}\phantom{\rule{1.em}{0ex}}\varphi \left(\psi \right)=-\sqrt{-\mu}tanh\left(\sqrt{-\mu}\psi \right),\phantom{\rule{1.em}{0ex}}\mathrm{or}\phantom{\rule{1.em}{0ex}}\varphi \left(\psi \right)=-\sqrt{-\mu}coth(\sqrt{-\mu}\psi $).
- (ii)
- $\phantom{\rule{1.em}{0ex}}\mathrm{If}\phantom{\rule{1.em}{0ex}}\mu >0,\phantom{\rule{1.em}{0ex}}\mathrm{then}\phantom{\rule{1.em}{0ex}}\varphi \left(\psi \right)=\sqrt{\mu}tan\left(\sqrt{\mu}\psi \right),\phantom{\rule{1.em}{0ex}}\mathrm{or}\phantom{\rule{1.em}{0ex}}\varphi \left(\psi \right)=-\sqrt{\mu}cot(\sqrt{\mu}\psi $).
- (iii)
- $\phantom{\rule{1.em}{0ex}}\mathrm{If}\phantom{\rule{1.em}{0ex}}\mu =0,\phantom{\rule{1.em}{0ex}}\mathrm{then}\phantom{\rule{1.em}{0ex}}\varphi \left(\psi \right)=\frac{-1}{\psi}$.

## 3. Execution of the Problem

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**In these graphical representations, variations are shown for the real and imaginary parts of the solution ${F}_{2}(x,y,t)$. (

**a**) A $2D$ plot representing the real part of ${F}_{2}(x,y,t)$ is presented. (

**b**) A $2D$ plot representing the img part of ${F}_{2}(x,y,t)$ is depicted.

**Figure 2.**In these graphical representations, variations are shown for the real and imaginary parts of the solution ${F}_{6}(x,t)$. (

**a**) A $2D$ plot depicting the real part of ${F}_{6}(x,y,t)$ is presented. (

**b**) A $2D$ plot representing the Img part of ${F}_{6}(x,y,t)$ is depicted.

**Figure 3.**In these graphical representations, variations are shown for the real and imaginary parts of the solution ${F}_{11}(x,y,t)$. (

**a**) A $2D$ plot representing the real part of ${F}_{11}(x,y,t)$ is presented. (

**b**) A $2D$ plot representing the img part of ${F}_{11}(x,y,t)$ is depicted.

**Figure 4.**In these graphical representations, variations are shown for the real and imaginary parts of the solution ${F}_{9}(x,y,t)$. (

**a**) A $2D$ plot representing the real part of ${F}_{9}(x,y,t)$ is presented. (

**b**) A $2D$ plot representing the real part of ${F}_{9}(x,y,t)$ is depicted for different values of $\alpha $. (

**c**) A $2D$ plot representing the img part of ${F}_{9}(x,t)$ is presented. (

**d**) A $2D$ plot representing the img part of ${F}_{9}(x,y,t)$ is depicted for different values of $\alpha $.

**Figure 5.**A $2D$ plot representing solution ${F}_{5}(x,y,t)$ is depicted for different values of $\alpha $.

**Figure 6.**A $2D$ plot representing solution ${F}_{15}(x,y,t)$ is depicted for different values of $\alpha $.

**Figure 7.**In these graphical representations, $\beta $ and $\gamma $ variations are shown for the solution ${F}_{1}(x,y,t)$. (

**a**) A $2D$ plot representing variation of ${F}_{1}(x,y,t)$ with respect to $\beta $. (

**b**) A $2D$ plot representing variation of ${F}_{4}(x,y,t)$ with respect to $\gamma $.

**Figure 8.**In these graphical representations, $\beta $ and $\gamma $ variations are shown for the solution ${F}_{4}(x,y,t)$. (

**a**) A $2D$ plot representing variation of ${F}_{4}(x,y,t)$ with respect to $\beta $. (

**b**) A $2D$ plot representing variation of ${F}_{4}(x,y,t)$ with respect to $\gamma $.

**Figure 9.**In these graphical representations, $\beta $ and $\gamma $ variations are shown for the solution ${F}_{14}(x,y,t)$. (

**a**) A $2D$ plot representing variation of ${F}_{14}(x,y,t)$ with respect to $\beta $. (

**b**) A $2D$ plot representing variation of ${F}_{14}(x,y,t)$ with respect to $\gamma $.

Case I: $\mathit{\tau}<0$ Present method |

$u={c}_{1}\left(-1/2\phantom{\rule{0.166667em}{0ex}}\frac{rb}{q{{c}_{1}}^{2}}-1/2\phantom{\rule{0.166667em}{0ex}}a\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{r}{q{{c}_{1}}^{2}}}tanh\left(1/2\phantom{\rule{0.166667em}{0ex}}\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{r}{q{{c}_{1}}^{2}}}\left(-\frac{{x}^{\beta}}{\beta}+\frac{q{{c}_{1}}^{2}{y}^{\gamma}}{\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)\right)\right)$ |

${\left(a-1/2\phantom{\rule{0.166667em}{0ex}}b\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{r}{q{{c}_{1}}^{2}}}tanh\left(1/2\phantom{\rule{0.166667em}{0ex}}\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{r}{q{{c}_{1}}^{2}}}\left(-\frac{{x}^{\beta}}{\beta}+\frac{q{{c}_{1}}^{2}{y}^{\gamma}}{\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)\right)\right)}^{-1}.$ |

Case I: ${\lambda}^{2}-4\mu <0$ ${G}^{\prime}/G$-expansion method |

$u=\frac{6/}{\beta}\left[\frac{-\lambda}{2}+{\vartheta}_{2}\left(\frac{-{c}_{1}sin\left({\vartheta}_{2}\zeta \right)+{c}_{2}cos\left({\vartheta}_{2}\zeta \right)}{{c}_{1}cos\left({\vartheta}_{2}\zeta \right)+{c}_{2}sin\left({\vartheta}_{2}\zeta \right)}\right)\right]$ |

Case I: Exp-function method |

${u}_{1}\left(\xi \right)=\frac{4{A}_{0}\left({\omega}^{2}-1\right)}{\left(4{\omega}^{2}-4-\omega {A}_{0}^{2}\right)cosh\left(\xi \right)+\left(4{\omega}^{2}-4+\omega {A}_{0}^{2}\right)sinh\left(\xi \right)}$ |

Case II: $\tau >0$ Present method |

$u={c}_{1}\left(-1/2\phantom{\rule{0.166667em}{0ex}}\frac{rb}{q{{c}_{1}}^{2}}+1/2\phantom{\rule{0.166667em}{0ex}}a\sqrt{2}\sqrt{\frac{r}{q{{c}_{1}}^{2}}}tan\left(1/2\phantom{\rule{0.166667em}{0ex}}\sqrt{2}\sqrt{\frac{r}{q{{c}_{1}}^{2}}}\left(-\frac{{x}^{\beta}}{\beta}+\frac{q{{c}_{1}}^{2}{y}^{\gamma}}{\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)\right)\right)$ |

${\left(a+1/2\phantom{\rule{0.166667em}{0ex}}b\sqrt{2}\sqrt{\frac{r}{q{{c}_{1}}^{2}}}tan\left(1/2\phantom{\rule{0.166667em}{0ex}}\sqrt{2}\sqrt{\frac{r}{q{{c}_{1}}^{2}}}\left(-\frac{{x}^{\beta}}{\beta}+\frac{q{{c}_{1}}^{2}{y}^{\gamma}}{\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)\right)\right)}^{-1}.$ |

$u(t,x,y)=-\frac{\sqrt{-m\left({l}^{2}-{k}^{2}\right)\left({B}^{2}+4D\psi \right)}}{2A\sqrt{l}}tan\left(\frac{\sqrt{-\left({B}^{2}+4D\psi \right)}}{2A}\left(kx+my+\frac{l{t}^{\alpha}}{\alpha}\right)\right).$ |

Case II: Exp-function method |

${u}_{2,3}\left(\xi \right)=\pm \frac{1}{2}\sqrt{\frac{{\omega}^{2}-1}{\omega}}\frac{1+tanh\left(\xi \right)-{B}_{0}sech\left(\xi \right)}{1+tanh\left(\xi \right)+{B}_{0}sech\left(\xi \right)}$ |

where $\xi =x+y-\omega t$ |

Case III: $\tau =0$ Present method |

$u={c}_{1}\left(-1/2\phantom{\rule{0.166667em}{0ex}}\frac{rb}{q{{c}_{1}}^{2}}-a{\left(-\frac{{x}^{\beta}}{\beta}+\frac{q{{c}_{1}}^{2}{y}^{\gamma}}{\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)}^{-1}\right){\left(a-b{\left(-\frac{{x}^{\beta}}{\beta}+\frac{q{{c}_{1}}^{2}{y}^{\gamma}}{\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)}^{-1}\right)}^{-1}.$ |

Case III: $B\ne 0,\psi =A-C$, and $\mu ={B}^{2}+4\psi D=0$ ${G}^{\prime}/G$-expansion method |

$u=\mp \frac{m\psi \left({l}^{2}-{k}^{2}\right)(B+2d\psi )}{2l\left|A\psi \right|}\sqrt{\frac{l}{m\left({l}^{2}-{k}^{2}\right)}}$ |

$\pm \sqrt{\frac{m\left({l}^{2}-{k}^{2}\right)}{l}}\frac{\left|\psi \right|}{\left|A\right|}\left[d+\frac{B}{2\psi}+\frac{{C}_{2}}{{C}_{1}+{C}_{2}\left(kx+my+\frac{l{t}^{\alpha}}{\alpha}\right)}\right]$ |

Case I: $\tau <0$ Present method |

$u={c}_{-1}\left(a-b\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{q{{c}_{-1}}^{2}}{r}}tanh\left(\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{q{{c}_{-1}}^{2}}{r}}\left(-\frac{{x}^{\beta}}{\beta}+1/4\phantom{\rule{0.166667em}{0ex}}\frac{{r}^{2}{y}^{\gamma}}{q{{c}_{-1}}^{2}\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)\right)\right)$ |

${\left(-2\phantom{\rule{0.166667em}{0ex}}\frac{q{{c}_{-1}}^{2}b}{r}-a\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{q{{c}_{-1}}^{2}}{r}}tanh\left(\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{q{{c}_{-1}}^{2}}{r}}\left(-\frac{{x}^{\beta}}{\beta}+1/4\phantom{\rule{0.166667em}{0ex}}\frac{{r}^{2}{y}^{\gamma}}{q{{c}_{-1}}^{2}\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)\right)\right)}^{-1}$ |

${\left(a-1/2\phantom{\rule{0.166667em}{0ex}}b\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{r}{q{{c}_{1}}^{2}}}tanh\left(1/2\phantom{\rule{0.166667em}{0ex}}\sqrt{-2\phantom{\rule{0.166667em}{0ex}}\frac{r}{q{{c}_{1}}^{2}}}\left(-\frac{{x}^{\beta}}{\beta}+\frac{q{{c}_{1}}^{2}{y}^{\gamma}}{\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)\right)\right)}^{-1}.$ |

Case I: $R<0$ MSE method |

$u=\pm \sqrt{\frac{R}{2\omega}}\times cot\left(\sqrt{\frac{R}{2\left({\omega}^{2}-1\right)}}(x+y-\omega t)\right)$ |

Case II: $\tau >0$ Present method |

$u={c}_{-1}\left(a+b\sqrt{2}\sqrt{\frac{q{{c}_{-1}}^{2}}{r}}tan\left(\sqrt{2}\sqrt{\frac{q{{c}_{-1}}^{2}}{r}}\left(-\frac{{x}^{\beta}}{\beta}+1/4\phantom{\rule{0.166667em}{0ex}}\frac{{r}^{2}{y}^{\gamma}}{q{{c}_{-1}}^{2}\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)\right)\right)$ |

${\left(-2\phantom{\rule{0.166667em}{0ex}}\frac{q{{c}_{-1}}^{2}b}{r}+a\sqrt{2}\sqrt{\frac{q{{c}_{-1}}^{2}}{r}}tan\left(\sqrt{2}\sqrt{\frac{q{{c}_{-1}}^{2}}{r}}\left(-\frac{{x}^{\beta}}{\beta}+1/4\phantom{\rule{0.166667em}{0ex}}\frac{{r}^{2}{y}^{\gamma}}{q{{c}_{-1}}^{2}\left(q-1\right)\left(q+1\right)\gamma}-\frac{q{t}^{\alpha}}{\alpha}\right)\right)\right)}^{-1}$ |

Case II: $R>0$ $MSE\phantom{\rule{3.33333pt}{0ex}}method$ |

$u=\pm \sqrt{\frac{R}{2\omega}}\times tanh\left(\sqrt{\frac{R}{2\left({\omega}^{2}-1\right)}}(x+y-\omega t)\right)$. |

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**MDPI and ACS Style**

Alshammari, M.; Moaddy, K.; Naeem, M.; Alsheekhhussain, Z.; Alshammari, S.; Al-Sawalha, M.M.
Solitary and Periodic Wave Solutions of Fractional Zoomeron Equation. *Fractal Fract.* **2024**, *8*, 222.
https://doi.org/10.3390/fractalfract8040222

**AMA Style**

Alshammari M, Moaddy K, Naeem M, Alsheekhhussain Z, Alshammari S, Al-Sawalha MM.
Solitary and Periodic Wave Solutions of Fractional Zoomeron Equation. *Fractal and Fractional*. 2024; 8(4):222.
https://doi.org/10.3390/fractalfract8040222

**Chicago/Turabian Style**

Alshammari, Mohammad, Khaled Moaddy, Muhammad Naeem, Zainab Alsheekhhussain, Saleh Alshammari, and M. Mossa Al-Sawalha.
2024. "Solitary and Periodic Wave Solutions of Fractional Zoomeron Equation" *Fractal and Fractional* 8, no. 4: 222.
https://doi.org/10.3390/fractalfract8040222