# Highly Dispersive Optical Solitons in Absence of Self-Phase Modulation by Laplace-Adomian Decomposition

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## Abstract

**:**

## 1. Introduction

#### Description of the Governing Model

## 2. Highly Dispersive Soliton Solutions for the Governing Model (1.1)

## 3. Application of Laplace Transform Combined with Adomian Decomposition Method

## 4. Test Examples

#### 4.1. Dark Highly Dispersive Optical Soliton

#### 4.2. Bright Highly Dispersive Optical Soliton

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Case 1: graphic representation: (

**a**) numerically generated dark soliton, (

**b**) 2D density plot.

**Figure 2.**Case 2: graphic representation: (

**a**) numerically generated dark soliton, (

**b**) 2D density plot.

**Figure 3.**Case 3: graphic representation: (

**a**) numerically generated dark soliton, (

**b**) 2D density plot.

**Figure 4.**Case 4: graphic representation: (

**a**) numerically generated bright soliton, (

**b**) 2D density plot.

**Figure 5.**Case 5: graphic representation: (

**a**) numerically generated bright soliton, (

**b**) 2D density plot.

**Figure 6.**Case 6: graphic representation: (

**a**) numerically generated bright soliton, (

**b**) 2D density plot.

**Table 1.**Equation (1) coefficients for dark highly dispersive optical solitons.

Cases | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{a}}_{5}$ | ${\mathit{a}}_{6}$ | $\mathit{\lambda}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\kappa}$ | $\mathit{\nu}$ | N |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | $1.20$ | $0.45$ | $3.05$ | $1.06$ | $2.22$ | $1.06$ | $0.05$ | $0.62$ | $0.06$ | $0.34$ | $0.09$ | 16 |

2 | $0.50$ | $0.64$ | $0.33$ | $0.02$ | $2.10$ | $2.03$ | $0.02$ | $0.01$ | $0.89$ | $0.17$ | $-0.81$ | 16 |

3 | $1.55$ | $1.04$ | $0.92$ | $2.22$ | $3.40$ | $-2.15$ | $4.02$ | $0.01$ | $0.90$ | $-0.26$ | $2.15$ | 16 |

x | $-3.0$ | $-2.0$ | $-1.0$ | $1.0$ | $2.0$ | $3.0$ | |
---|---|---|---|---|---|---|---|

t | |||||||

$0.1$ | $5.21\times {10}^{-9}$ | $3.45\times {10}^{-9}$ | $2.09\times {10}^{-9}$ | $2.16\times {10}^{-9}$ | $3.64\times {10}^{-9}$ | $5.02\times {10}^{-9}$ | |

$0.3$ | $6.01\times {10}^{-8}$ | $4.32\times {10}^{-8}$ | $3.34\times {10}^{-8}$ | $3.32\times {10}^{-8}$ | $4.14\times {10}^{-8}$ | $6.88\times {10}^{-8}$ | |

$0.5$ | $4.02\times {10}^{-7}$ | $3.62\times {10}^{-7}$ | $1.92\times {10}^{-7}$ | $2.02\times {10}^{-7}$ | $3.22\times {10}^{-7}$ | $5.10\times {10}^{-7}$ |

x | $-3.0$ | $-2.0$ | $-1.0$ | $1.0$ | $2.0$ | $3.0$ | |
---|---|---|---|---|---|---|---|

t | |||||||

$0.1$ | $9.01\times {10}^{-9}$ | $7.12\times {10}^{-9}$ | $6.33\times {10}^{-10}$ | $1.02\times {10}^{-9}$ | $6.09\times {10}^{-9}$ | $8.62\times {10}^{-9}$ | |

$0.3$ | $7.13\times {10}^{-8}$ | $3.02\times {10}^{-8}$ | $1.11\times {10}^{-8}$ | $2.01\times {10}^{-8}$ | $3.98\times {10}^{-8}$ | $7.08\times {10}^{-8}$ | |

$0.5$ | $2.62\times {10}^{-7}$ | $2.88\times {10}^{-7}$ | $1.52\times {10}^{-7}$ | $1.28\times {10}^{-7}$ | $3.01\times {10}^{-7}$ | $2.87\times {10}^{-7}$ |

x | $-3.0$ | $-2.0$ | $-1.0$ | $1.0$ | $2.0$ | $3.0$ | |
---|---|---|---|---|---|---|---|

t | |||||||

$0.1$ | $6.71\times {10}^{-9}$ | $4.45\times {10}^{-9}$ | $9.04\times {10}^{-10}$ | $9.32\times {10}^{-10}$ | $5.16\times {10}^{-9}$ | $7.23\times {10}^{-9}$ | |

$0.3$ | $5.59\times {10}^{-8}$ | $5.01\times {10}^{-8}$ | $2.71\times {10}^{-8}$ | $3.09\times {10}^{-8}$ | $4.25\times {10}^{-8}$ | $8.00\times {10}^{-8}$ | |

$0.5$ | $1.36\times {10}^{-7}$ | $1.08\times {10}^{-7}$ | $1.01\times {10}^{-7}$ | $1.33\times {10}^{-7}$ | $2.21\times {10}^{-7}$ | $2.07\times {10}^{-7}$ |

**Table 5.**Equation (1) coefficients for bright highly dispersive optical solitons.

Cases | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{a}}_{5}$ | ${\mathit{a}}_{6}$ | $\mathit{\lambda}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\kappa}$ | $\mathit{\nu}$ | N |
---|---|---|---|---|---|---|---|---|---|---|---|---|

4 | $0.55$ | $0.15$ | $2.22$ | $3.76$ | $1.05$ | $3.00$ | $0.01$ | $0.90$ | $1.22$ | $0.05$ | $0.53$ | 16 |

5 | $0.20$ | $0.94$ | $2.09$ | $1.98$ | $4.20$ | $-1.15$ | $3.02$ | $1.04$ | $2.06$ | $-0.60$ | $17.87$ | 16 |

6 | $-2.55$ | $2.00$ | $1.76$ | $2.87$ | $1.22$ | $1.45$ | $0.02$ | $2.36$ | $0.64$ | $0.14$ | $3.18$ | 16 |

x | $-3.0$ | $-2.0$ | $-1.0$ | $1.0$ | $2.0$ | $3.0$ | |
---|---|---|---|---|---|---|---|

t | |||||||

$0.1$ | $6.43\times {10}^{-9}$ | $4.15\times {10}^{-9}$ | $2.58\times {10}^{-9}$ | $2.09\times {10}^{-9}$ | $3.99\times {10}^{-9}$ | $5.89\times {10}^{-9}$ | |

$0.3$ | $8.11\times {10}^{-8}$ | $5.76\times {10}^{-8}$ | $2.88\times {10}^{-8}$ | $2.62\times {10}^{-8}$ | $4.88\times {10}^{-8}$ | $7.79\times {10}^{-8}$ | |

$0.5$ | $6.22\times {10}^{-7}$ | $5.02\times {10}^{-7}$ | $3.93\times {10}^{-7}$ | $3.34\times {10}^{-7}$ | $4.85\times {10}^{-7}$ | $6.94\times {10}^{-7}$ |

x | $-3.0$ | $-2.0$ | $-1.0$ | $1.0$ | $2.0$ | $3.0$ | |
---|---|---|---|---|---|---|---|

t | |||||||

$0.1$ | $8.89\times {10}^{-9}$ | $6.27\times {10}^{-9}$ | $3.98\times {10}^{-9}$ | $3.01\times {10}^{-9}$ | $6.08\times {10}^{-9}$ | $8.09\times {10}^{-9}$ | |

$0.3$ | $9.53\times {10}^{-8}$ | $4.29\times {10}^{-8}$ | $4.88\times {10}^{-9}$ | $1.07\times {10}^{-8}$ | $4.13\times {10}^{-8}$ | $8.44\times {10}^{-8}$ | |

$0.5$ | $9.71\times {10}^{-7}$ | $5.42\times {10}^{-7}$ | $1.93\times {10}^{-7}$ | $6.34\times {10}^{-8}$ | $3.65\times {10}^{-7}$ | $8.04\times {10}^{-7}$ |

x | $-3.0$ | $-2.0$ | $-1.0$ | $1.0$ | $2.0$ | $3.0$ | |
---|---|---|---|---|---|---|---|

t | |||||||

$0.1$ | $7.97\times {10}^{-9}$ | $6.01\times {10}^{-9}$ | $4.03\times {10}^{-9}$ | $3.98\times {10}^{-9}$ | $6.24\times {10}^{-9}$ | $8.00\times {10}^{-9}$ | |

$0.3$ | $8.37\times {10}^{-8}$ | $6.19\times {10}^{-8}$ | $5.18\times {10}^{-8}$ | $5.99\times {10}^{-8}$ | $6.43\times {10}^{-8}$ | $8.40\times {10}^{-8}$ | |

$0.5$ | $9.11\times {10}^{-7}$ | $7.32\times {10}^{-7}$ | $3.58\times {10}^{-7}$ | $3.81\times {10}^{-7}$ | $6.95\times {10}^{-7}$ | $8.89\times {10}^{-7}$ |

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**MDPI and ACS Style**

González-Gaxiola, O.; Biswas, A.; Moraru, L.; Moldovanu, S.
Highly Dispersive Optical Solitons in Absence of Self-Phase Modulation by Laplace-Adomian Decomposition. *Photonics* **2023**, *10*, 114.
https://doi.org/10.3390/photonics10020114

**AMA Style**

González-Gaxiola O, Biswas A, Moraru L, Moldovanu S.
Highly Dispersive Optical Solitons in Absence of Self-Phase Modulation by Laplace-Adomian Decomposition. *Photonics*. 2023; 10(2):114.
https://doi.org/10.3390/photonics10020114

**Chicago/Turabian Style**

González-Gaxiola, Oswaldo, Anjan Biswas, Luminita Moraru, and Simona Moldovanu.
2023. "Highly Dispersive Optical Solitons in Absence of Self-Phase Modulation by Laplace-Adomian Decomposition" *Photonics* 10, no. 2: 114.
https://doi.org/10.3390/photonics10020114