# Highly Dispersive Optical Solitons in Birefringent Fibers with Polynomial Law of Nonlinear Refractive Index by Laplace–Adomian Decomposition

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equation

#### Bright and Dark Solitons

## 3. Description and Application of the LADM

#### Convergence of the Proposed Method

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Theorem**

**2.**

## 4. Graphical Representations

#### 4.1. Dark Soliton Simulation

**Case A:**Let us consider the following:

**Case B**: Let us consider the following:

#### 4.2. Bright Soliton Simulation

**Case C:**Let us consider the following:

**Case D**: Let us consider the following:

## 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.**(

**Above**) Three-dimensional illustrations of the numerical simulation and exact solution, and two-dimensional illustration of the approximation of $|u{|}^{2}$; (

**Below**) three-dimensional illustrations of the numerical simulation and exact solution, and two-dimensional illustration of the approximation of $|v{|}^{2}$ for Case A.

**Figure 2.**(

**Above**) Three-dimensional illustrations of the numerical simulation and exact solution, and two-dimensional illustration of the approximation of $|u{|}^{2}$; (

**Below**) three-dimensional illustrations of the numerical simulation and exact solution, and two-dimensional illustration of the approximation of $|v{|}^{2}$ for Case B.

**Figure 3.**(

**Above**) Three-dimensional illustrations of the numerical simulation and exact solution, and two-dimensional illustration of the approximation of $|u{|}^{2}$; (

**Below**) three-dimensional illustrations of the numerical simulation and exact solution, and two-dimensional illustration of the approximation of $|v{|}^{2}$ for Case C.

**Figure 4.**(

**Above**) Three-dimensional illustrations of the numerical simulation and exact solution, and two-dimensional illustration of the approximation of $|u{|}^{2}$; (

**Below**) three-dimensional illustrations of the numerical simulation and exact solution, and two-dimensional illustration of the approximation of $|v{|}^{2}$ for Case D.

**Table 1.**Case A: Absolute error for different values of $\left(x,t\right)$ considering $N=15$ steps.

$\left(\mathit{t},\mathit{x}\right)$ | $-2.0$ | $-1.0$ | $0$ | $1.0$ | $2.0$ |
---|---|---|---|---|---|

$0.1$ | $4.7\times {10}^{-8}$ | $3.5\times {10}^{-8}$ | $2.3\times {10}^{-9}$ | $3.9\times {10}^{-8}$ | $5.2\times {10}^{-8}$ |

$0.3$ | $5.0\times {10}^{-7}$ | $4.6\times {10}^{-7}$ | $3.7\times {10}^{-8}$ | $4.9\times {10}^{-7}$ | $6.1\times {10}^{-7}$ |

$0.5$ | $5.2\times {10}^{-7}$ | $5.6\times {10}^{-7}$ | $4.9\times {10}^{-7}$ | $5.8\times {10}^{-7}$ | $7.0\times {10}^{-6}$ |

$0.8$ | $6.1\times {10}^{-5}$ | $4.8\times {10}^{-5}$ | $5.5\times {10}^{-7}$ | $4.3\times {10}^{-5}$ | $6.9\times {10}^{-5}$ |

**Table 2.**Case B: Absolute error for different values of $\left(x,t\right)$ considering $N=15$ steps.

$\left(\mathit{t},\mathit{x}\right)$ | $-2.0$ | $-1.0$ | $0$ | $1.0$ | $2.0$ |
---|---|---|---|---|---|

$0.1$ | $3.2\times {10}^{-8}$ | $3.0\times {10}^{-8}$ | $2.1\times {10}^{-9}$ | $3.3\times {10}^{-8}$ | $3.8\times {10}^{-8}$ |

$0.3$ | $6.1\times {10}^{-7}$ | $5.1\times {10}^{-7}$ | $3.4\times {10}^{-8}$ | $5.6\times {10}^{-7}$ | $6.7\times {10}^{-7}$ |

$0.5$ | $6.8\times {10}^{-7}$ | $6.0\times {10}^{-7}$ | $2.9\times {10}^{-7}$ | $6.2\times {10}^{-7}$ | $6.9\times {10}^{-6}$ |

$0.8$ | $7.2\times {10}^{-5}$ | $6.4\times {10}^{-5}$ | $3.5\times {10}^{-7}$ | $6.6\times {10}^{-5}$ | $8.0\times {10}^{-5}$ |

**Table 3.**Case C: Absolute error for different values of $\left(x,t\right)$ considering $N=15$ steps.

$\left(\mathit{t},\mathit{x}\right)$ | $-2.0$ | $-1.0$ | $0$ | $1.0$ | $2.0$ |
---|---|---|---|---|---|

$0.1$ | $4.5\times {10}^{-8}$ | $3.7\times {10}^{-8}$ | $1.8\times {10}^{-9}$ | $3.2\times {10}^{-8}$ | $4.9\times {10}^{-8}$ |

$0.3$ | $4.4\times {10}^{-7}$ | $4.7\times {10}^{-7}$ | $2.3\times {10}^{-9}$ | $4.6\times {10}^{-7}$ | $4.0\times {10}^{-7}$ |

$0.5$ | $8.8\times {10}^{-7}$ | $5.7\times {10}^{-7}$ | $3.3\times {10}^{-8}$ | $5.2\times {10}^{-7}$ | $8.3\times {10}^{-6}$ |

$0.8$ | $7.2\times {10}^{-5}$ | $3.4\times {10}^{-5}$ | $7.5\times {10}^{-8}$ | $2.9\times {10}^{-5}$ | $7.0\times {10}^{-5}$ |

**Table 4.**Case D: Absolute error for different values of $\left(x,t\right)$ considering $N=15$ steps.

$\left(\mathit{t},\mathit{x}\right)$ | $-2.0$ | $-1.0$ | $0$ | $1.0$ | $2.0$ |
---|---|---|---|---|---|

$0.1$ | $7.2\times {10}^{-8}$ | $4.4\times {10}^{-8}$ | $5.2\times {10}^{-9}$ | $4.2\times {10}^{-8}$ | $6.9\times {10}^{-8}$ |

$0.3$ | $6.3\times {10}^{-7}$ | $4.7\times {10}^{-7}$ | $6.3\times {10}^{-9}$ | $4.6\times {10}^{-7}$ | $5.3\times {10}^{-7}$ |

$0.5$ | $7.8\times {10}^{-7}$ | $5.9\times {10}^{-7}$ | $7.7\times {10}^{-8}$ | $5.5\times {10}^{-7}$ | $8.0\times {10}^{-6}$ |

$0.8$ | $8.3\times {10}^{-5}$ | $2.4\times {10}^{-5}$ | $9.0\times {10}^{-7}$ | $3.1\times {10}^{-5}$ | $9.1\times {10}^{-5}$ |

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**MDPI and ACS Style**

González-Gaxiola, O.; Biswas, A.; Yıldırım, Y.; Moraru, L.
Highly Dispersive Optical Solitons in Birefringent Fibers with Polynomial Law of Nonlinear Refractive Index by Laplace–Adomian Decomposition. *Mathematics* **2022**, *10*, 1589.
https://doi.org/10.3390/math10091589

**AMA Style**

González-Gaxiola O, Biswas A, Yıldırım Y, Moraru L.
Highly Dispersive Optical Solitons in Birefringent Fibers with Polynomial Law of Nonlinear Refractive Index by Laplace–Adomian Decomposition. *Mathematics*. 2022; 10(9):1589.
https://doi.org/10.3390/math10091589

**Chicago/Turabian Style**

González-Gaxiola, Oswaldo, Anjan Biswas, Yakup Yıldırım, and Luminita Moraru.
2022. "Highly Dispersive Optical Solitons in Birefringent Fibers with Polynomial Law of Nonlinear Refractive Index by Laplace–Adomian Decomposition" *Mathematics* 10, no. 9: 1589.
https://doi.org/10.3390/math10091589