# Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization

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## Abstract

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## 1. Introduction

## 2. A Short Overview of (${\mathit{G}}^{\mathbf{\prime}}\mathbf{/}\mathit{G}$, $\mathbf{1}\mathbf{/}\mathit{G}$)-Expansion Technique

**I**: For $\lambda >$ 0.

## 3. Implementation of Nonlinearities Laws

#### 3.1. Ker Law

#### 3.2. Power Law

## 4. Method’s Application with Nonlinearities Laws

#### 4.1. Application for Kerr Law

#### 4.2. Application for Power Law

## 5. Graphs and Meanings

#### 5.1. Visualization of the BME with Kerr Law Nonlinearity

#### 5.2. Sensitivity Analysis

#### 5.3. The Graphical Representation of the BME with Power Law Nonlinearity

#### 5.4. Sensitivity Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Graphs of the solution $\left|\mathcal{M}\left(x,y,t\right)\right|$ of Equation (25) for $\lambda =1.95,\mathsf{{\rm K}}=-3.05,\mathcal{B}=2.6,\eta =2.9,c=0.2,d=0.2,{l}_{1}=1.2$ and ${l}_{2}=0.75$

**:**(

**a**) A three-dimensional visualization showcasing the periodic bright soliton, (

**b**) a two-dimensional visualization showcasing the periodic bright soliton, and (

**c**) a contour depiction of periodic bright soliton.

**Figure 2.**Graphs of the solution $\left|\mathcal{M}\left(x,y,t\right)\right|$ of Equation (28) for $\lambda =-5.5,\mathsf{{\rm K}}=-0.05,\mathcal{B}=3.35,\eta =0.2,c=0.2,d=0.2,{l}_{1}=0.1$ and ${l}_{2}=-3.41$: (

**a**) A 3D depiction of kink-type soliton solution, (

**b**) a 2D illustration of kink-type soliton solution, and (

**c**) a contour portrayal of kink-type soliton solution.

**Figure 3.**Graphs of the solution $\left|\mathcal{M}\left(x,y,t\right)\right|$ of Equation (30) for $\mathsf{{\rm K}}=0.25,\mathcal{B}=1.1,\eta =-5.02,c=-0.01,d=-5.1,{l}_{1}=0.5$ and ${l}_{2}=0.02$

**:**(

**a**) A 3D representation of dark-type soliton, (

**b**) a 2D representation of dark-type soliton, and (

**c**) a contour representation of dark-type soliton.

**Figure 4.**Sensitivity analysis:(

**a**) $\lambda >0$ for Equation (25), (

**b**) $\lambda <0$ for Equation (27), and (

**c**) $\lambda =0$, for Equation (30).

**Figure 5.**Graphical representations of the solutions $\left|\mathcal{M}\left(x,y,t\right)\right|$ of Equation (38) for $\lambda =1.2,\mathsf{{\rm K}}=-1.05,\mathcal{B}=1.5,c=0.3,d=-2.2,\tau =0.1,{l}_{1}=0.2$ and ${l}_{2}=-0.75$: (

**a**) A 3D depiction of singular soliton, (

**b**) a 2D depiction of singular soliton, and (

**c**) contour interpretation of singular soliton.

**Figure 6.**Graphs of the solution $\left|\mathcal{M}\left(x,y,t\right)\right|$ of Equation (40) for $\lambda =-1.2,\mathsf{{\rm K}}=1.05,\mathcal{B}=1.5,\eta =3.92,\tau =0.1,c=0.3,d=2.2,{l}_{1}=0.2$ and ${l}_{2}=-0.75$: (

**a**) A 3D depiction of anti-kink soliton, (

**b**) a 2D illustration of anti-kink soliton, and (

**c**) contour demonstration of anti-kink soliton.

**Figure 7.**Graphical representations of the solutions $\left|\mathcal{M}\left(x,y,t\right)\right|$ of Equation (43) for $\mathsf{{\rm K}}=-0.02,\mathcal{B}=1.1,\tau =0.1,n=1,\eta =-0.2,c=-0.2,d=-0.04,{l}_{1}=5.0$ and ${l}_{2}=0.2$

**:**(

**a**) A 3D depiction of dark soliton (

**b**) A 2D illustration of dark soliton and (

**c**) Contour demonstration of dark soliton.

**Figure 8.**Sensitivity analysis:(

**a**) $\lambda >0$ for Equation (36), (

**b**) $\lambda <0$ for Equation (40), and (

**c**) $\lambda =0$, for Equation (43).

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**MDPI and ACS Style**

Hossain, M.N.; Alsharif, F.; Miah, M.M.; Kanan, M.
Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization. *Mathematics* **2024**, *12*, 1585.
https://doi.org/10.3390/math12101585

**AMA Style**

Hossain MN, Alsharif F, Miah MM, Kanan M.
Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization. *Mathematics*. 2024; 12(10):1585.
https://doi.org/10.3390/math12101585

**Chicago/Turabian Style**

Hossain, Md Nur, Faisal Alsharif, M. Mamun Miah, and Mohammad Kanan.
2024. "Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization" *Mathematics* 12, no. 10: 1585.
https://doi.org/10.3390/math12101585