# Variational Principle and Diverse Wave Structures of the Modified Benjamin-Bona-Mahony Equation Arising in the Optical Illusions Field

## Abstract

**:**

## 1. Introduction

## 2. The Two Methods

#### 2.1. The VDM

**Note:**It can be seen that the VDM is based on the stationary condition, so we can obtain the optimal solution. Moreover, the VDM is very simple and can construct the different solutions in two steps.

#### 2.2. The HFFM

**Note:**It can be easily found that this method is very simple and can construct the periodic solution in one step.

## 3. Applications

#### 3.1. The VDM

**The bright wave solution**

**The bright-dark wave solution**

**The bright-like wave solution**

**The kinky-bright wave solution**

#### 3.2. The HFFM

**The periodic wave solution**

## 4. Results and Discussion

## 5. Conclusions and Future Recommendations

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The dynamic characteristics of ${\psi}_{1}^{+}\left(x,t\right)$ with $k=1$, $c=6$. (

**a**) 3-D plot, (

**b**) 2-D curve for $t=0$.

**Figure 2.**The dynamic characteristics of ${\psi}_{2}^{+}\left(x,t\right)$ with $k=1$, $c=6$. (

**a**) 3-D plot, (

**b**) 2-D curve for $t=0$.

**Figure 3.**The dynamic characteristics of ${\psi}_{3}^{+}\left(x,t\right)$. with $k=1$, $c=6$. (

**a**) 3-D plot, (

**b**) 2-D curve for $t=0$.

**Figure 4.**The dynamic characteristics of ${\psi}_{4}^{+}\left(x,t\right)$ with $k=1$, $c=6$. (

**a**) 3-D plot, (

**b**) 2-D curve for $t=0$.

**Figure 5.**The dynamic characteristics of ${\psi}_{5}^{+}\left(x,t\right)$ with ${A}_{5}=1$, $k=1$, $c=6$. (

**a**) 3-D plot, (

**b**) 2-D curve for $t=0$.

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**MDPI and ACS Style**

Wang, K.-J.
Variational Principle and Diverse Wave Structures of the Modified Benjamin-Bona-Mahony Equation Arising in the Optical Illusions Field. *Axioms* **2022**, *11*, 445.
https://doi.org/10.3390/axioms11090445

**AMA Style**

Wang K-J.
Variational Principle and Diverse Wave Structures of the Modified Benjamin-Bona-Mahony Equation Arising in the Optical Illusions Field. *Axioms*. 2022; 11(9):445.
https://doi.org/10.3390/axioms11090445

**Chicago/Turabian Style**

Wang, Kang-Jia.
2022. "Variational Principle and Diverse Wave Structures of the Modified Benjamin-Bona-Mahony Equation Arising in the Optical Illusions Field" *Axioms* 11, no. 9: 445.
https://doi.org/10.3390/axioms11090445