# Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Symmetry Analysis

## 3. Optimal System of Subalgebras

#### 3.1. ${l}^{2}\ne 0$

#### 3.2. ${l}^{2}=0$

**Theorem**

**1.**

## 4. Symmetry Reductions and Exact Solutions

#### 4.1. Solutions through ${V}_{1}$

#### 4.2. Solutions through ${V}_{2}$

#### 4.3. Solutions through ${V}_{3}$

#### 4.4. Solutions through ${V}_{2}+{V}_{3}$

#### 4.5. Solutions through ${V}_{2}-{V}_{3}$

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**). Spatial structure of the exact solution $q$ of (38) for Equation (1), with the parameters as ${A}_{1}=1,$ ${C}_{1}=2,$ ${C}_{2}=1,$ $\lambda =3,$ $\mu =1$, and ${\alpha}_{0}=\mathrm{tan}t.$ (

**b**). Spatial structure of the exact solution $r$ of (38), in which the parameters are the same as (

**a**).

**Figure 2.**(

**a**). Spatial structure of the exact solution $q$ of (48) for Equation (1), with the parameters as ${B}_{0}=-3,$ $a=1,$ $b=3,$ $c=1,$ ${\alpha}_{0}=0.05$ and ${\alpha}_{3}=1$. (

**b**). Spatial structure of the exact solution $r$ of (48), in which the parameters are the same as (

**a**).

**Figure 3.**(

**a**). Spatial structure of the exact solution $q$ of (48) for Equation (1), with the parameters as ${B}_{0}=-3,$ $a=1,$ $b=3,$ $c=1,$ ${\alpha}_{0}=0.05$, and ${\alpha}_{3}=\mathrm{cos}t.$ (

**b**). Spatial structure of the exact solution $r$ of (48), in which the parameters are the same as (

**a**).

**Figure 4.**(

**a**). Spatial structure of the exact solution $q$ of (49) for Equation (1), with the parameters as ${B}_{0}=-3,$ $a=1,$ $b=3,$ $c=1,$ ${\alpha}_{0}=-\mathrm{sin}t$, and ${\alpha}_{3}=t.$ (

**b**). Spatial structure of the exact solution $r$ of (49), in which the parameters are the same as (

**a**).

**Figure 5.**(

**a**). Spatial structure of the exact solution $q$ of (55) for Equation (1), with the parameters as ${A}_{1}=1,$ $a=-1,$ $b=1,$ $C=0,$ ${\alpha}_{0}=\mathrm{sin}t$, and ${\alpha}_{3}=1$. (

**b**). Spatial structure of the exact solution $r$ of (55), in which the parameters are the same as (

**a**).

**Figure 6.**(

**a**). Spatial structure of the exact solution $q$ of (55) for Equation (1), with the parameters as ${A}_{1}=1,$ $a=-1,$ $b=1,$ $C=0,$ ${\alpha}_{0}=\mathrm{sin}t$, and ${\alpha}_{3}=t.$ (

**b**). Spatial structure of the exact solution $r$ of (55), in which the parameters are the same as (

**a**).

$[{\mathit{V}}_{\mathit{i}},{\mathit{V}}_{\mathit{j}}]$ | ${\mathit{V}}_{1}$ | ${\mathit{V}}_{2}$ | ${\mathit{V}}_{3}$ |
---|---|---|---|

${V}_{1}$ | 0 | $-{V}_{2}$ | $-3{V}_{3}$ |

${V}_{2}$ | ${V}_{2}$ | 0 | 0 |

${V}_{3}$ | $3{V}_{3}$ | 0 | 0 |

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Liu, N.
Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients. *Symmetry* **2020**, *12*, 522.
https://doi.org/10.3390/sym12040522

**AMA Style**

Liu N.
Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients. *Symmetry*. 2020; 12(4):522.
https://doi.org/10.3390/sym12040522

**Chicago/Turabian Style**

Liu, Na.
2020. "Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients" *Symmetry* 12, no. 4: 522.
https://doi.org/10.3390/sym12040522