# Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Lax Representations of the Equations (3) and (4)

## 3. Local and Nonlocal Reductions of the nAKNS(3) Equation (3)

#### 3.1. Double Wronskian Solutions

**Theorem**

**1.**

#### 3.2. Real Local and Nonlocal Reductions

#### 3.2.1. Reduction Procedure

**Theorem**

**2.**

**Proof.**

#### 3.2.2. Some Examples of Solutions

**Soliton solutions:**Let ${\Lambda}_{1}\left(t\right)$ be the diagonal matrix, that is,

**Remark**

**1.**

**Remark**

**2.**

**Jordan block solutions:**To present elements of the basic Wronskian column vector of this case, we first introduce lower triangular Toeplitz (LTT) matrices which are defined as

#### 3.2.3. Dynamics

#### 3.3. Complex Local and Nonlocal Reductions

**Theorem**

**3.**

## 4. Local and Nonlocal Reductions of the nAKNS(-1) Equation (4)

#### 4.1. Double Wronskian Solutions

**Theorem**

**4.**

#### 4.2. Real Local and Nonlocal Reductions

**Remark**

**3.**

**Theorem**

**5.**

#### 4.2.1. Some Examples of Solutions

**Soliton solutions:**Let ${\Omega}_{1}\left(t\right)$ be the diagonal matrix

**Jordan block solutions:**We take

#### 4.2.2. Dynamics

#### 4.3. Complex Local and Nonlocal Reductions

**Theorem**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) shape and motion with u given by (42a) for ${a}_{1}=2$ and ${\theta}_{1}=2$. (

**b**) waves in solid line and dotted line stand for plot (a) at $t=0.6$ and $t=0.9$, respectively. (

**c**) shape and motion with u given by (42c) for ${a}_{1}=3$ and ${b}_{1}=4$. (

**d**) shape and motion with u given by (43a) for ${a}_{1}=2$.

**Figure 2.**(

**a**) shape and motion with u given by (44a) for ${a}_{1}=2$, ${a}_{2}=1$, ${\theta}_{1}=1.5$ and ${\theta}_{2}=1$. (

**b**) waves in solid and dotted line stand for plot (a) at $t=0.4$ and $t=-0.2$, respectively. (

**c**) shape and motion with u given by (44c) for ${a}_{1}=1$ and ${a}_{2}=2$. (

**d**) waves in solid and dotted line stand for plot (c) at $t=0$ and $t=-2$, respectively.

**Figure 3.**(

**a**) shape and motion with u given by (48a) for ${a}_{1}=1$ and ${\theta}_{1}=2$. (

**b**) waves in solid and dotted line stand for plot (

**a**) at $t=-0.5$ and $t=-2$, respectively. (

**c**) waves in solid and dotted line stand for (48b) for ${a}_{1}=1,\phantom{\rule{3.33333pt}{0ex}}{b}_{1}=8$ at $t=0$ and $t=-0.2$, respectively. (

**d**) waves in solid and dotted line stand for (48c) for ${a}_{1}=2$ at $t=-1$ and $t=0.5$, respectively.

**Figure 4.**Shape and motion of (62). (

**a**) a stationary rogue wave with ${c}_{1}=2i$ and ${\alpha}_{1}={\tilde{\alpha}}_{1}=1$. (

**b**) waves in solid and dotted line stand for plot (

**a**) at $t=-3$ and $t=0$, respectively. (

**c**) a moving rogue wave with ${c}_{1}=1+2i$, ${\alpha}_{1}=5$ and ${\tilde{\alpha}}_{1}=1$. (

**d**) waves in solid and dotted line stand for plot (c) at $t=-3$ and $t=0$, respectively.

**Figure 5.**${\left|u\right|}^{2}$ given by (65) for ${c}_{1}=0.01+i$. (

**a**) shape and motion with ${\alpha}_{1}=1+i$ and ${\tilde{\alpha}}_{1}=1$; (

**b**) waves in solid and dotted line stand for plot (a) at $t=2$ and $t=2.4$, respectively. (

**c**) wave shape with ${\alpha}_{1}={\tilde{\alpha}}_{1}=1$ at $t=2$.

**Figure 6.**(

**a**) shape and motion with u given by (85a) for ${d}_{1}=10$ and ${\theta}_{1}=6$. (

**b**) waves in solid line and dotted line stand for plot (

**a**) at $t=-4$ and $t=-2$, respectively. (

**c**) shape and motion with u given by (85c) for ${d}_{1}=8$. (

**d**) shape and motion with u given by (85e) for ${d}_{1}=4$ and ${e}_{1}=3$.

**Figure 7.**(

**a**) shape and motion with u given by (86a) for ${d}_{1}=8$, ${d}_{2}=12$, ${\theta}_{1}=1$ and ${\theta}_{2}=1.5$. (

**b**) waves in solid and dotted line stand for plot (

**a**) at $t=1$ and $t=-3$, respectively. (

**c**) shape and motion with u given by (86c) for ${d}_{1}=5$ and ${d}_{2}=4$. (

**d**) waves in solid and dotted line stand for plot (

**c**) at $t=5$ and $t=-1$, respectively.

**Figure 8.**(

**a**) shape and motion with u given by (90a) for ${d}_{1}=4$ and ${\theta}_{1}=0.1$. (

**b**) waves in solid and dotted line stand for plot (

**a**) at $t=-1$ and $t=2$, respectively. (

**c**) waves in solid and dotted line stand for (90b) for ${d}_{1}=8$ at $t=-1$ and $t=6$, respectively. (

**d**) waves in solid and dotted line stand for (90c) for ${d}_{1}=4,\phantom{\rule{3.33333pt}{0ex}}{e}_{1}=32$ at $t=0$ and $t=-2$, respectively.

**Figure 9.**Shape and motion of (100). (

**a**) a stationary soliton wave with ${w}_{1}=4+i$ and ${\alpha}_{1}={\tilde{\alpha}}_{1}=1$. (

**b**) waves in solid and dotted line stand for plot (

**a**) at $t=-1.5$ and $t=-1$, respectively. (

**c**) a moving soliton wave with ${w}_{1}=4i$, ${\alpha}_{1}=5$ and ${\tilde{\alpha}}_{1}=1$. (

**d**) waves in solid and dotted line stand for plot (

**c**) at $t=-1$ and $t=0.5$, respectively.

**Figure 10.**${\left|u\right|}^{2}$ given by (101) for ${c}_{1}=0.05+0.2i$. (

**a**) shape and motion with ${\alpha}_{1}=1+i$ and ${\tilde{\alpha}}_{1}=1$; (

**b**) wave shape at $t=-3$ and $t=-2$, respectively. (

**c**) wave shape with ${\alpha}_{1}={\tilde{\alpha}}_{1}=1$ at $t=-5$.

$(\mathit{\sigma},\mathit{\delta})$ | $\mathbf{\Lambda}\left(\mathit{t}\right)$ | T |
---|---|---|

(1, −1) | (38) with ${\Lambda}_{2}\left(t\right)=-{\Lambda}_{1}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=I$ |

(1, 1) | (38) with ${\Lambda}_{2}\left(t\right)=-{\Lambda}_{1}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}=-{T}_{2}=I$ |

(−1, −1) | (38) | ${T}_{1}=-{T}_{4}=I,\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=\mathbf{0}$ |

(−1, 1) | (38) | ${T}_{1}=-{T}_{4}=iI,\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=\mathbf{0}$ |

$(\mathit{\sigma},\mathit{\delta})$ | $\mathbf{\Lambda}\left(\mathit{t}\right)$ | T |
---|---|---|

(1, −1) | (38) with ${\Lambda}_{2}\left(t\right)=-{\Lambda}_{1}^{*}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=I$ |

(1, 1) | (38) with ${\Lambda}_{2}\left(t\right)=-{\Lambda}_{1}^{*}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}=-{T}_{2}=I$ |

(−1, −1) | (38) with ${\Lambda}_{2}\left(t\right)={\Lambda}_{1}^{*}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=I$ |

(−1, 1) | (38) with ${\Lambda}_{2}\left(t\right)={\Lambda}_{1}^{*}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}=-{T}_{2}=I$ |

$(\mathit{\sigma},\mathit{\delta})$ | $\mathbf{\Omega}\left(\mathit{t}\right)$ | T |
---|---|---|

(1,1) | (80) with ${\Omega}_{2}\left(t\right)=-{\Omega}_{1}\left(t\right)$ | ${T}_{1}={T}_{4}={\mathbf{0}}_{n+1},\phantom{\rule{1.em}{0ex}}{T}_{3}=-{T}_{2}=I$ |

(1,−1) | (80) with ${\Omega}_{2}\left(t\right)=-{\Omega}_{1}\left(t\right)$ | ${T}_{1}={T}_{4}={\mathbf{0}}_{n+1},\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=I$ |

(−1,1) | (80) | ${T}_{1}=-{T}_{4}=I,\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=\mathbf{0}$ |

(−1,−1) | (80) | ${T}_{1}=-{T}_{4}=iI,\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=\mathbf{0}$ |

$(\mathit{\sigma},\mathit{\delta})$ | $\mathbf{\Omega}\left(\mathit{t}\right)$ | T |
---|---|---|

(1,1) | (80) with ${\Omega}_{2}\left(t\right)=-{\Omega}_{1}^{*}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}=-{T}_{2}=I$ |

(1,−1) | (80) with ${\Omega}_{2}\left(t\right)=-{\Omega}_{1}^{*}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=I$ |

(−1,1) | (80) with ${\Omega}_{2}\left(t\right)={\Omega}_{1}^{*}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}={T}_{2}=I$ |

(−1,−1) | (80) with ${\Omega}_{2}\left(t\right)={\Omega}_{1}^{*}\left(t\right)$ | ${T}_{1}={T}_{4}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{T}_{3}=-{T}_{2}=I$ |

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**MDPI and ACS Style**

Xu, H.J.; Zhao, S.L.
Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions. *Symmetry* **2021**, *13*, 23.
https://doi.org/10.3390/sym13010023

**AMA Style**

Xu HJ, Zhao SL.
Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions. *Symmetry*. 2021; 13(1):23.
https://doi.org/10.3390/sym13010023

**Chicago/Turabian Style**

Xu, Hai Jing, and Song Lin Zhao.
2021. "Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions" *Symmetry* 13, no. 1: 23.
https://doi.org/10.3390/sym13010023