# Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP)

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## Abstract

**:**

## 1. Introduction

## 2. Lie Symmetry Group Method

#### Invariant Solution Generated by ${X}_{1}$

## 3. Travelling Wave Solution

#### 3.1. The Extended Improved $\left({G}^{\prime}/G\right)$ Method

#### 3.1.1. Case

**Hyperbolic function solutions**: When $\frac{\gamma {\beta}^{2}}{C{\alpha}^{3}}0$, we obtain

**Trigonometric function solutions**: When $\frac{\gamma {\beta}^{2}}{C{\alpha}^{3}}0$, we obtain

**Rational function solutions**: When $\frac{\gamma {\beta}^{2}}{C{\alpha}^{3}}=0$, we obtain

#### 3.1.2. Case

#### 3.1.3. Case

**Hyperbolic function solutions**: When $\frac{\gamma {\beta}^{2}}{C{\alpha}^{3}}0$, we obtain

**Trigonometric function solutions**: When $\frac{\gamma {\beta}^{2}}{C{\alpha}^{3}}0$, we obtain

**Rational function solutions**: When $\frac{\gamma {\beta}^{2}}{C{\alpha}^{3}}=0$, yields $\beta =0$, we obtain

#### 3.1.4. Case

**Hyperbolic function solutions**: When $\frac{\gamma {\beta}^{2}}{4C{\alpha}^{3}}0$, we obtain

**Trigonometric function solutions**: When $\frac{\gamma {\beta}^{2}}{4C{\alpha}^{3}}0$, we obtain

**Rational function solutions**: When $\frac{\gamma {\beta}^{2}}{4C{\alpha}^{3}}=0$, yields $\beta =0$, we get

#### 3.2. The Modified Extended Tanh Method with Riccati Equation

#### 3.2.1. Case

#### 3.2.2. Case

#### 3.2.3. Case

#### 3.2.4. Case

#### 3.3. Connection between the Extended Tanh Method and Extended Improved $\left({G}^{\prime}/G\right)$ Method

**To proof Equation (73)**, first divide Equation (41) by $G(\xi )$ and after some simplification, we get

## 4. Results and Discussion

**Soliton solutions:**Setting $A\ne 0$ but $B=0$ in Equation (78), we obtain

**Singular soliton solutions:**Setting $A=0$ but $B\ne 0$ in Equation (78), we obtain

**Periodic solutions**

**Singular rational soliton**: When $\frac{\gamma {\beta}^{2}}{{\alpha}^{3}C}=0$, for simplicity, consider $\lambda =0$ in Equation (48) yields the following traveling wave solution:

## 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.**(

**a**) Represents $v(x,0,t)$ in (80) for $\gamma =1$, $\beta =3$, $C=4$ and $\alpha =1$, and (

**b**) represents $v(x,0,t)$ in Equation (80) for $\gamma =1$, $\beta =3$, $C=4$ and $\alpha =1$ and $t=0,1,2.$

**Figure 2.**(

**a**) Represents $v(x,0,t)$ in Equation (84) for $\gamma =-1$, $\beta =3$, $C=0.25$ and $\alpha =1$, and (

**b**) represents $v(x,0,t)$ in Equation (84) for $\gamma =-1$, $\beta =3$, $C=0.25$, $\alpha =1$, and $t=0,1,2.$

**Figure 3.**(

**a**) Represents $v(x,0,t)$ in Equation (86) for $\beta =0$, $C=0.25$, $\alpha =1$, and (

**b**) represents $v(x,0,t)$ in Equation (86) for $\gamma =-1$, $\beta =3$, $C=0.25$, $\alpha =1$ and $t=0,1,2.$

${X}_{1}$ | ${X}_{2}$ | ${X}_{3}$ | ${X}_{4}$ | ${X}_{5}$ | ${X}_{6}$ | ${X}_{7}$ | ${X}_{8}$ | |

${X}_{1}$ | 0 | $-{X}_{2}$ | $-2{X}_{3}$ | $-{X}_{4}$ | $4{X}_{5}$ | $2{X}_{6}$ | $3{X}_{7}$ | ${X}_{8}$ |

${X}_{2}$ | ${X}_{2}$ | 0 | 0 | 0 | $-{X}_{7}$ | $-{X}_{8}$ | 0 | 0 |

${X}_{3}$ | $2{X}_{3}$ | 0 | 0 | 0 | ${X}_{6}$ | 0 | ${X}_{8}$ | 0 |

${X}_{4}$ | ${X}_{4}$ | 0 | 0 | 0 | $2{X}_{7}$ | $2{X}_{8}$ | 0 | 0 |

${X}_{5}$ | $-4{X}_{5}$ | ${X}_{7}$ | $-{X}_{6}$ | $-2{X}_{7}$ | 0 | 0 | 0 | 0 |

${X}_{6}$ | $-2{X}_{6}$ | ${X}_{8}$ | 0 | $-2{X}_{8}$ | 0 | 0 | 0 | 0 |

${X}_{7}$ | $-3{X}_{7}$ | 0 | $-{X}_{8}$ | 0 | 0 | 0 | 0 | 0 |

${X}_{8}$ | $-{X}_{8}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Ad | ${X}_{1}$ | ${X}_{2}$ | ${X}_{3}$ | ${X}_{4}$ | ${X}_{5}$ | ${X}_{6}$ | ${X}_{7}$ | ${X}_{8}$ |

${X}_{1}$ | ${X}_{1}$ | ${X}_{2}{e}^{\epsilon}$ | ${X}_{3}{e}^{2\epsilon}$ | ${X}_{4}{e}^{\epsilon}$ | ${X}_{5}{e}^{-4\epsilon}$ | ${X}_{6}{e}^{-2\epsilon}$ | ${X}_{3}{e}^{-3\epsilon}$ | ${X}_{8}{e}^{-\epsilon}$ |

${X}_{2}$ | ${X}_{1}-\epsilon {X}_{2}$ | ${X}_{2}$ | ${X}_{3}$ | ${X}_{4}$ | ${X}_{5}+\epsilon {X}_{7}$ | ${X}_{6}+\epsilon {X}_{8}$ | ${X}_{7}$ | ${X}_{8}$ |

${X}_{3}$ | ${X}_{1}-2\epsilon {X}_{3}$ | ${X}_{2}$ | ${X}_{3}$ | ${X}_{4}$ | ${X}_{5}-\epsilon {X}_{6}$ | ${X}_{6}$ | ${X}_{7}-\epsilon {X}_{8}$ | ${X}_{8}$ |

${X}_{4}$ | ${X}_{1}-\epsilon {X}_{4}$ | ${X}_{2}$ | ${X}_{3}$ | ${X}_{4}$ | ${X}_{5}-2\epsilon {X}_{7}$ | ${X}_{6}-2\epsilon {X}_{8}$ | ${X}_{7}$ | ${X}_{8}$ |

${X}_{5}$ | ${X}_{1}+4\epsilon {X}_{5}$ | ${X}_{2}-\epsilon {X}_{7}$ | ${X}_{3}+\epsilon {X}_{6}$ | ${X}_{4}+2\epsilon {X}_{7}$ | ${X}_{5}$ | ${X}_{6}$ | ${X}_{7}$ | ${X}_{8}$ |

.. | ${X}_{1}+2\epsilon {X}_{6}$ | ${X}_{2}-\epsilon {X}_{8}$ | ${X}_{3}$ | ${X}_{4}+2\epsilon {X}_{8}$ | ${X}_{5}$ | ${X}_{6}$ | ${X}_{7}$ | ${X}_{8}$ |

${X}_{7}$ | ${X}_{1}+3\epsilon {X}_{7}$ | ${X}_{2}$ | ${X}_{3}+\epsilon {X}_{8}$ | ${X}_{4}$ | ${X}_{5}$ | ${X}_{6}$ | ${X}_{7}$ | ${X}_{8}$ |

${X}_{8}$ | ${X}_{1}+\epsilon {X}_{8}$ | ${X}_{2}$ | ${X}_{3}$ | ${X}_{4}$ | ${X}_{5}$ | ${X}_{6}$ | ${X}_{7}$ | ${X}_{8}$ |

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**MDPI and ACS Style**

Laouini, G.; Amin, A.M.; Moustafa, M.
Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP). *Symmetry* **2021**, *13*, 224.
https://doi.org/10.3390/sym13020224

**AMA Style**

Laouini G, Amin AM, Moustafa M.
Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP). *Symmetry*. 2021; 13(2):224.
https://doi.org/10.3390/sym13020224

**Chicago/Turabian Style**

Laouini, Ghaylen, Amr M. Amin, and Mohamed Moustafa.
2021. "Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP)" *Symmetry* 13, no. 2: 224.
https://doi.org/10.3390/sym13020224