# New (3+1)-Dimensional Kadomtsev–Petviashvili–Sawada– Kotera–Ramani Equation: Multiple-Soliton and Lump Solutions

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of a New (3+1)-Dimensional KP–SK–R Equation

## 3. Painlevé Analysis to a Related Equation

#### Painlevé Analysis

**(i)****Leading order behavior and coefficients:**

**(ii)****Resonant points:**

- (i)
- The principal branch: $k=-1,2,3,6,7,10$;
- (ii)
- The secondary branch: $k=-2,-1,5,6,7,12$.

**(iii)****Verifying compatibility conditions**

## 4. Multiple-Soliton Solutions

## 5. Lump Solutions (LSs)

**Case 1.**

**Case 2.**

**Case 3.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**One-soliton solution (15) is plotted in $(x,t)$-plane for (

**a**) $(y,z)=(0,0)$ and (

**b**) $(y,z)=(5,5)$.

**Figure 3.**Lump solution (35) is plotted in $(x,y)$-plane for (

**a**) $(t,z)=(0,0)$ and (

**b**) $(t,z)=(1,1)$.

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**MDPI and ACS Style**

Wazwaz, A.-M.; Abu Hammad, M.; Al-Ghamdi, A.O.; Alshehri, M.H.; El-Tantawy, S.A.
New (3+1)-Dimensional Kadomtsev–Petviashvili–Sawada– Kotera–Ramani Equation: Multiple-Soliton and Lump Solutions. *Mathematics* **2023**, *11*, 3395.
https://doi.org/10.3390/math11153395

**AMA Style**

Wazwaz A-M, Abu Hammad M, Al-Ghamdi AO, Alshehri MH, El-Tantawy SA.
New (3+1)-Dimensional Kadomtsev–Petviashvili–Sawada– Kotera–Ramani Equation: Multiple-Soliton and Lump Solutions. *Mathematics*. 2023; 11(15):3395.
https://doi.org/10.3390/math11153395

**Chicago/Turabian Style**

Wazwaz, Abdul-Majid, Ma’mon Abu Hammad, Ali O. Al-Ghamdi, Mansoor H. Alshehri, and Samir A. El-Tantawy.
2023. "New (3+1)-Dimensional Kadomtsev–Petviashvili–Sawada– Kotera–Ramani Equation: Multiple-Soliton and Lump Solutions" *Mathematics* 11, no. 15: 3395.
https://doi.org/10.3390/math11153395