# Periodic and Solitary Wave Solutions of the Long Wave–Short Wave Yajima–Oikawa–Newell Model

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## Abstract

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## 1. Introduction

## 2. Solutions of the YON Model

#### 2.1. Jacobi Elliptic Sine Solution

#### 2.2. Jacobi Elliptic Cosine Solution

#### 2.3. Jacobi Delta Amplitude Solution

#### 2.4. Traveling Waves: Solitons

#### 2.5. Traveling Waves: Rational Solution

## 3. Conservation Laws

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Conserved Vectors Depending on Derivatives up to the Second Order

## References

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**Figure 1.**Elliptic cosine solution with $\alpha =1$, $\beta =2$, $V=1.2$, $b=5$, $a=1.3$, ${z}_{0}=0$, $m=0.5$. (

**a**). Short wave $\left|S\right|$. (

**b**). Long wave L.

**Figure 2.**Delta amplitude solution with $\alpha =1$, $\beta =1$, $V=2$, $b=9$, $a=3$, ${z}_{0}=-2$, $m=0.5$. (

**a**). Short wave $\left|S\right|$. (

**b**). Long wave L.

**Figure 3.**Bright soliton solution, with $\alpha =0.5$, $\beta =2$, $a=0.25$, $b=0$, $V=4$, ${z}_{0}=-20$. (

**a**). Short wave $\left|S\right|$. (

**b**). Long wave L.

**Figure 4.**Rational solution with $\alpha =2$, $\beta =1$, $V=1$. (

**a**). Short wave $\left|S\right|$. (

**b**). Long wave L.

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**MDPI and ACS Style**

Caso-Huerta, M.; Degasperis, A.; Leal da Silva, P.; Lombardo, S.; Sommacal, M.
Periodic and Solitary Wave Solutions of the Long Wave–Short Wave Yajima–Oikawa–Newell Model. *Fluids* **2022**, *7*, 227.
https://doi.org/10.3390/fluids7070227

**AMA Style**

Caso-Huerta M, Degasperis A, Leal da Silva P, Lombardo S, Sommacal M.
Periodic and Solitary Wave Solutions of the Long Wave–Short Wave Yajima–Oikawa–Newell Model. *Fluids*. 2022; 7(7):227.
https://doi.org/10.3390/fluids7070227

**Chicago/Turabian Style**

Caso-Huerta, Marcos, Antonio Degasperis, Priscila Leal da Silva, Sara Lombardo, and Matteo Sommacal.
2022. "Periodic and Solitary Wave Solutions of the Long Wave–Short Wave Yajima–Oikawa–Newell Model" *Fluids* 7, no. 7: 227.
https://doi.org/10.3390/fluids7070227