# Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering

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## Abstract

**:**

## 1. Introduction

## 2. Lump Solution

## 3. Lump One Stripe Solution

## 4. Lump Two Stripe Solution

## 5. Ma-Breather (MB) and Its Relating Rogue Wave

## 6. Kuznetsov-Ma Breather (KMB) and Its Relating Rogue Wave

## 7. Generalized Breathers (GB)

## 8. Akhmediev Breathers (AB)

## 9. Standard Rogue Wave (SRW) Solutions

## 10. Multiwaves Solutions (MS)

## 11. Homoclinic Breather (HB)

## 12. M-Shaped Rational Solitons

## 13. Interactional Solutions with Double Exponential Form

## 14. Kink Cross-Rational (KCR) Solutions

## 15. Periodic Cross-Rational (PCR) Solutions

## 16. Result and Discussions

## 17. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The lump profiles of the solution ${\mathsf{\Delta}}_{1}$ in Equation (7) are presented via distinct parameters $\omega =1$. Three-dimensional profiles at (i) ${a}_{1}=5,$ (ii) ${a}_{1}=10$ and (iii) ${a}_{1}=-3$.

**Figure 2.**Contours graphs for Figure 1.

**Figure 3.**The lump one stripe profiles of the solution ${\mathsf{\Delta}}_{2}$ in Equation (10) are interpreted via distinct values of $\omega =5,{k}_{1}=2,\rho =4,\gamma =-1,p=2,q=-1,\nu =1,{b}_{0}=3$. Three-dimensional profiles are shown in (i) ${a}_{1}=5,$ (ii) ${a}_{1}=10$ and (iii) ${a}_{1}=-2$.

**Figure 4.**Contour displays for Figure 3.

**Figure 5.**The lump two stripe graphs of the solution ${\mathsf{\Delta}}_{3}$ in Equation (13) are interpreted via distinct values of $\omega =5,{k}_{1}=2,\rho =4,\gamma =-1,{k}_{2}=1,{k}_{3}=1,{k}_{4}=2,{b}_{0}=3$. Three-dimensional profiles at (i) ${a}_{0}=5,$ (ii) ${a}_{0}=10$ and (iii) ${a}_{0}=-2$.

**Figure 6.**Contour graphs for Figure 5.

**Figure 7.**The MB graphs of the solution ${\mathsf{\Delta}}_{4}$ in Equation (16) are interpreted via distinct values of $\omega =5,{p}_{1}=2,{p}_{2}=3,{\alpha}_{1}=1,{\alpha}_{2}=2.5,{\lambda}_{1}=1,{\lambda}_{2}=2,{\beta}_{2}=3,\rho =4,{\gamma}_{2}=1$. Three-dimensional profiles at (i) ${a}_{0}=5,$ (ii) ${a}_{0}=10$ and (iii) ${a}_{0}=-1$.

**Figure 8.**Contour graphs for Figure 7.

**Figure 9.**The KMB graphs of the solution ${\mathsf{\Delta}}_{5}$ in Equation (19) are interpreted through values of ${a}_{1}=2$, ${a}_{3}=1$, ${a}_{4}=3$, $\omega =5$, ${p}_{1}=2$, ${p}_{2}=3$, ${b}_{1}=1$, ${b}_{2}=2.5$, ${b}_{3}=1$, ${b}_{4}=2$ and $\rho =4$. Three-dimensional profiles at (i) ${a}_{2}=5,$ (ii) ${a}_{2}=20$ and (iii) ${a}_{2}=-1$.

**Figure 10.**Contour graphs for Figure 9.

**Figure 11.**The GB profiles of the solution ${\mathsf{\Delta}}_{6}$ in Equation (24) are made through values of $\kappa =5$, $b=1$ and $\rho =4$. Three-dimensional graphs at (i) $\sigma =0.2,$ (ii) $\sigma =0.8$ and (iii) $\sigma =-0.1$.

**Figure 12.**Contour graphs for Figure 11.

**Figure 13.**The AB profiles of the solution ${\mathsf{\Delta}}_{7}$ in Equation (27) are made through values of $a=5$, $b=1$, $c=3$, ${p}_{0}=4$ and $m=5$. Three-dimensional graphs at (i) $\beta =5,$ (ii) $\beta =10$ and (iii) $\beta =-3$.

**Figure 14.**Contour slots for Figure 13.

**Figure 15.**The SRW profiles of the solution ${\mathsf{\Delta}}_{8}$ in Equation (30) are made for values of $\u03f5=0.5$, $b=1$, $c=3$, $\gamma =0.1$, $\nu =10$, $\mu =2$ and $\kappa =2$. Three-dimensional graphs at (i) $\delta =-5,$ (ii) $\delta =10$ and (iii) $\delta =20$.

**Figure 16.**Contour slots for Figure 15.

**Figure 17.**The MS graphs of the solution ${\mathsf{\Delta}}_{9}$ in Equation (37) are made for values of ${a}_{6}=0.2$, ${c}_{1}=1$, ${k}_{1}=3,{a}_{4}=0.1,{b}_{1}=10,{b}_{2}=2,{c}_{2}=2,{k}_{2}=3$. Three-dimensional graphs at (i) ${a}_{5}=-5,$ (ii) ${a}_{5}=3$ and (iii) ${a}_{5}=7$.

**Figure 18.**Contour slots for Figure 17.

**Figure 19.**The HB profiles of the solution ${\mathsf{\Delta}}_{11}$ in Equation (43) are constructed for values of ${c}_{1}=2$, ${c}_{2}=1$, ${k}_{1}=3$, ${k}_{2}=0.1$, $\gamma =1$ and $\beta =5$. Three-dimensional graphs at (i) ${d}_{3}=-5,$ (ii) ${d}_{3}=3$ and (iii) ${d}_{3}=15$.

**Figure 20.**Contour slots for Figure 19.

**Figure 21.**The MS profiles of the solution ${\mathsf{\Delta}}_{11}$ in Equation (40) are constructed for values of ${b}_{1}=0.5$, $p=1$, ${a}_{4}=3$, ${a}_{2}=2$, ${k}_{1}=0.1$, $\nu =1$ and ${T}_{r}=2$. Three-dimensional graphs at (i) ${a}_{5}=-5,$ (ii) ${a}_{5}=3$ and (iii) ${a}_{5}=10$.

**Figure 22.**Contour slots for Figure 21.

**Figure 23.**The soliton profiles of the solution ${\mathsf{\Delta}}_{12}$ in Equation (46) are made for values of ${a}_{4}=2$, ${b}_{2}=1$, ${b}_{1}=3$, ${a}_{2}=0.1$, ${T}_{r}=1$, ${c}_{2}=5$, ${k}_{2}=2$ and $\nu =3$. Three-dimensional graphs at (i) ${a}_{3}=-5,$ (ii) ${a}_{3}=0.1$ and (iii) ${a}_{5}=4$.

**Figure 24.**Contour profiles for Figure 23.

**Figure 25.**The KCR profiles of the solution ${\mathsf{\Delta}}_{13}$ in Equation (49) are formed for values of ${b}_{2}=1$, ${b}_{1}=3$, ${a}_{2}=0.1$, $\mu =2$, ${g}_{0}=3$, ${b}_{3}=2$, ${b}_{4}=5$, ${c}_{2}=4$ and ${k}_{2}=2$. Three-dimensional graphs at (i) ${a}_{2}=-5,$ (ii) ${a}_{2}=1$ and (iii) ${b}_{2}=40$.

**Figure 26.**Contour profiles for Figure 25.

**Figure 27.**The PCR graphs of the solution ${\mathsf{\Delta}}_{14}$ in Equation (52) are formed for particular values of ${b}_{2}=1$, ${a}_{4}=2$, ${a}_{2}=-3$, $\mu =2$, ${g}_{0}=7$, ${b}_{2}=1$, ${c}_{1}=3$, ${k}_{1}=0.1$, ${c}_{2}=6$ and $\gamma =2,{k}_{2}=1$. Three-dimensional graphs at (i) ${a}_{1}=-4,$ (ii) ${a}_{1}=0$ and (iii) ${a}_{1}=30$.

**Figure 28.**Shows contour shapes for Figure 27.

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Seadawy, A.R.; Zahed, H.; Rizvi, S.T.R.
Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering. *Mathematics* **2022**, *10*, 1818.
https://doi.org/10.3390/math10111818

**AMA Style**

Seadawy AR, Zahed H, Rizvi STR.
Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering. *Mathematics*. 2022; 10(11):1818.
https://doi.org/10.3390/math10111818

**Chicago/Turabian Style**

Seadawy, Aly R., Hanadi Zahed, and Syed T. R. Rizvi.
2022. "Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering" *Mathematics* 10, no. 11: 1818.
https://doi.org/10.3390/math10111818