# Quiescent Optical Solitons with Quadratic-Cubic and Generalized Quadratic-Cubic Nonlinearities

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## Abstract

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## 1. Introduction

## 2. The Enhanced Kudryashov’s Procedure

- Step–1: The reduced model (3) admits the explicit solution$$U\left(\xi \right)={\lambda}_{0}+\sum _{l=1}^{N}\sum _{i+j=l}{\lambda}_{ij}{Q}^{i}\left(\xi \right){R}^{j}\left(\xi \right),$$$${{R}^{\prime}\left(\xi \right)}^{2}={R\left(\xi \right)}^{2}(1-\chi {R\left(\xi \right)}^{2}),$$$${Q}^{\prime}\left(\xi \right)=Q\left(\xi \right)(\eta Q\left(\xi \right)-1),$$

## 3. Quadratic-Cubic Nonlinearity

#### 3.1. Linear Temporal Evolution

- Result–1:$$\begin{array}{c}{\lambda}_{0}=-\frac{5{b}_{1}}{8{b}_{2}},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{01}={\lambda}_{20}={\lambda}_{11}={\lambda}_{10}=0,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{02}=-\chi {\lambda}_{0},\hfill \\ k=\pm \frac{1}{4}\sqrt{\frac{{b}_{1}}{2a}},\phantom{\rule{3.33333pt}{0ex}}\omega =-\frac{15{b}_{1}^{2}}{64{b}_{2}}.\hfill \end{array}$$$$q(x,t)=-\frac{5{b}_{1}}{8{b}_{2}}\left[1-\chi {\left(\frac{4c}{4{c}^{2}exp\left[\pm \frac{1}{4}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]+\chi exp\left[\mp \frac{1}{4}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]}\right)}^{2}\right]\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{15{b}_{1}^{2}}{64{b}_{2}}t+{\theta}_{0}\right)}.$$$$q(x,t)=-\frac{5{b}_{1}}{8{b}_{2}}{tanh}^{2}\left[\frac{1}{4}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{15{b}_{1}^{2}}{64{b}_{2}}t+{\theta}_{0}\right)},$$$$q(x,t)=-\frac{5{b}_{1}}{8{b}_{2}}{coth}^{2}\left[\frac{1}{4}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{15{b}_{1}^{2}}{64{b}_{2}}t+{\theta}_{0}\right)}.$$
- Result–2:$$\begin{array}{c}{\lambda}_{0}=-\frac{5{b}_{1}}{8{b}_{2}},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{01}={\lambda}_{11}={\lambda}_{02}=0,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{20}=4{\eta}^{2}{\lambda}_{0},\hfill \\ {\lambda}_{10}=-4\eta {\lambda}_{0},\phantom{\rule{3.33333pt}{0ex}}k=\pm \frac{1}{2}\sqrt{\frac{{b}_{1}}{2a}},\phantom{\rule{3.33333pt}{0ex}}\omega =-\frac{15{b}_{1}^{2}}{64{b}_{2}}.\hfill \end{array}$$$$q(x,t)=-\frac{5{b}_{1}}{8{b}_{2}}\left\{1-\frac{4b\eta exp\left[\frac{1}{2}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]}{{\left(bexp\left[\frac{1}{2}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]+\eta \right)}^{2}}\right\}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{15{b}_{1}^{2}}{64{b}_{2}}t+{\theta}_{0}\right)}.$$$$q(x,t)=-\frac{5{b}_{1}}{8{b}_{2}}{tanh}^{2}\left[\frac{1}{4}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{15{b}_{1}^{2}}{64{b}_{2}}t+{\theta}_{0}\right)},$$$$q(x,t)=-\frac{5{b}_{1}}{8{b}_{2}}{coth}^{2}\left[\frac{1}{4}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{15{b}_{1}^{2}}{64{b}_{2}}t+{\theta}_{0}\right)}.$$

#### 3.2. Generalized Temporal Evolution

- Result–1:$$\begin{array}{c}{\lambda}_{0}=-\frac{{b}_{1}(2l+3)}{4{b}_{2}(l+1)},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{01}={\lambda}_{20}={\lambda}_{11}={\lambda}_{10}=0,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{02}=-\chi {\lambda}_{0},\hfill \\ k=\pm \frac{1}{2(l+1)}\sqrt{\frac{{b}_{1}}{2a}},\phantom{\rule{3.33333pt}{0ex}}\omega =-\frac{{b}_{1}^{2}(4l(l+2)+3)}{16{b}_{2}l{(l+1)}^{2}}.\hfill \end{array}$$$$\begin{array}{c}q(x,t)=-\frac{{b}_{1}(2l+3)}{4{b}_{2}(l+1)}\left\{1-\chi {\left(\frac{4c}{4{c}^{2}exp\left[\pm \frac{1}{2(l+1)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]+\chi exp\left[\mp \frac{1}{2(l+1)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]}\right)}^{2}\right\}\hfill \\ \times {e}^{i\left(-\frac{{b}_{1}^{2}(4l(l+2)+3)}{16{b}_{2}l{(l+1)}^{2}}t+{\theta}_{0}\right)}.\hfill \end{array}$$$$q(x,t)=-\frac{{b}_{1}(2l+3)}{4{b}_{2}(l+1)}{tanh}^{2}\left[\frac{1}{2(l+1)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(4l(l+2)+3)}{16{b}_{2}l{(l+1)}^{2}}t+{\theta}_{0}\right)},$$$$q(x,t)=-\frac{{b}_{1}(2l+3)}{4{b}_{2}(l+1)}{coth}^{2}\left[\frac{1}{2(l+1)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(4l(l+2)+3)}{16{b}_{2}l{(l+1)}^{2}}t+{\theta}_{0}\right)}.$$
- Result–2:$$\begin{array}{c}{\lambda}_{0}=-\frac{{b}_{1}(2l+3)}{4{b}_{2}(l+1)},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{01}={\lambda}_{11}={\lambda}_{02}=0,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{20}=4{\eta}^{2}{\lambda}_{0},\hfill \\ {\lambda}_{10}=-4\eta {\lambda}_{0},\phantom{\rule{3.33333pt}{0ex}}k=\pm \frac{1}{l+1}\sqrt{\frac{{b}_{1}}{2a}},\phantom{\rule{3.33333pt}{0ex}}\omega =-\frac{{b}_{1}^{2}(2l+1)(2l+3)}{16{b}_{2}l{(l+1)}^{2}}.\hfill \end{array}$$$$q(x,t)=-\frac{{b}_{1}(2l+3)}{4{b}_{2}(l+1)}\left\{1-\frac{4b\eta exp\left[\frac{1}{l+1}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]}{{\left(bexp\left[\frac{1}{l+1}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]+\eta \right)}^{2}}\right\}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2l+1)(2l+3)}{16{b}_{2}l{(l+1)}^{2}}t+{\theta}_{0}\right)}.$$$$q(x,t)=-\frac{{b}_{1}(2l+3)}{4{b}_{2}(l+1)}{tanh}^{2}\left[\frac{1}{2(l+1)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2l+1)(2l+3)}{16{b}_{2}l{(l+1)}^{2}}t+{\theta}_{0}\right)},$$$$q(x,t)=-\frac{{b}_{1}(2l+3)}{4{b}_{2}(l+1)}{coth}^{2}\left[\frac{1}{2(l+1)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2l+1)(2l+3)}{16{b}_{2}l{(l+1)}^{2}}t+{\theta}_{0}\right)}.$$

## 4. Generalized Quadratic-Cubic Nonlinearity

#### 4.1. Linear Temporal Evolution

- Result–1:$$\begin{array}{c}{\lambda}_{0}=-\frac{{b}_{1}(2+3m)}{4{b}_{2}(1+m)},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{01}={\lambda}_{20}={\lambda}_{11}={\lambda}_{10}=0,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{02}=-\chi {\lambda}_{0},\hfill \\ k=\pm \frac{m}{2(1+m)}\sqrt{\frac{{b}_{1}}{2a}},\phantom{\rule{3.33333pt}{0ex}}\omega =-\frac{{b}_{1}^{2}(2+m)(2+3m)}{16{b}_{2}{(1+m)}^{2}}.\hfill \end{array}$$$$\begin{array}{c}q(x,t)={\left[-\frac{{b}_{1}(2+3m)}{4{b}_{2}(1+m)}\left\{1-\chi {\left(\frac{4c}{4{c}^{2}exp\left[\pm \frac{m}{2(1+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]+\chi exp\left[\mp \frac{m}{2(1+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]}\right)}^{2}\right\}\right]}^{\frac{1}{m}}\hfill \\ \times {e}^{i\left(-\frac{{b}_{1}^{2}(2+m)(2+3m)}{16{b}_{2}{(1+m)}^{2}}t+{\theta}_{0}\right)}.\hfill \end{array}$$$$q(x,t)={\left\{-\frac{{b}_{1}(2+3m)}{4{b}_{2}(1+m)}{tanh}^{2}\left[\frac{m}{2(1+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2+m)(2+3m)}{16{b}_{2}{(1+m)}^{2}}t+{\theta}_{0}\right)},$$$$q(x,t)={\left\{-\frac{{b}_{1}(2+3m)}{4{b}_{2}(1+m)}{coth}^{2}\left[\frac{m}{2(1+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2+m)(2+3m)}{16{b}_{2}{(1+m)}^{2}}t+{\theta}_{0}\right)}.$$
- Result–2:$$\begin{array}{c}{\lambda}_{0}=-\frac{{b}_{1}(2+3m)}{4{b}_{2}(1+m)},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{01}={\lambda}_{11}={\lambda}_{02}=0,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{20}=4{\eta}^{2}{\lambda}_{0},\hfill \\ {\lambda}_{10}=-4\eta {\lambda}_{0},\phantom{\rule{3.33333pt}{0ex}}k=\pm \frac{m}{1+m}\sqrt{\frac{{b}_{1}}{2a}},\phantom{\rule{3.33333pt}{0ex}}\omega =-\frac{{b}_{1}^{2}(2+m)(2+3m)}{16{b}_{2}{(1+m)}^{2}}.\hfill \end{array}$$$$q(x,t)={\left\{-\frac{{b}_{1}(2+3m)}{4{b}_{2}(1+m)}\left(1-\frac{4b\eta exp\left[\frac{m}{1+m}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]}{{\left(bexp\left[\frac{m}{1+m}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]+\eta \right)}^{2}}\right)\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2+m)(2+3m)}{16{b}_{2}{(1+m)}^{2}}t+{\theta}_{0}\right)}.$$$$q(x,t)={\left\{-\frac{{b}_{1}(2+3m)}{4{b}_{2}(1+m)}{tanh}^{2}\left[\frac{m}{2(1+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2+m)(2+3m)}{16{b}_{2}{(1+m)}^{2}}t+{\theta}_{0}\right)},$$$$q(x,t)={\left\{-\frac{{b}_{1}(2+3m)}{4{b}_{2}(1+m)}{coth}^{2}\left[\frac{m}{2(1+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2+m)(2+3m)}{16{b}_{2}{(1+m)}^{2}}t+{\theta}_{0}\right)}.$$

#### 4.2. Generalized Temporal Evolution

- Result–1:$$\begin{array}{c}{\lambda}_{0}=-\frac{{b}_{1}(2l+3m)}{4{b}_{2}(l+m)},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{01}={\lambda}_{20}={\lambda}_{11}={\lambda}_{10}=0,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{02}=-\chi {\lambda}_{0},\hfill \\ k=\pm \frac{m}{2(l+m)}\sqrt{\frac{{b}_{1}}{2a}},\phantom{\rule{3.33333pt}{0ex}}\omega =-\frac{{b}_{1}^{2}(2l+m)(2l+3m)}{16{b}_{2}l{(l+m)}^{2}}.\hfill \end{array}$$$$\begin{array}{c}q(x,t)={\left[-\frac{{b}_{1}(2l+3m)}{4{b}_{2}(l+m)}\left\{1-\chi {\left(\frac{4c}{4{c}^{2}exp\left[\pm \frac{m}{2(l+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]+\chi exp\left[\mp \frac{m}{2(l+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]}\right)}^{2}\right\}\right]}^{\frac{1}{m}}\hfill \\ \times {e}^{i\left(-\frac{{b}_{1}^{2}(2l+m)(2l+3m)}{16{b}_{2}l{(l+m)}^{2}}t+{\theta}_{0}\right)}.\hfill \end{array}$$$$q(x,t)={\left\{-\frac{{b}_{1}(2l+3m)}{4{b}_{2}(l+m)}{tanh}^{2}\left[\frac{m}{2(l+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2l+m)(2l+3m)}{16{b}_{2}l{(l+m)}^{2}}t+{\theta}_{0}\right)},$$$$q(x,t)={\left\{-\frac{{b}_{1}(2l+3m)}{4{b}_{2}(l+m)}{coth}^{2}\left[\frac{m}{2(l+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2l+m)(2l+3m)}{16{b}_{2}l{(l+m)}^{2}}t+{\theta}_{0}\right)}.$$
- Result–2:$$\begin{array}{c}{\lambda}_{0}=-\frac{{b}_{1}(2l+3m)}{4{b}_{2}(l+m)},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{01}={\lambda}_{11}={\lambda}_{02}=0,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{20}=4{\eta}^{2}{\lambda}_{0},\hfill \\ {\lambda}_{10}=-4\eta {\lambda}_{0},\phantom{\rule{3.33333pt}{0ex}}k=\pm \frac{m}{l+m}\sqrt{\frac{{b}_{1}}{2a}},\phantom{\rule{3.33333pt}{0ex}}\omega =-\frac{{b}_{1}^{2}(2l+m)(2l+3m)}{16{b}_{2}l{(l+m)}^{2}}.\hfill \end{array}$$$$q(x,t)={\left\{-\frac{{b}_{1}(2l+3m)}{4{b}_{2}(l+m)}\left(1-\frac{4b\eta exp\left[\frac{m}{l+m}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]}{{\left(bexp\left[\frac{m}{l+m}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]+\eta \right)}^{2}}\right)\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2l+m)(2l+3m)}{16{b}_{2}l{(l+m)}^{2}}t+{\theta}_{0}\right)}.$$$$q(x,t)={\left\{-\frac{{b}_{1}(2l+3m)}{4{b}_{2}(l+m)}{tanh}^{2}\left[\frac{m}{2(l+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2l+m)(2l+3m)}{16{b}_{2}l{(l+m)}^{2}}t+{\theta}_{0}\right)},$$$$q(x,t)={\left\{-\frac{{b}_{1}(2l+3m)}{4{b}_{2}(l+m)}{coth}^{2}\left[\frac{m}{2(l+m)}\sqrt{\frac{{b}_{1}}{2a}}\phantom{\rule{3.33333pt}{0ex}}x\right]\right\}}^{\frac{1}{m}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{e}^{i\left(-\frac{{b}_{1}^{2}(2l+m)(2l+3m)}{16{b}_{2}l{(l+m)}^{2}}t+{\theta}_{0}\right)}.$$

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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## Share and Cite

**MDPI and ACS Style**

Arnous, A.H.; Biswas, A.; Yıldırım, Y.; Moraru, L.; Moldovanu, S.; Iticescu, C.; Khan, S.; Alshehri, H.M.
Quiescent Optical Solitons with Quadratic-Cubic and Generalized Quadratic-Cubic Nonlinearities. *Telecom* **2023**, *4*, 31-42.
https://doi.org/10.3390/telecom4010003

**AMA Style**

Arnous AH, Biswas A, Yıldırım Y, Moraru L, Moldovanu S, Iticescu C, Khan S, Alshehri HM.
Quiescent Optical Solitons with Quadratic-Cubic and Generalized Quadratic-Cubic Nonlinearities. *Telecom*. 2023; 4(1):31-42.
https://doi.org/10.3390/telecom4010003

**Chicago/Turabian Style**

Arnous, Ahmed H., Anjan Biswas, Yakup Yıldırım, Luminita Moraru, Simona Moldovanu, Catalina Iticescu, Salam Khan, and Hashim M. Alshehri.
2023. "Quiescent Optical Solitons with Quadratic-Cubic and Generalized Quadratic-Cubic Nonlinearities" *Telecom* 4, no. 1: 31-42.
https://doi.org/10.3390/telecom4010003