# On the Effect of Electron Streaming and Existence of Quasi-Solitary Mode in a Strongly Coupled Quantum Dusty Plasma—Far and Near Critical Nonlinearity

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## Abstract

**:**

## 1. Introduction

## 2. Formulation

## 3. Linear Propagation and Streaming

- $A=\frac{b}{3}+{\left(\frac{q}{2}\pm \frac{1}{2}\sqrt{{q}^{2}-\frac{4}{27}{p}^{3}}\right)}^{1/3}+{\left(\frac{q}{2}\mp \frac{1}{2}\sqrt{{q}^{2}-\frac{4}{27}{p}^{3}}\right)}^{1/3}$
- $p=3{\chi}^{2}-2\chi b+\theta $; $q={\chi}^{3}-b{\chi}^{2}+\theta \chi +\lambda $; $\chi =b/3$; $\theta =(1/4)(2ca-4d)$; $(1/4)(bd-{a}^{2}d-{c}^{2})$; $B=\sqrt{\frac{{a}^{2}}{4}+A-b}$; $E=\sqrt{{A}^{2}-d}$

## 4. Reductive Perturbation Approach and the Derivation of the Kdv Equation

#### 4.1. Soliton Cnoidal Wave Solution

## 5. Critical Nonlinearity and MKdv Equation

- ${\chi}_{1}=-\frac{{\alpha}_{1}{\mu}_{1}}{{Z}_{d}{(\lambda -{u}_{e0})}^{2}}+\frac{\lambda ({d}_{22}-1)}{[{\lambda}^{2}-({d}_{0}+{d}_{11})]}+\beta $
- ${\chi}_{2}=\frac{3{\mu}_{1}{({\alpha}_{1}/{Z}_{d})}^{2}}{2{(\lambda -{u}_{e0})}^{4}}+\frac{\lambda ({d}_{23}+{d}_{11}){({d}_{12}-1)}^{2}}{{[{\lambda}^{2}-({d}_{0}+{d}_{11})]}^{3}}+\frac{\lambda ({d}_{12}-1)({d}_{12}+{d}_{24})}{{[{\lambda}^{2}-({d}_{0}+{d}_{11})]}^{2}}+\frac{{({d}_{12}-1)}^{2}}{{[{\lambda}^{2}-({d}_{0}+{d}_{11})]}^{2}}+\frac{\lambda {d}_{25}}{[{\lambda}^{2}-({d}_{0}+{d}_{11})]}-\beta $

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RPT | Reductive Perturbation Technique; |

KdV | Korteweg–deVries; |

MKdV | Modified Korteweg–deVries. |

## Appendix A

## References

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**Figure 1.**The plot of the dispersion relation in the absence of strong coupling and electron streaming velocity.

**Figure 2.**The plot of the dispersion relation in the presence of strong coupling but in the absence of electron streaming velocity.

**Figure 3.**The variation of the wave frequency with the electron streaming velocity. In this case, we see that the wave frequency varies quite clearly with the quantum parameter ‘H’.

**Figure 4.**The variation of the wave frequency with the electron streaming velocity. In this case, we see that the wave frequency varies quite clearly with the quantum parameter ‘H’.

**Figure 5.**The figure depicts the ‘fast’ and ‘slow’ modes of excitation in absence of strong coupling and for the quantum situation.

**Figure 6.**The figure depicts the ‘fast’ and ‘slow’ modes of excitation in absence of strong coupling but for the classical situation.

**Figure 7.**The figure depicts the ‘fast’ and ‘slow’ modes of excitation in presence of strong coupling but for the classical situation.

**Figure 8.**The figure depicts the ‘fast’ and ‘slow’ modes of excitation in presence of strong coupling in the quantum situation.

**Figure 9.**This is the form of quasi-soliton or the nanopteron soliton for which its analytical form is given by (40).

**Figure 14.**The variation of the nonlinear coefficient (${A}^{\prime}$) in the case of a modified KdV equation with $\beta \phantom{\rule{3.33333pt}{0ex}}(={n}_{i0}/\left({Z}_{d}{n}_{d0}\right))$.

**Figure 15.**The graphical representation of quasi-soliton for MKdV equation for which its analytical form is represented by Equation (54).

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**MDPI and ACS Style**

Chaudhuri, S.; Chowdhury, A.R.
On the Effect of Electron Streaming and Existence of Quasi-Solitary Mode in a Strongly Coupled Quantum Dusty Plasma—Far and Near Critical Nonlinearity. *Plasma* **2021**, *4*, 408-425.
https://doi.org/10.3390/plasma4030030

**AMA Style**

Chaudhuri S, Chowdhury AR.
On the Effect of Electron Streaming and Existence of Quasi-Solitary Mode in a Strongly Coupled Quantum Dusty Plasma—Far and Near Critical Nonlinearity. *Plasma*. 2021; 4(3):408-425.
https://doi.org/10.3390/plasma4030030

**Chicago/Turabian Style**

Chaudhuri, Shatadru, and Asesh Roy Chowdhury.
2021. "On the Effect of Electron Streaming and Existence of Quasi-Solitary Mode in a Strongly Coupled Quantum Dusty Plasma—Far and Near Critical Nonlinearity" *Plasma* 4, no. 3: 408-425.
https://doi.org/10.3390/plasma4030030