# Dust-Ion-Acoustic Rogue Waves in a Dusty Plasma Having Super-Thermal Electrons

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Derivation of the NLSE

## 4. Modulational Instability and Rogue Waves

## 5. Results and Discussion

## 6. Conclusions

- Fast and slow DIA modes have been observed.
- Both modulationally stable (i.e., $P/Q<0$) and unstable (i.e., $P/Q>0$) parametric regimes of the DIAWs can exist for both fast and slow modes.
- The amplitude of the first and second-order DIARWs decreases (increases) with ion (electron) temperature.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Eslami, P.; Mottaghizadeh, M.; Pakzad, H.R. Nonplanar dust-ion acoustic solitary waves in warm plasma with superthermal electrons. IEEE Trans. Plasma Sci.
**2013**, 41, 3589. [Google Scholar] [CrossRef] - Shahmansouri, M.; Tribeche, M. Dust acoustic localized structures in an electron depleted dusty plasma with two-suprathermal ion-temperature. Astrophys. Space Sci.
**2012**, 342, 87. [Google Scholar] [CrossRef] - Saini, N.S.; Singh, K. Head-on collision of two dust ion acoustic solitary waves in a weakly relativistic multicomponent superthermal plasma. Phys. Plasmas
**2016**, 23, 103701. [Google Scholar] [CrossRef] - Shukla, P.K.; Mamun, A.A. Introduction to Dusty Plasma Physics; Institute of Physics: Bristol, UK, 2002. [Google Scholar]
- Zaman, D.M.S.; Mannan, A.; Chowdhury, N.A.; Mamun, A.A. Dust-acoustic rogue waves in opposite polarity dusty plasma featuring nonextensive statistics. High Temp.
**2020**, 58, 789. [Google Scholar] [CrossRef] - Boufendi, L.; Bouchoule, A. Industrial developments of scientific insights in dusty plasmas. Plasma Sources Sci. Technol.
**2002**, 11, 211. [Google Scholar] [CrossRef] - Gill, T.S.; Bains, A.S.; Bedi, C. Modulational instability of dust acoustic solitons in multicomponent plasma with kappa-distributed electrons and ions. Phys. Plasmas
**2010**, 17, 013701. [Google Scholar] [CrossRef] - Alinejad, H.; Mahdavi, M.; Shahmansouri, M. Modulational instability of ion–acoustic waves in a plasma with two-temperature kappa-distributed electrons. Astrophys. Space Sci.
**2014**, 352, 571. [Google Scholar] [CrossRef] - Gharaee, H.; Afghah, S.; Abbasi, H. Modulational instability of ion-acoustic waves in plasmas with superthermal electrons. Phys. Plasmas
**2011**, 18, 032116. [Google Scholar] [CrossRef] - Ferdousi, M.; Sultana, S.; Hossen, M.M.; Miah, M.R.; Mamun, A.A. Dust-acoustic shock excitations in κ-nonthermal electron depleted dusty plasmas. Eur. Phys. J. D
**2017**, 71, 102. [Google Scholar] [CrossRef] - Rahman, M.; Chowdhury, N.A.; Mannan, A.; Mamun, A.A. Dust-acoustic rogue waves in an electron-positron-ion-dust plasma medium. Galaxies
**2021**, 9, 31. [Google Scholar] [CrossRef] - Jahan, S.; Haque, M.N.; Chowdhury, N.A.; Mannan, A.; Mamun, A.A. Ion–acoustic rogue waves in double pair plasma having non-extensive particles. Universe
**2021**, 7, 63. [Google Scholar] [CrossRef] - Sikta, J.; Chowdhury, N.A.; Mannan, A.; Sultana, S.; Mamun, A.A. Electrostatic dust-acoustic rogue waves in an electron depleted dusty plasma. Plasma
**2021**, 4, 230–238. [Google Scholar] [CrossRef] - Maksimovic, M.; Gary, S.P.; Skoug, R.M. Solar wind electron suprathermal strength and temperature gradients: Ulysses observations. J. Geophys. Res.
**2000**, 105, 18337. [Google Scholar] [CrossRef] - Pierrard, V.; Lazar, M. Kappa distributions: Theory and applications in space plasmas. Sol. Phys.
**2010**, 267, 153. [Google Scholar] [CrossRef] [Green Version] - Vasyliunas, V.M. A survey of low-energy electrons in the evening sector of the magnetosphere with OGO 1 and OGO 3. J. Geophys. Res.
**1968**, 73, 2839. [Google Scholar] [CrossRef] - Amin, M.R.; Morfill, G.E.; Shukla, P.K. Modulational instability of dust-acoustic and dust-ion-acoustic waves. Phys. Rev. E
**1998**, 58, 6517. [Google Scholar] [CrossRef] - Jukui, X.; He, L. Modulational instability of cylindrical and spherical dust ion–acoustic waves. Phys. Plasmas
**2003**, 10, 339. [Google Scholar] [CrossRef] - Saini, N.S.; Kourakis, I. Dust-acoustic wave modulation in the presence of superthermal ions. Phys. Plasmas
**2008**, 15, 123701. [Google Scholar] [CrossRef] - Guo, S.; Mei, L. Three-dimensional dust-ion-acoustic rogue waves in a magnetized dusty pair-ion plasma with nonthermal nonextensive electrons and opposite polarity dust grainsa. Phys. Plasmas
**2014**, 21, 082303. [Google Scholar] [CrossRef] - Guo, S.; Mei, L.; Sun, A. Nonlinear ion–acoustic structures in a nonextensive electron–positron–ion–dust plasma: Modulational instability and rogue waves. Ann. Phys.
**2012**, 332, 38. [Google Scholar] [CrossRef] [Green Version] - Guo, S.; Mei, L.; Shi, W. Rogue wave triplets in an ion-beam dusty plasma with superthermal electrons and negative ions. Phys. Lett. A
**2013**, 377, 2118. [Google Scholar] [CrossRef] - Almutalk, S.A.; El-Tantawy, S.A.; El-Awady, E.I.; El-Labany, S.K. On the numerical solution of nonplanar dust-acoustic super rogue waves in a strongly coupled dusty plasma. Phys. Lett. A
**2019**, 16, 1937. [Google Scholar] [CrossRef] - Denra, R.; Paul, S.; Ghosh, U.; Sarkar, S. Nonlinear dust-acoustic wave propagation in a Lorentzian dusty plasma in presence of negative ions. J. Plasma Phys.
**2018**, 84, 5. [Google Scholar] [CrossRef] - Dubinov, A.E.; Kitayev, I.N. Nonlinear periodic backward dust acoustic waves. Planet. Space Sci.
**2021**, 195, 105142. [Google Scholar] [CrossRef] - El-Labany, S.K.; El-Taibany, W.F.; El-Tantawy, A.A.; Zedan, N.A. Effects of double spectral electron distribution and polarization force on dust acoustic waves in a negative dusty plasma. Contrib. Plasma Phys.
**2020**, 60, e202000049. [Google Scholar] [CrossRef] - El-Taibany, W.F.; El-Labany, S.K.; Behery, E.E.; Abdelghany, A.M. Nonlinear dust acoustic waves in a self-gravitating and opposite-polarity complex plasma medium. Eur. Phys. J. Plus
**2019**, 134, 457. [Google Scholar] [CrossRef] - Chahal, B.S.; Singh, M.; Saini, N.S. Dust ion acoustic freak waves in a plasma with two temperature electrons featuring Tsallis distribution. Phys. A
**2018**, 491, 935. [Google Scholar] [CrossRef] - El-Labany, S.K.; El-Shewy, E.K.; Abd El-Razek, H.N.; El-Rahman, A.A. Dust-Ion Acoustic Freak Wave Propagation in a Nonthermal Mesospheric Dusty Plasma. Plasma Phys. Rep.
**2017**, 43, 576. [Google Scholar] [CrossRef] - Mandi, L.; Mondal, K.K.; Chatterjee, P. Analytical solitary wave solution of the dust ion acoustic waves for the damped forced modified Korteweg-de Vries equation in q-nonextensive plasmas. Eur. Phys. J. Spec. Top.
**2019**, 228, 2753. [Google Scholar] [CrossRef] - Paul, A.; Bandyopadhyay, A.; Das, K.P. Dust ion acoustic solitary structures at the acoustic speed in the presence of nonthermal electrons and isothermal positrons. Plasma Phys. Rep.
**2019**, 45, 466. [Google Scholar] [CrossRef] [Green Version] - Sinha, A.; Sahu, B. dust-ion-acoustic waves in unmagnetized 4-component plasma. Adv. Space Res.
**2021**, 67, 1244. [Google Scholar] [CrossRef] - Farooq, M.; Ahmad, M. Dust ion acoustic waves in four component magnetized dusty plasma with effect of slow rotation and superthermal electrons. Phys. Plasmas
**2017**, 24, 123707. [Google Scholar] [CrossRef] - Das, R.; Karmakar, K. Fast and slow modes on dust ion acoustic solitary waves in a warm plasma. Int. J. Sci. Res.
**2013**, 3, 63. [Google Scholar] - Mowafy, A.E. Propagation of the dust ion acoustic waves in inhomogeneous warm dusty plasma. Afr. Rev. Phys.
**2012**, 7, 0032. [Google Scholar] - Heera, N.M.; Akter, J.; Tamanna, N.K.; Chowdhury, N.A.; Rajib, T.I.; Sultana, S.; Mamun, A.A. Ion–acoustic shock waves in a magnetized plasma featuring super-thermal distribution. AIP Adv.
**2021**, 11, 055117. [Google Scholar] [CrossRef] - Banik, S.; Shikha, R.K.; Noman, A.A.; Chowdhury, N.A.; Mannan, A.; Roy, T.S.; Mamun, A.A. First and second-order dust-ion-acoustic rogue waves in non-thermal plasma. Eur. Phys. J. D
**2021**, 75, 43. [Google Scholar] [CrossRef] - Shikha, R.K.; Chowdhury, N.A.; Mannan, A.; Mamun, A.A. Electrostatic dust-acoustic envelope solitons in an electron-depleted plasma. Contrib. Plasma Phys.
**2021**, 61, e202000117. [Google Scholar] [CrossRef] - Khondaker, S.; Mannan, A.; Chowdhury, N.A.; Mamun, A.A. Rogue waves in multi-pair plasma medium. Contrib. Plasma Phys.
**2019**, 59, e201800125. [Google Scholar] [CrossRef] [Green Version] - Dubinov, A.E. On a widespread inaccuracy in defining the Mach number of solitons in a plasma. Plasma Phys. Rep.
**2009**, 35, 991. [Google Scholar] [CrossRef] - Saberian, E.; Esfandyari-Kalejahi, A.; Afsari-Ghazi, M. Nonlinear dust-acoustic structures in space plasmas with superthermal electrons, positrons, and ions. Plasma Phys. Rep.
**2017**, 43, 83. [Google Scholar] [CrossRef] - Shukla, P.K.; Silin, V. Dust ion–acoustic wave. Phys. Scr.
**1992**, 45, 508. [Google Scholar] [CrossRef] - Barkan, A.; Angelo, N.D.; Merlino, R.L. Experiments on ion–acoustic waves in dusty plasmas. Planet. Space Sci.
**1996**, 44, 239. [Google Scholar] [CrossRef] - Kourakis, I.; Shukla, P.K. Modulational instability and localized excitations of dust-ion acoustic waves. Phys. Plasmas
**2003**, 10, 3459. [Google Scholar] [CrossRef] [Green Version] - Kourakis, I.; Shukla, P.K. Exact theory for localized envelope modulated electrostatic wavepackets in space and dusty plasmas. Nonlinear Proc. Geophys.
**2005**, 12, 407. [Google Scholar] [CrossRef] - Fedele, R. Envelope solitons versus solitons. Phys. Scr.
**2002**, 65, 502. [Google Scholar] [CrossRef] - Noman, A.A.; Chowdhury, N.A.; Mannan, A.; Mamun, A.A. Dust-acoustic envelope solitons in super-thermal plasmas. Contrib. Plasma Phys.
**2019**, 59, e201900023. [Google Scholar] [CrossRef] [Green Version] - Akhmediev, N.; Ankiewicz, A.; Soto-Crespo, J.M. Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E
**2009**, 80, 026601. [Google Scholar] [CrossRef] [Green Version] - Ankiewicz, A.; Devine, N.; Akhmediev, N. Are rogue waves robust against perturbations? Phys. Lett. A
**2009**, 373, 3997. [Google Scholar] [CrossRef] - Bandyopadhyay, P.; Konopka, U.; Khrapak, S.A.; Morfill, G.E.; Sen, A. Effect of polarization force on the propagation of dust acoustic solitary waves. New J. Phys.
**2010**, 12, 073002. [Google Scholar] [CrossRef] - Zadorozhny, A.M. Effects of charged dust on mesospheric electrical structure. Adv. Space Res.
**2001**, 28, 1095. [Google Scholar] [CrossRef] - Wang, Y.L.; Guo, X.Y.; Li, Q.S. Nonlinear dust acoustic waves in strongly coupled dusty plasmas with charged dust particles. Commun. Theor. Phys.
**2016**, 65, 247. [Google Scholar] [CrossRef] - Bandyopadhyay, P.; Prasad, G.; Sen, A.; Kaw, P.K. Experimental study of nonlinear dust acoustic solitary waves in a dusty plasma. Phys. Rev. Lett.
**2012**, 101, 065006. [Google Scholar] [CrossRef] [Green Version] - Shalini; Saini, N.S. Dust ion acoustic rogue waves in superthermal warm ion plasma. J. Plasma Phys.
**2015**, 81, 905810316. [Google Scholar]

**Figure 1.**The variation of ${\omega}_{f}$ vs. k (

**left panel**) and ${\omega}_{s}$ vs. k (

**right panel**) when other plasma parameters are $\kappa =1.8$, $\rho =1\times {10}^{3}$, $\mu =3\times {10}^{-6}$, ${\mu}_{2}=0.3$, and ${\mu}_{3}=0.05$.

**Figure 2.**The variation of $P/Q$ with k for different values of $\kappa $ for $\omega \equiv {\omega}_{f}$ (

**left panel**) and $\omega \equiv {\omega}_{s}$ (

**right panel**) when other plasma parameters are $\rho =1\times {10}^{3}$, $\mu =3\times {10}^{-6}$, ${\mu}_{2}=0.3$, and ${\mu}_{3}=0.05$.

**Figure 3.**The variation of ${\mathsf{\Gamma}}_{g}$ with $\tilde{k}$ for different values of ${\mu}_{3}$ (

**left panel**) and the variation of $|{\Phi}_{1}|$ with $\xi $ for different values of ${\mu}_{2}$ (

**right panel**) when other plasma parameters are $k=1.3$, ${\tilde{\Phi}}_{0}=0.5$, $\kappa =1.8$, $\rho =1\times {10}^{3}$, $\mu =3\times {10}^{-6}$, and $\omega \equiv {\omega}_{f}$.

**Figure 4.**The variation of $|{\Phi}_{2}|$ with $\xi $ for different values of ${\mu}_{2}$ (

**left panel**) and the comparison between the first-order (dashed green curve) and second-order (solid blue curve) rational solutions of NLSE (

**right panel**) when other plasma parameters are $\tau =0$, $k=1.3$, ${\tilde{\Phi}}_{0}=0.5$, $\kappa =1.8$, $\rho =1\times {10}^{3}$, $\mu =3\times {10}^{-6}$, ${\mu}_{3}=0.05$, and $\omega \equiv {\omega}_{f}$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Noman, A.A.; Islam, M.K.; Hassan, M.; Banik, S.; Chowdhury, N.A.; Mannan, A.; Mamun, A.A.
Dust-Ion-Acoustic Rogue Waves in a Dusty Plasma Having Super-Thermal Electrons. *Gases* **2021**, *1*, 106-116.
https://doi.org/10.3390/gases1020009

**AMA Style**

Noman AA, Islam MK, Hassan M, Banik S, Chowdhury NA, Mannan A, Mamun AA.
Dust-Ion-Acoustic Rogue Waves in a Dusty Plasma Having Super-Thermal Electrons. *Gases*. 2021; 1(2):106-116.
https://doi.org/10.3390/gases1020009

**Chicago/Turabian Style**

Noman, Akib Al, Md Khairul Islam, Mehedi Hassan, Subrata Banik, Nure Alam Chowdhury, Abdul Mannan, and A. A. Mamun.
2021. "Dust-Ion-Acoustic Rogue Waves in a Dusty Plasma Having Super-Thermal Electrons" *Gases* 1, no. 2: 106-116.
https://doi.org/10.3390/gases1020009