# Low relativistic effects on the modulational instability of rogue waves in electronegative plasmas

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## Abstract

Relativistic ion-acoustic waves are investigated in an electronegative plasma. The use of the reductive perturbation method summarizes the hydrodynamic model to a nonlinear Schrödinger equation which supports the occurrence of modulational instability (MI). From the MI criterion, we derive a critical value for the relativistic parameter \(\alpha _{1}\), below which MI may develop in the system. The MI analysis is then conducted considering the presence and absence of negative ions, coupled to effects of relativistic parameter and the electron-to-negative ion temperature ratio. Under high values of the latter, additional regions of instability are detected, and their spatial expansion is very sensitive to the change in \(\alpha _{1}\) and may support the appearance of rogue waves whose behaviors are discussed. The parametric analysis of super-rogue wave amplitude is performed, where its enhancement is debated relatively to changes in \(\alpha _{1}\), in the presence and absence of negative ions.

## Keywords

Relativistic electronegative plasma Rogue waves Modulational instability## Introduction

Envelope solitons, generic solutions of the nonlinear Schrödinger (NLS) equation, have been extensively studied during the past 30 years, due to their fundamental importance in nonlinear physics. Based on their localization properties, breather solitons have been used as models of rogue waves (RWs) whose behaviors and characteristics are not yet fully unmasked, mainly because they may appear suddenly, propagate within short times, destroy everything on their way and disappear without any trace [1, 2]. For instance, it has been well established that they may appear in physical systems as the consequence of the interplay between nonlinear and dispersive effects, under the activation of the so-called MI phenomenon [3, 4, 5, 6, 7, 8]. Recently, interest in studying RWs has gone beyond oceanography and hydrodynamics [9, 10] to reach some other areas related to optics and photonics [11, 12, 13, 14], Bose–Einstein condensation [15, 16, 17], biophysics [18, 19, 20, 21], plasma physics [22, 23], just to name a few. Particularly, ion-acoustic super-RWs were found in an ultra-cold neutral plasma in the presence of ion-fluid and nonextensive electron distribution [24]. In the same direction, magnetosonic RWs, of first and second order, were investigated numerically in a magnetized plasma [25]. The occurrence of fundamental and second-order RWs was also investigated in a relativistically degenerate plasma using the NLS equation [23]. Comparison between experimental and theoretical occurrences of RWs was proposed recently and applied to multicomponent plasmas with negative ions [26]. A comprehensive analysis by El-Tantawy et al. [27] once more brought out the close relationship between the existence of ion-acoustic RWs and MI in electronegative plasmas (ENPs) in the presence of Maxwellian negative ions, where the dynamical behaviors of the Akhmediev breather (AB), Kuznetsov–Ma (KM) breather and super-RWs were compared.

ENPs and their applications have become an active research direction, mainly due to their particular properties related to the simultaneous presence of positive and negative ions, and electrons. Many different processes have been used to experimentally produce ENPs, including plasma processing reactors [28] and low-temperature experiments [29, 30]. Obviously, from recent contributions, when only positive ions are taken into consideration, the nonlinear terms of the Korteweg–de Vries (KdV) equation are positive, and one may obtain only compressive solitary waves [31], whereas in the presence of both positive and negative ions, soliton characteristics considerably change, due to the nonlinear response of the system to the presence of negative ions [32, 33]. This is indubitably related to the charge neutrality condition which changes, leading to a decrease in the number of electrons and a decrease in their subsequent shielding effect. Quite a limited number of works have been devoted to ENPs, including the contributions by Ghim and Hershkowitz [34] and Mamun et al. [35], where the existence of ion-acoustic waves (IAWs) and dust-acoustic waves (DAWs) was addressed in ENPs containing Boltzmann negative ions, Boltzmann electrons and cold mobile positive ions. The response of such waves, solutions of the KdV equation, to external magnetic fields was also studied [32, 33]. Panguetna et al. [36] proposed a comprehensive study of IAWs and their dependence to electronegative parameters such as the negative ion concentration ratio (*α*) and the electron-to-negative ion temperature ratio (\(\sigma _{n}\)). In two-space dimensions, beyond the study of MI, dromion solutions and their collision scenario were also studied [37]. More recently, the ENP model was extended to its three-dimensional version, giving Tabi et al. [38] the room to study the effect of the modulation angle on the onset of MI, with application to the three-dimensional Davey–Stewartson equations. Obviously, none of the above-cited works includes relativistic effects which should be considered in the emergence of IAWs when the speed a plasma particle approaches that of light. The nonlinear behaviors of plasma waves may importantly be modified by relativistic effects and lead to fascinating spectra of results, exploitable in the laboratory and in the space. IAWs in weakly relativistic plasmas were studied by Das et al. [39, 40], via the KdV equation, and applied to both nonisothermal and isothermal plasmas. El-Labany [41] reported on the existence of modulated weakly relativistic IAWs in a collisionless, unmagnetized, warm plasma with nonthermal electrons using a NLS equation. The latter was also derived recently by Abdikian [42], in three dimensions, to study the emergence of IAWs, under the activation of MI, in a magnetoplasma with pressure of relativistic electrons. Further confirmation was given on the effect of relativistic parameter to bring about new instability and dynamical regimes in the generation mechanism of modulated IAWs via MI.

The main purpose of the present work is to investigate IAWs properties in an ENP, under weak relativistic effects, in one dimension. One of our main results suggests that there is a critical value, \(\alpha _\mathrm{1,cr}\), of the relativistic parameter below which MI and its subsequent nonlinear regime (RWs) may appear in the system.

The layout of the paper goes as follows. In Sect. 2, the relativistic ENP model is presented and a reductive perturbation method (RPM) is employed to derive a NLS equation which describes the evolution of modulated wave packets. In Sect. 3, the criterion for MI is derived, from which we find a critical expression for the relativistic parameter. Importance is then given to the effect of negative ions on such instabilities. The response from RWs solutions to relativistic effects is investigated in the same context, followed by a parametric analysis of the super-RW maximum amplitude when ENP and relativistic parameters are varied. The paper ends with concluding remarks in Sect. 4.

## Model and amplitude equation

*e*is the magnitude of the electron charge. The time and space variables are normalized by the ion Debye length \(\lambda _D =\sqrt{k_{B}T_{e}/4 \pi e^2n}\) and the ion plasma period \(\omega ^{-1}=1/\sqrt{4\pi e^2n_0/m}\), respectively. Here, \(\sigma _n=T_e/T_n\) is the electrons-to-negative ion temperature ratio, \(\mu _e=n_{e0}/n_0\) and \(\mu _n=n_{n0}/n_0\), where \(n_0\) , \(n_{n0}\) and \(n_{e0}\), are the unperturbed densities of the positive ions, negative ions and electrons, respectively. At equilibrium, the neutrality condition of the plasma reads \(\mu _e+\mu _n = 1\), where \(\mu _e=n_{e0}/n_0=1/({1+\alpha })\), with \(\alpha =n_{n0}/n_{e0}\). Using the power series expansion of the exponential function around zero, Eq. (1c) becomes

*p*. These are generated by the nonlinear terms, which means that the corresponding coefficients are of maximum order \(\epsilon ^{p}\). Then, the relations \(n_{l}^{(p)*}= n_{-l}^{(p)}\), \(u_{l}^{(p)*}= u_{-l}^{(p)}\) and \(\phi _l^{(p)*}= \phi _{-l}^{(p)}\) should be satisfied because of reality condition of physical variables. The asterisk denotes the complex conjugate. Substituting the trial solutions (4) into basic Eqs. (1a), (1b) and (3) and equating the quantities with equal power of \(\epsilon\), one obtains several coupled equations in different orders of \(\epsilon\).

*P*and

*Q*are the dispersion and nonlinearity coefficients, respectively, and their expressions are given by

*n*is the number of harmonics of the fundamental frequency and \(\tau _{0j}\) is the initial phase of the

*j*th harmonic. This, after linearizing around the unperturbed wave, leads to coefficients \(a_j\) of the form \(a_j(\xi )=A_je^{(i\alpha _j+\delta _i\xi )}+B_je^{(-i\alpha _j-\delta _i\xi )}\), where \(\tan \alpha _j=2\delta _j/(j\zeta )^2\), with \(\delta _j=j\zeta \sqrt{1-j^2\zeta ^2/4}\) being the growth rate of the

*j*th harmonic of the perturbation which remains positive for frequencies in the range \(0<j\zeta <2\). For the first harmonic case, the MI is established and the instability growth rate has a maximum at \(\zeta =\sqrt{2}\). However, for arbitrary values of \(\zeta\), different cases of RWs were proposed and extensively studied, among which the generalized form [12, 13, 27, 69, 70, 71]

*a*determines the physical behavior of the solution through the function arguments \(b=\sqrt{8a(1-2a)}\) and \(c=\frac{2\pi }{L}=2\sqrt{1-2a}\), with

*L*being the periodicity length of the solution [69, 70]. We should stress that solution (13) can describe three different kinds of breather solutions, depending on the value of

*a*. The super-RW solutions of the focusing NLS equation (11) are localized in both time and space. There are, in fact, two such solutions, the Peregrine soliton and the second-order rogue wave soliton, which are obtained from the general theory of Akhmediev et al. [68, 69] when \(\zeta \rightarrow 0\)

*j*is the order of the solution and \({\bar{\tau }}=2P\tau\). The functions \(G_j(\xi ,{\bar{\tau }})\), \(H_j(\xi ,{\bar{\tau }})\) and \(F_j(\xi ,{\bar{\tau }})\) are polynomials in variables of \({\bar{\tau }}\) and \(\xi\), with \(F_j(\xi ,{\bar{\tau }})\) not having no real zero. We should, however, stress that the Peregrine RW can be derived as a limiting case of the KM breather, especially when \(a\rightarrow 1/2\) [72]. In order to get the two solutions, we will restrict our study to the cases \(j\le 2\).

## Modulational instability and rogue waves

*P*and

*Q*determines the stability of plane-wave solutions to small periodic perturbations. This also implies that any solution whose amplitude is governed by Eq. (11) depends on the sign of

*PQ*, which, in most of plasma systems, depends on system parameters. To this effect, let us assume plane solutions for Eq. (11) to be of the form \(\psi (\xi ,\tau )=\psi _{0}(\xi )e^{iQ\psi _0^{2}\tau }\) that is subjected to a small perturbation \(\delta \psi (\xi ,\tau )=(U_0+iV_0)e^{i(K\zeta -\Omega \tau )}\), with

*K*and \(\Omega\) being, respectively, the wave number and frequency of the perturbation. Following the standard calculations of MI, one obtains the nonlinear dispersion relation

*P*, which can be rewritten in the form

*Q*that was positive or negative for some values of the wave number

*k*. We should stress that coefficient of nonlinearity in the present study keeps the same expression and therefore keeps the same features as in Ref. [36]. For its part, the relativistic contribution in the expression of

*P*is such that

*k*and \(\alpha _1\) that may lead to the formation of bright, or NLS, envelope solitons as the consequence of MI. One should remember that for intervals of

*k*where \(\alpha _1<0\), no MI should be expected. Such regions in Fig. 1 are indicated by the label (MS). Some of the values of \(\alpha _1\) appearing in those areas have been chosen to plot the product

*PQ*in Fig. 2. Figure 2a is obtained for the value \(\sigma _n=5\) of the electron-to-negative ion temperature ratio. There, the instability domain is very sensitive to the change in \(\alpha _1\), and there exists only one region of instability for a value \(k>k_\mathrm{cr}\) of the wave number. However, the region of stability expands with increasing \(\alpha _1\).

*K*obtained for the nonrelativistic case [36]. Equation (22) suggests that if \(P_0\rightarrow \infty\), the nonrelativistic problem will be retried. Otherwise, relativistic effects will be present in the system and influence the features of \(K_\mathrm{cr}\) as shown in Fig. 3, where the two curves give information both in the absence and presence of negative ions. In general, \(K_\mathrm{cr}\) is an increasing function of \(\alpha _1\), but the range for instability to occur is larger when negative ions are absent. In such intervals, one may expect the appearance of RWs.

More interestingly, such waves appear in regions of parameters where modulated IAWs are expected as the result of the interplay between nonlinear and dispersive effects; this because they have in common the term \(\sqrt{\frac{2P}{Q}}\) which should be positive. It is for example shown in Fig. 8 that the negative ion concentration ratio \(\alpha\) has influence on the RW amplitude, where the different panels correspond, respectively, to \(\alpha =0.1\), 0.5 and 0.85. Depending on such values, the regions of instability, related to the RW appearance, display different features. For \(\alpha =0\), it is obvious from Fig. 8a that the RW solutions exist in regions of high \(\sigma _n\), i.e., \(25\le \sigma _n\le 50\), where the highest \(|\psi _{S,{\max }}|=|\psi _{S}(0,0)|=4\sqrt{\frac{2P}{Q}}\) belongs to \(k=2\), while high-amplitude RWs are expected for \(k=1.8\) in the case of \(\alpha =0.5\) as depicted in Fig. 8b. Of course, \(\alpha =0\) corresponds to the case where there are no negative ions. The result is therefore not surprising because Fig. 1 reveals the appearance of modulated waves even in the absence of negative ions, where \(0<\alpha _1<\alpha _\mathrm{1,cr}\). Comparing these two cases, one clearly sees that the wave amplitude in Fig. 8b has decreased and the zone of instability gets delocalized, with the highest MI growth rate appearing in the interval \(30\le \sigma _n\le 45\). For \(\alpha =0.85\), \(|\psi _{S,{\max }}|\) is shown in Fig. 8c. Obviously, \(|\psi _{S,{\max }}|\) has increased and regions of instability are expanded, compared to what is observed in Fig. 8b. It should be noted that the calculations of Fig. 8 have been made for a relativistic parameter \(\alpha _1=0.1\). The same calculations are repeated in Fig. 9, but for \(\alpha _1=0.3\), with \(\alpha\) keeping the same values as previously. Although the detected regions of instability display the same behaviors as in Fig. 8, it is nevertheless obvious that the wave amplitude is lower which shows that against \(\alpha _1\), \(\alpha\) can influence the appearance and formation of RW in the studied weakly relativistic plasma system. The dynamical behaviors of RWs were discussed in the nonrelativistic model of ENPs, and a critical value for \(\alpha\) was proposed [25], below which the wave amplitude decreases or increases, depending on the other plasma parameter values. However, in our context, it is highly ostensible that the relativistic character of the studied system contributes to change such behaviors, therefore leading to much richer comportments.

## Concluding remarks

In conclusion, a weakly relativistic model of ENP has been proposed in this work, and we have addressed the dynamics of ion-acoustic waves. In fact, after reducing the proposed model to a NLS equation, we have studied the MI, through its growth rate, and its response to plasma parameters such as \(\alpha\), \(\sigma _n\) and \(\alpha _1\). One of the main results was the determination of the critical value of the relativistic parameter \(\alpha _1\) under which MI may take place. Based on this, we have characterized the appearance of MI both in the presence (\(\alpha \ne 0\)) and absence (\(\alpha =0\)) of negative ions. The influence of the electron-to-negative ion temperature ratio on MI has also been discussed, where additional regions of instability have been detected due to the interplay between \(\alpha _1\) and \(\sigma _n\). Moreover, the link between instability and the appearance of RWs has been discussed along with their response to both negative ion concentration and relativistic effects. The parametric analysis of the RW amplitude has been performed, showing that it may be enhanced or reduced, depending on the balance between ENP parameters and the introduced relativistic effects.

## Notes

### Acknowledgements

This work is supported by the Botswana International University of Science and Technology under the Grant No. DVC/RDI/2/1/16I (25). CBT thanks the Kavli Institute for Theoretical Physics (KITP), University of California Santa Barbara (USA), where this work was supported in part by the National Science Foundation Grant No. NSF PHY-1748958 and NIH Grant No. R25GM067110.

## References

- 1.Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A
**373**, 675 (2009)ADSCrossRefzbMATHGoogle Scholar - 2.Akhmediev, N., Soto-Crespo, J.M., Ankiewicz, A.: Extreme waves that appear from nowhere: on the nature of rogue waves. Phys. Lett. A
**373**, 2137 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 3.Maïna, I., Tabi, C.B., Mohamadou, A., Ekobena, H.P.F., Kofané, T.C.: Discrete impulses in ephaptically coupled nerve fibers. Chaos
**25**, 043118 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 4.Tabi, C.B., Maïna, I., Mohamadou, A., Ekobena, H.P.F., Kofané, T.C.: Long-range intercellular \(\text{ Ca }^{2+}\) wave patterns. Physica A
**435**, 1 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 5.Etémé, A.S., Tabi, C.B., Mohamadou, A.: Long-range patterns in Hindmarsh–Rose networks. Commun. Nonlinear Sci. Numer. Simul.
**43**, 211 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 6.Tabi, C.B., Ondoua, R.Y., Ekobena, H.P., Mohamadou, A., Kofané, T.C.: Energy patterns in coupled α-helix protein chains with diagonal and off-diagonal couplings. Phys. Lett. A
**380**, 2374 (2016)ADSCrossRefGoogle Scholar - 7.Mefire, G.R.Y., Tabi, C.B., Mohamadou, A., Ekobena, H.P.F., Kofané, T.C.: Modulated pressure waves in large elastic tubes. Chaos
**23**, 033128 (2013)ADSCrossRefzbMATHGoogle Scholar - 8.Madiba, S.E., Tabi, C.B., Ekobena, H.P.F., Kofané, T.C.: Long-range energy modes in α-helix lattices with inter-spine coupling. Physica A
**514**, 298 (2019)ADSMathSciNetCrossRefGoogle Scholar - 9.Kharif, C., Pelinovsky, E., Slunyaev, A.: Rogue Waves in the Ocean. Springer, Heidelberg (2009)zbMATHGoogle Scholar
- 10.Chabchoub, A., Hoffmann, N., Onorato, M., Slunyaev, A., Sergeeva, A., Pelinovsky, E., Akhmediev, N.: Observation of a hierarchy of up to fifth-order rogue waves in a water tank. Phys. Rev. E
**86**, 056601 (2012)ADSCrossRefGoogle Scholar - 11.Dudley, J.M., Genty, G., Dias, F., Kibler, B., Akhmediev, N.: Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation. Opt. Express
**17**, 21497 (2009)ADSCrossRefGoogle Scholar - 12.Kibler, B., Fatome, J., Finot, C., Millot, G., Genty, G., Wetzel, B., Akhmediev, N., Dias, F., Dudley, J.M.: Observation of Kuznetsov–Ma soliton dynamics in optical fibre. Sci. Rep.
**2**, 463 (2012)ADSCrossRefGoogle Scholar - 13.Sun, W.-R., Tian, B., Sun, Y., Chai, J., Jiang, Y.: Akhmediev breathers, Kuznetsov–Ma solitons and rogue waves in a dispersion varying optical fiber. Laser Phys.
**26**, 035402 (2016)ADSCrossRefGoogle Scholar - 14.Frisquet, B., Kibler, B., Millot, G.: Collision of Akhmediev breathers in nonlinear fiber optics. Phys. Rev. X
**3**, 041032 (2013)Google Scholar - 15.Li, S., Prinari, B., Biondini, G.: Solitons and rogue waves in spinor Bose–Einstein condensates. Phys. Rev. E
**97**, 022221 (2018)ADSMathSciNetCrossRefGoogle Scholar - 16.Li, L., Yu, F.: Non-autonomous multi-rogue waves for spin-1 coupled nonlinear Gross–Pitaevskii equation and management by external potentials. Sci. Rep.
**7**, 10638 (2017)ADSCrossRefGoogle Scholar - 17.Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Vector rogue waves in binary mixtures of Bose–Einstein condensates. Eur. Phys. J. Spec. Top.
**185**, 169 (2010)CrossRefGoogle Scholar - 18.Tabi, C.B.: Fractional unstable patterns of energy in α-helix proteins with long-range interactions. Chaos Solitons Fract.
**116**, 092114 (2018)MathSciNetCrossRefGoogle Scholar - 19.Tchinang, J.D.T., Togueu, A.B.M., Tchawoua, C.: Biological multi-rogue waves in discrete nonlinear Schrödinger equation with saturable nonlinearities. Phys. Lett. A
**380**, 3057 (2016)ADSMathSciNetCrossRefGoogle Scholar - 20.Jia, H.-X., Liu, Y.-J., Wang, Y.-N.: Rogue-wave interaction of a nonlinear Schrödinger model for the alpha-helical protein. Z. Naturforsch. A
**27**, 71 (2015)Google Scholar - 21.Du, Z., Tian, B., Qu, Q.-X., Chai, H.-P., Wu, X.-Y.: Semirational rogue waves for the three-coupled fourth-order nonlinear Schrödinger equations in an alpha helical protein. Superlattice Microstrust.
**112**, 362 (2017)ADSCrossRefGoogle Scholar - 22.Sultana, S., Islam, S., Mamun, A.A., Schlickeiser, R.: Modulated heavy nucleus-acoustic waves and associated rogue waves in a degenerate relativistic quantum plasma system. Phys. Plasmas
**25**, 012113 (2018)ADSCrossRefGoogle Scholar - 23.Pathak, P., Sharma, S.K., Nakamura, Y., Bailung, H.: Observation of second order ion acoustic Peregrine breather in multicomponent plasma with negative ions. Phys. Plasmas
**23**, 022107 (2016)ADSCrossRefGoogle Scholar - 24.El-Tantawy, S.A., El-Bedwehy, N.A., Moslem, W.M.: Super rogue waves in ultracold neutral nonextensive plasmas. J. Plasma Phys.
**79**, 1049 (2013)ADSCrossRefGoogle Scholar - 25.El-Tantawy, S.A., El-Bedwehy, N.A., El-Labany, S.K.: Ion-acoustic super rogue waves in ultracold neutral plasmas with nonthermal electrons. Phys. Plasmas
**20**, 072102 (2013)ADSCrossRefGoogle Scholar - 26.Bailung, H., Sharma, S.K., Nakamura, Y.: Observation of Peregrine solitons in a multicomponent plasma with negative ions. Phys. Rev. Lett.
**107**, 255005 (2011)ADSCrossRefGoogle Scholar - 27.El-Tantawy, S.A., Wazwaz, A.M., Ali Shan, S.: On the nonlinear dynamics of breathers waves in electronegative plasmas with Maxwellian negative ions. Phys. Plasmas
**24**, 022105 (2017)ADSCrossRefGoogle Scholar - 28.Gottscho, R.A., Gaebe, C.E.: Negative ion kinetics in RF glow discharges. IEEE Trans. Plasma Sci.
**14**, 92 (1986)ADSCrossRefGoogle Scholar - 29.Jacquinot, J., McVey, B.D., Scharer, J.E.: Mode conversion of the fast magnetosonic wave in a deuterium-hydrogen tokamak plasma. Phys. Rev. Lett.
**39**, 88 (1977)ADSCrossRefGoogle Scholar - 30.Ichiki, R., Yoshimura, S., Watanabe, T., Nakamura, Y., Kawai, Y.: Experimental observation of dominant propagation of the ion-acoustic slow mode in a negative ion plasma and its application. Phys. Plasmas
**9**, 4481 (2002)ADSCrossRefGoogle Scholar - 31.Ikezi, H., Taylor, R., Baker, D.: Formation and interaction of ion-acoustic solitions. Phys. Rev. Lett.
**25**, 11 (1970)ADSCrossRefGoogle Scholar - 32.Anowar, M.G., Ashrafi, K.S., Mamun, A.A.: Dust ion-acoustic solitary waves in a magnetized dusty electronegative plasma. J. Plasma Phys.
**77**, 133 (2011)ADSCrossRefGoogle Scholar - 33.Duha, S.S., Rahman, M.S., Mamun, A.A., Anowar, G.M.: Multidimensional instability of dust ion-acoustic solitary waves in a magnetized dusty electronegative plasma. J. Plasma Phys.
**78**, 279 (2012)ADSCrossRefGoogle Scholar - 34.Ghim, Y.K., Hershkowitz, N.: Experimental verification of Boltzmann equilibrium for negative ions in weakly collisional electronegative plasmas. Appl. Phys. Lett.
**94**, 151503 (2009)ADSCrossRefGoogle Scholar - 35.Mamun, A.A., Shukla, P.K., Eliasson, B.: Solitary waves and double layers in a dusty electronegative plasma. Phys. Rev. E
**80**, 046406 (2009)ADSCrossRefGoogle Scholar - 36.Panguetna, C.S., Tabi, C.B., Kofané, T.C.: Electronegative nonlinear oscillating modes in plasmas. Commun. Nonlinear Sci. Numer. Simul.
**55**, 326 (2018)ADSCrossRefGoogle Scholar - 37.Panguetna, C.S., Tabi, C.B., Kofané, T.C.: Two-dimensional modulated ion-acoustic excitations in electronegative plasmas. Phys. Plasmas
**24**, 092114 (2017)ADSCrossRefGoogle Scholar - 38.Tabi, C.B., Panguetna, C.S., Kofané, T.C.: Electronegative (3+1)-dimensional modulated excitations in plasmas. Physica B
**545**, 70 (2018)CrossRefGoogle Scholar - 39.Das, G.C., Paul, S.N.: Ion?acoustic solitary waves in relativistic plasmas. Phys. Fluids
**28**, 823 (1985)ADSCrossRefzbMATHGoogle Scholar - 40.Das, G.C., Karmakar, B., Paul, S.: Propagation of solitary waves in relativistic plasmas. IEEE Trans. Plasma Sci.
**16**, 22 (1988)ADSCrossRefGoogle Scholar - 41.El-Labany, S., Krim, M.A., El-Warraki, S., El-Taibany, W.: Modulational instability of a weakly relativistic ion acoustic wave in a warm plasma with nonthermal electrons. Chin. Phys.
**12**, 759 (2003)ADSCrossRefGoogle Scholar - 42.Abdikian, A.: Modulational instability of ion-acoustic waves in magnetoplasma with pressure of relativistic electrons. Phys. Plasmas
**24**, 052123 (2017)ADSCrossRefGoogle Scholar - 43.Zheng, X., Chen, Y., Hu, H., Wang, G., Huang, F., Dong, C., Yu, M.Y.: Dust voids in collision-dominated plasmas with negative ions. Phys. Plasmas
**16**, 023701 (2009)ADSCrossRefGoogle Scholar - 44.Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999)zbMATHGoogle Scholar
- 45.Pitaevskii, L.P.: Vortex lines in an imperfect bose gas. Sov. Phys. JETP
**13**, 451 (1961)Google Scholar - 46.Gaididei, Y.B., Rasmussen, K.O., Christiansen, P.L.: Nonlinear excitations in two-dimensional molecular structures with impurities. Phys. Rev. E
**52**, 2951 (1995)ADSCrossRefGoogle Scholar - 47.Ondoua, R.Y., Tabi, C.B., Ekobena, H.P., Mohamadou, A., Kofané, T.C.: Discrete energy transport in the perturbed Ablowitz–Ladik equation for Davydov model of α-helix proteins. Eur. Phys. J. B
**86**, 374 (2012)Google Scholar - 48.Ekobena, H.P.F., Tabi, C.B., Mohamadou, A., Kofané, T.C.: Intramolecular vibrations and noise effects on pattern formation in a molecular helix. J. Phys. Condens. Matter
**23**, 375104 (2011)CrossRefGoogle Scholar - 49.Tabi, C.B., Mimshe, J.C.F., Ekobena, H.P.F., Mohamadou, A., Kofané, T.C.: Nonlinear wave trains in three-strand α-helical protein models. Eur. Phys. J. B
**86**, 374 (2013)ADSMathSciNetCrossRefGoogle Scholar - 50.Mimshe, J.C.F., Tabi, C.B., Edongue, H., Ekobena, H.P.F., Mohamadou, A., Kofané, T.C.: Wave patterns in α-helix proteins with interspine coupling. Phys. Scr.
**87**, 025801 (2013)ADSCrossRefzbMATHGoogle Scholar - 51.Chin, S.L., Hosseini, S.A., Liu, W., Luo, Q., Théberge, F., Aközbek, N., Becker, A., Kandidov, V.P., Kosareva, O.G., Schroeder, H.: The propagation of powerful femtosecond laser pulses in opticalmedia: physics, applications, and new challenges. Can. J. Phys.
**83**, 863 (2005)ADSCrossRefGoogle Scholar - 52.Sprangle, P., Penano, J.R., Hafizi, B.: Propagation of intense short laser pulse in the atmosphere, propagation of intense short laser pulses in the atmosphere. Phys. Rev. E
**66**, 046418 (2002)ADSCrossRefGoogle Scholar - 53.Shim, B., Schrauth, S.E., Gaeta, A.L.: Filamentation in air with ultrashort mid-infrared pulses. Opt. Express
**19**, 9118 (2001)ADSCrossRefGoogle Scholar - 54.Long, R.R.: Solitary waves in the westerlies. J. Atmos. Sci.
**21**, 197 (1964)ADSCrossRefGoogle Scholar - 55.Benny, D.J.: Long nonlinear waves in fluid flows. Appl. Math.
**45**, 52 (1966)Google Scholar - 56.Ruvinski, K.D., Feldstein, F.I., Freidman, G.I.: Effect of nonlinear damping due to the generation of capillary-gravity ripples on the stability of short wind waves and their modulation by an internal-wave train. Izv. Atmos. Ocean. Phys.
**22**, 219 (1986)Google Scholar - 57.Franken, P., Hill, A.E., Peters, C.W., Weinrich, G.: Generation of optical harmonics. Phys. Rev. Lett.
**7**, 118 (1961)ADSCrossRefGoogle Scholar - 58.Kapron, F.P., Maurer, R.D., Teter, M.P.: Theory of backscattering effects in waveguides. Appl. Opt.
**11**, 1352 (1972)ADSCrossRefGoogle Scholar - 59.Miya, T., Hanawa, F., Chida, K., Ohmori, Y.: Dispersion-free VAD single-mode fibers in the 1.5-μm wavelength region. Appl. Opt.
**22**, 372 (1983)ADSCrossRefGoogle Scholar - 60.Agrawal, G.P.: Nonlinear Fiber Optics, Optics and Photonics, 5th edn. Academic Press, Cambridge (2013)Google Scholar
- 61.Wamba, E., Mohamadou, A., Kofané, T.C.: Modulational instability of a trapped Bose–Einstein condensate with two- and three-body interactions. Phys. Rev. E
**77**, 046216 (2008)ADSCrossRefGoogle Scholar - 62.Tamilthiruvalluvar, R., Wamba, E., Subramaniyan, S., Porsezian, K.: Impact of higher-order nonlinearity on modulational instability in two-component Bose–Einstein condensates. Phys. Rev. E
**99**, 032202 (2019)ADSCrossRefGoogle Scholar - 63.Belobo Belobo, D., Ben-Bolie, G.H., Kofané, T.C.: Dynamics of matter-wave condensates with time-dependent two- and three-body interactions trapped by a linear potential in the presence of atom gain or loss. Phys. Rev. E
**89**, 042913 (2014)ADSCrossRefGoogle Scholar - 64.Belobo Belobo, D., Ben-Bolie, G.H., Kofané, T.C.: Dynamics of kink, antikink, bright, generalized Jacobi elliptic function solutions of matter-wave condensates with time-dependent two- and three-body interactions. Phys. Rev. E
**91**, 042902 (2015)ADSCrossRefGoogle Scholar - 65.Hsu, H.C., Kharif, C., Abid, M., Chen, Y.Y.: A nonlinear Schrödinger equation for gravity? Capillary water waves on arbitrary depth with constant vorticity. Part 1. J. Fluid Mech.
**854**, 146 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 66.Toenger, S., Godin, T., Billet, C., Dias, F., Erkintalo, M., Genty, G., Dudley, J.M.: Emergent rogue wave structures and statistics in spontaneous modulation instability. Sci. Rep.
**5**, 10380 (2015)ADSCrossRefGoogle Scholar - 67.Sun, W.R., Wang, L.: Vector rogue waves, rogue wave-to-soliton conversions and modulation instability of the higher-order matrix nonlinear Schrödinger equation. Eur. Phys. J. Plus
**133**, 495 (2018)CrossRefGoogle Scholar - 68.Akhmediev, N.N., Eleonskii, V.M., Kulagin, N.E.: Generation of periodic trains of picoseconld pulses in an optical fiber: exact solutions. Sov. Phys. JETP
**62**, 894 (1985)Google Scholar - 69.Akhmediev, N.N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Teor. Mat. Fiz.
**69**, 189 (1986)CrossRefzbMATHGoogle Scholar - 70.Wang, L.H., Porsezian, K., He, J.S.: Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation. Phys. Rev. E
**87**, 053202 (2013)ADSCrossRefGoogle Scholar - 71.Abdikian, A., Ismaeel, S.: Ion-acoustic rogue waves and breathers in relativistically degenerate electron-positron plasmas. Eur. Phys. J. Plus
**132**, 368 (2017)CrossRefGoogle Scholar - 72.Ankiewicz, A., Clarkson, P.A., Akhmediev, N.: Rogue waves, rational solutions, the patterns of their zeros and integral relations. J. Phys. A Math. Theor.
**43**, 122002 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 73.Akhmediev, N., Eleonskii, V., Kulagin, N.: Exact first-order solutions of the nonlinear Schrödinger equation. Theor. Math. Phys.
**72**, 809 (1987)CrossRefzbMATHGoogle Scholar - 74.Kuznetsov, E.: Solitons in a parametrically unstable plasma. Akad. Nauk SSSR Dokl.
**236**, 575 (1977)ADSGoogle Scholar - 75.Dysthe, K.B., Trulsen, K.: Note on breather type solutions of the NLS as models for freak-waves. Phys. Scr.
**T82**, 48 (1999)ADSCrossRefGoogle Scholar - 76.Osborne, A., Onorato, M., Serio, M.: The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys. Lett. A
**275**, 386 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 77.Ma, Y.C.: The perturbed plane? Wave solutions of the cubic Schrödinger equation. Stud. Appl. Math.
**60**, 43 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 78.Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B Appl. Math.
**25**, 16 (1983)CrossRefzbMATHGoogle Scholar - 79.Ankiewicz, A., Devine, N., Akhmediev, N.: Are rogue waves robust against perturbations? Phys. Lett. A
**373**, 3997 (2009)ADSCrossRefzbMATHGoogle Scholar - 80.Abdelwahed, H.G., El-Shewy, E.K., Zahran, M.A., Elwakil, S.A.: On the rogue wave propagation in ion pair superthermal plasma. Phys. Plasmas
**23**, 022102 (2016)ADSCrossRefGoogle Scholar

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