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Effect of Tsallis–Gurevich distributed ions on nonlinear dust‐acoustic oscillations in collisionless nonextensive plasma
Abstract
Both linear and weakly nonlinear dust‐acoustic (DA) solitons propagation are revisited in the presence of adiabatically trapped‐nonextensive ions. A physically relevant distribution (called Tsallis‐Gurevich ions distribution) is outlined here for the first time. The effect of particle trapping has been taken into account, resulting in a new expression for the ion density. The role a background ion nonextensivity may play on the main proprieties (viz., dispersion relation and soliton's profile) of DA mode, inherent to space dusty plasma, is then analysed. In the q > 1 case, we have shown that as the nonextensive character of trapped ions increases in the plasma, the potential pulse amplitude increases while its width is narrowed. The modifications prompted by the presence of adiabatically trapped‐nonextensive ions on both DA energy and solitary wave's electric field are also analysed. Interestingly, we have found that DA soliton energy increases as the ions evolve far away from their Maxwellian extensive trapping. Also, it is found that, for q > 1, the stronger the inter‐ions correlation, the stronger the electric field. Our investigation, motivated by space and laboratory plasma observations of plasmas containing non‐Maxwellian particles alongside trapped particles, may complement and provide new insight into previously published works dealing with solitary waves in plasma.
... where T i is the ion temperature, m i is its mass, and f (f < 0) is the electrostatic potential. Therefore, to model the trapped nonextensive ions, we refer to the following one-dimensional Tsallis like ions distribution [29], ...
... As for the integral (5) associated with q > 1 case, it leads to the following normalized ions density expression [29], ...
The investigation focuses on examining the characteristics of dust acoustic double-layers (DA-DLs) in the presence of adiabatically trapped-nonextensive ions featuring Tsallis-Gurevich distribution. Indeed, the ion-density and pseudo-potential expressions are formulated in terms of transcendental functions, allowing for a detailed analysis of Dust Acoustic Double Layers associated with complex plasmas in the presence of non-extensive adiabatically trapped ions. The study then delves into the impact of background ion nonextensivity on key DA-DLs properties. The findings reveal that our plasma model can exhibits both compressive and rarefactive DA-DLs, depending on the nonextensive parameter q. Also, the results indicate that in both sub-sonic and supra-sonic scenarios, the amplitude of the DA-DLs structure rises as the nonextensive parameter decreases. Additionally, it is observed that, for a constant value of the nonextensive parameter q, the amplitude of the DA-DLs structure increases with higher Mach numbers. This investigation, prompted by observations in space and laboratory plasmas containing non-Maxwellian particles alongside trapped particles, has the potential to complement and offer fresh insights into previous works on solitary waves in plasma.
... In this section, we numerically discuss the evolution of the magnitude (amplitude) of the polarization force, denoted by the parameter R, across different ranges of q and α. For typical space plasma parameters [66,67], i.e., n i0 = 5 × 10 7 cm −3 , n e0 = 4 × 10 7 cm −3 , n d0 = 0.33 × 10 4 cm −3 , Z d = 3 × 10 3 , T e = 50 eV, T i = 0.05 eV, σ = 10 −3 , and f = 1.25. It is worth noting that these values have been used in all our calculations here and throughout this paper because the electron temperature (T e = 50 eV) is much higher compared to the nonthermal, nonextensive ion temperature (T i = 0.05 eV); i.e., the electron-dust interaction remains weak, but the ion-dust interaction is strong in the grain vicinity [68]. ...
We investigate the effects of non-extensivity (q), non-thermality (α), obliqueness (l z ), the strength of the magnetic field (ω c ), and dust grain temperature (σ d ) on the basic features (viz., amplitude, width, velocity, and soliton energy) of obliquely propagating dust-acoustic solitary waves (DASWs) in a magnetized dusty plasma, which consists of highly negatively charged dust grains, Boltzmann-distributed electrons, and nonthermal non-extensive Cairns-Tsallis(C-T)-distributed ions. First, we derived the expression of the C-T polarization force and analyzed the combined effects of the ions’ non-extensivity (q) and non-thermality (α) parameters on the magnitude (R) of this polarization force. Our results show that R strongly depends on both the q-parameter and the α-parameter. Specifically, for q < 1, the ions’ non-extensivity and non-thermality weaken the polarization force, whereas for q > 1, R shifts toward higher values. Thus, the obliquely propagating DASWs are more likely to form in a magnetized non-extensive plasma rather than in a magnetized extensive plasma q = 1. Subsequent key findings are as follows: The wave phase velocity increases linearly as the obliquity (l z ) decreases. This implies that a reduced obliqueness results in faster soliton motion and spikier solitary structures. Moreover, the amplitude (width) of DASWs decreases (increases) with increasing l z . An increase in the magnetic field magnitude (ω c ) affects only the width of the DASWs. The amplitude (width) of DASWs decreases (increases) with higher dust grain temperature (σ d ). This indicates that dust temperature significantly affects wave excitation. Specifically, at higher dust temperatures, dispersion dominates over nonlinear effects, resulting in smoother solitary structures. The soliton’s energy increases with α and becomes more pronounced as q decreases (from 1 to 0.75). It increases also with higher ω c and dust temperature (σ d ), especially in the presence of nonthermal energetic particles. This investigation provides valuable insights into the propagation mechanisms of nonlinear DASWs in both space and laboratory plasmas containing non-extensive, nonthermal C-T-distributed ions and dust grains.
In this article, oblique propagation of coherent and dissipative ion-acoustic solitary waves (IASWs) in the presence of
-Gurevich distributed electrons in two-component magnetized viscous plasma is investigated. Therefore, a new
-Gurevich distribution function describing the evolution of adiabatically trapped suprathermal electrons is outlined. Using the properties of hypergeometric functions, the associated electron density is given in a simple form. The modified Burger KdV-like equation and its appropriate analytical solutions, with
-Gurevich electron density, are also illustrated. Numerical results reveal that the main properties (wave profile, electric field, and transported energy) of ion-acoustic solitons (IASs) and ion-acoustic shock waves (IASWs) are significantly affected by the intrinsic plasma parameters such as the adiabatically trapped suprathermal electron parameter, obliquity, external magnetic field, and viscosity. In particular, the suprathermality of adiabatically trapped electrons makes the IASW profile larger and steeper. Moreover, an increase in the external magnetic field magnitude increases (slightly decreases) the amplitude (width) of IASWs. In addition, the electric field associated with IASWs, which adopt a localized profile, is stronger for high propagation angles, intense magnetic fields, and strong viscosity.
In this paper, the problem of large amplitude dust acoustic (DA) solitons has been addressed in a charge varying dusty plasma with ions following a Cairns-Gurevich distribution. Based on the orbit motion limited approach, the correct Cairns-Gurevich ion charging current is presented for the first time. The expression relating the variable dust charge to the plasma potential is given in terms of the Lambert function and we take advantage of this transcendental function to, carefully, analyse DA solitons in a charge varying dusty plasma with trapped nonthermal ions. Our results show that the spatial patterns of the variable charge solitary wave are significantly changed due to the presence of ion population modelled by the Cairns-Gurevich distribution. An addition of a small concentration of trapped nonthermal ions makes the solitary structure less spiky, grows the net negative charge residing on the dust grain surface, and contributes to the electron depletion. Finally, our investigation is extended to highlight the effect of the grain dust charge variation. We have shown that under certain conditions, the impact of dust charge fluctuation may furnish an alternate physical mechanism rasing anomalous dissipation, which becomes more strong and may predominate over the dispersion as the nonthermal character of ions following the Cairns-Gurevich distribution increases.
The effect of polarization force acting on oppositely charged dust grains in magnetized plasma is analysed for obliquely propagating low frequency electrostatic waves with Maxwellian ions, and non‐Maxwellian electrons following the hybrid Cairns–Tsallis velocity distribution. It is shown that the polarization force acting on positively charged dust grain is different from the one acting on negatively charged dust grain in the presence of hybrid Cairns–Tsallis distributed electrons. A nonlinear equation in the form of Zakharov–Kuznetsov equation is derived via perturbation analysis for the description of electrostatic solitary structures in the limit of weak‐amplitude. A parametric investigation determines the modifications arising in the propagation of dust acoustic soliton due to the presence of polarization force and hybrid Cairns–Tsallis distributed electrons. It is also found that the effect of polarization force is more pronounced for heavier dust grain with high charge number. Finally, there is a lower limit found for the q‐nonextensive parameter, below which modifications in soliton profile cease to exist due to the presence of hybrid Cairns–Tsallis distributed electrons.
In the present article, we studied the effect of nonthermal electrons on the formation and existence of double‐layer structures in a three‐species plasma consisting of positive ions, nonthermal electrons, and immobile negative dust‐charged grains. Employing the reductive perturbations, a modified Korteweg–de Vries (mKdV) type equation is derived for the dust‐ion‐acoustic waves (DIAWs) bearing nonthermality. We found that both positive and negative polarity shock structures (double layer) can exist such that it switches polarity while changing the dust charge concentrations. However, strong nonthemality favours only rarefactive structures irrespective of the ion temperature. It is also found that increasing the nonthermal electron in the system the width of the double layer is increased; furthermore, the shock structure forms with small dust charge concentration. For small ionic temperature, increasing the nonthermal electrons in the system makes the double layer potential to increase; however, for σ = 1 reverse phenomena occurs. Our results are relevant to the shock observations in Q machine experiments and in the ionospheric regime of the earth.
The effects of the self-gravitational and the polarization forces on dust acoustic waves structure in a two-fluid complex plasma model are studied. The electrons and the ions obey the non-extensive distribution. The nonlinear dynamics of the massive positive and negative dust grains are considered. The reductive perturbation technique is applied to obtain a Zakharov-Kuznetsov equation. The nonlinear characteristics of the dust acoustic cnoidal (periodic) and solitary waves are investigated using the bifurcation theory. It is found that the dust acoustic wave properties depend strongly on the gravitational and the polarization parameters as well as the non-extensive indexes for electrons and ions. The possible applications of the present work to space plasma situations are discussed.
Linear and nonlinear properties of electrostatic waves on the ion time scale in a collisional rotating magnetoplasma with warm relativistically streaming ions and non-Maxwellian electrons have been investigated here. In the weak nonlinearity limit, we have derived Zakharov-Kuznetsov-Burgers equation to study the shock wave propagation in dissipative magneto-rotating plasmas with non-Maxwellian electrons. It has been found that ion acoustic shock waves with kappa distributed electrons admit only compressive shock structures, however, the ones with Cairns distributed electrons have been shown to allow for the formation of both compressive and rarefactive structures. This change in behavior has been found to be closely linked with the difference in the shapes of both distribution functions. The dependence of the characteristics of ion acoustic shock structures on rotation, obliqueness, relativistic streaming, kinematic viscosity and non-Maxwellian electrons has also been explored in detail. The relevance of the work with regard to planetary magnetospheres and pulsars has also been pointed out.
The problem of dust acoustic solitons is addressed in an unmagnetized collisionless dusty plasma with ions satisfying a Cairns–Gurevich distribution. A new type of density that describes, simultaneously, the evolution of the energetic ions and those trapped in the plasma potential well is outlined. Both highly and weakly nonlinear waves are investigated by deriving the Sagdeev potential for the large amplitude limit, and establishing the nonlinear partial differential equations (mK–dV equation) for the small but finite amplitude limit. It is shown that the effect of the nonthermal ion trapping on DA waves can be quite important. In particular, an increase of nonthermal character of ions following the Cairns–Gurevich distribution lead to an increase of the main quantities (amplitude and width) of dust acoustic solitons. Our investigation will be helpful in understanding the nonlinear DA waves in the presence of nonthermal trapped ions which may exist in space.
Solitary electrostatic structures involving density depletions have been observed in the upper ionosphere by the Freja satellite [Dovner et al., 1994]. If these are interpreted as ion sound solitons, the difficulty arises that the standard Korteweg-de Vries description predicts structures with enhanced rather than depleted density. Here we show that the presence of non-thermal electrons may change the nature of ion sound solitary structures and allow the existence of structures very like those observed.
An investigation of the propagation of ion acoustic waves in nonthermal plasmas in the presence of trapped electrons has been undertaken. This has been motivated by space and laboratory plasma observations of plasmas containing energetic particles, resulting in long-tailed distributions, in combination with trapped particles, whereby some of the plasma particles are confined to a finite region of phase space. An unmagnetized collisionless electron-ion plasma is considered, featuring a non-Maxwellian-trapped electron distribution, which is modelled by a kappa distribution function combined with a Schamel distribution. The effect of particle trapping has been considered, resulting in an expression for the electron density. Reductive perturbation theory has been used to construct a KdV-like Schamel equation, and examine its behaviour. The relevant configurational parameters in our study include the superthermality index kappa and the characteristic trapping parameter beta. A pulse-shaped family of solutions is proposed, also depending on the weak soliton speed increment u(0). The main modification due to an increase in particle trapping is an increase in the amplitude of solitary waves, yet leaving their spatial width practically unaffected. With enhanced superthermality, there is a decrease in both amplitude and width of solitary waves, for any given values of the trapping parameter and of the incremental soliton speed. Only positive polarity excitations were observed in our parametric investigation. (C) 2014 AIP Publishing LLC.
Weak ion-acoustic (IA) solitary wave propagation is investigated in the presence of electron trapping and background nonextensivity. A physically meaningful distribution is outlined and a Schamel-like equation is derived. The role a background electron nonextensivity may play on the energy carried by the IA soliton is then examined. It is found that nonextensivity may cause a soliton energy depletion. An increase of the amount of electron trapping leads to a net shift towards higher values of the soliton energy.
Solitary ion-acoustic wave propagation in the presence of electron trapping is investigated within the theoretical framework of the Tsallis statistical mechanics. A physically meaningful Schamel-like distribution is outlined. In the small amplitude limit, the nonlinear dispersion relation is derived to analyze the global dependency of the main solitary wave quantities. It is found that for a given amplitude and trapping state, the solitary potential structure speeds up and broadens as the electron nonextensivity strengthens. Our results may be of basic interest for experiments that involve particle trapping. The flexibility provided by the nonextensive q-parameter enables one to obtain a good agreement between theory and experiment.
The effects of vortex-like and non-thermal ion distributions are incorporated in the study of nonlinear dust-acoustic waves in an unmagnetized dusty plasma. It is found that owing to the departure from the Boltzmann ion distribution to a vortex-like phase space distribution, the dynamics of small but finite amplitude dust-acoustic waves is governed by a modified Kortweg–de Vries equation. The latter admits a stationary dust-acoustic solitary wave solution, which has larger amplitude, smaller width, and higher propagation velocity than that involving adiabatic ions. On the other hand, consideration of a non-thermal ion distribution provides the possibility of coexistence of large amplitude rarefactive as well as compressive dust-acoustic solitary waves, whereas these structures appear independently when the wave amplitudes become infinitely small. The present investigation should help us to understand the salient features of the non-linear dust-acoustic waves that have been observed in a recent numerical simulation study.
The dispersion relations for electrostatic plane-wave propagation in a collisionless thermal plasma are discussed in the context of the nonextensive statistics proposed by Tsallis. Analytic formulas both for the undamped (Bohm-Gross) and Landau damped waves are derived and compared with the standard results. In the extensive limiting case (q=1), the classical dispersion relations based on the Maxwellian distribution are recovered. It is shown that the experimental results points to a class of Tsallis's velocity distribution described by a nonextensive q-parameter smaller than unity.
By using the generalized (r,q) distribution function, the effect of particle trapping on the linear and nonlinear evolution of an ion-acoustic wave in an electron-ion plasma has been discussed. The spectral indices q and r contribute to the high-energy tails and flatness on top of the distribution function respectively. The generalized Korteweg–de Vries equations with associated solitary wave solutions for different ranges of parameter r are derived by employing a reductive perturbation technique. It is shown that spectral indices r and q affect the trapping of electrons and subsequently the dynamics of the ion acoustic solitary wave significantly.
The seminal paper of
Mamun et al. [Phys. Plasmas 3, 702 (1996)]
is revisited within the theoretical framework of the Tsallis statistical mechanics. The nonextensivity may originate from the correlation or long-range interactions in the dusty plasma. It is found that depending on whether the nonextensive parameter q is positive or negative, the dust-acoustic (DA) soliton exhibits compression for q<0 and rarefaction for q>0. The lower limit of the Mach number for the existence of DA solitary waves is greater (smaller) than its Maxwellian counterpart in the case of superextensivity (subextensivity).
The correct nonextensive electron and ion charging currents are presented for the first time based on the orbit motion limited approach. For −1<q<1, where q measures the amount of plasma nonextensivity, the nonextensive electron charging current is expressed in terms of the hypergeometric function. The variable dust charge is expressed in terms of the Lambert function and we take advantage of this transcendental function to investigate succinctly the effects of nonextensive charge carriers. The obtained formulas bring a possibility to build theories on nonlinear collective process in variable charge nonextensive dusty plasmas.
The dependence of the asymptotic behaviour of small ion-acoustic waves on the number of resonant electrons is investigated by assuming an electron equation of state corresponding to the observed flat-topped electron distribution functions. The result is a modified Korteweg-de Vries equation with a stronger nonlinearity.
Stix's treatment of zero-damped electrostatic waves in a Maxwellian plasma is extended to the nonlinear regime. Stationary Bernstein-Greene-Krusk almodes which propagate with ion acoustic speed are constructed. This subclass consists of solitary, snoidal (=periodical waves, like ocean waves, which can be written in terms of Jacobian elliptic functions) and sinusoidal waves. A discrimination of those waves can be given by a single parameter, the steepness parameter, which contains nonlinearity, trapping of particles and dispersion. It turns out that Sadgeev's soliton represents a special case of the class solitons having the largest width and the lowest velocity. Hence a modified Korteweg-de-Vries equation must exist with a stronger nonlinearity.
The possibility of the nonlinear decay of a localized perturbation into the ion-acoustic solitons is studied. This corresponds to the formation of several electrons–holes in the phase space. The plasma is assumed to contain a population of super-thermal electrons and therefore the κ distribution is used to model the high energy tail in the electron distribution function. The formalism is derived near the ion plasma frequency. In this range of frequency, the ion dynamics is considerable and the ion-acoustic solitons are the stationary solutions of the governing equations. It is shown that a slowly varying dynamics of the order of ion motions causes an initial Gaussian hole to be disintegrated into a number of electron–holes. The non-stationary process of the hole formation is adiabatic. The hole velocities, which are of the order of the ion-acoustic velocity, are slightly different. The set of ion-acoustic solitons forms neighboring (neighboring in phase space) holes. The influence of both trapped and super-thermal electrons on this process is studied.
With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS
q
k [1 –
i=1
W
p
i
q
]/(q-1), whereq characterizes the generalization andp
i are the probabilities associated withW (microscopic) configurations (W). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq1 limit.
In this article, we have investigated the effect of the Gurevich polarization force on both large-amplitude dust-acoustic (DA) solitary waves and double-layer (DL) structures in a collisionless complex plasma. To do this, we have redefined the 3-D Gurevich distribution, which describes the evolution of the adiabatically trapped ions in the plasma potential well and derived the associated density and polarization force expressions. As an application, we have reviewed, using the pseudo-potential approach, the nonlinear DA solitons in an unmagnetized polarized complex plasma containing inertial dust grains, Maxwellian electrons, and adiabatically trapped ions. We have shown that the spatial profile of DA solitary waves is drastically affected by the Gurevich polarization force. In particular, we have found that due to the presence of this polarization force, the potential pulse amplitude increases while its width decreases, thus making the DA structure more spiky. For the sake of completeness, we have also analyzed the effect of the polarization force on DA-DL structures. Our numerical results have shown that both amplitude and width of rarefactive (compressive) DL DA structure decrease (increase) in the presence of polarization force.
This study is essentially an attempt to investigate the effects of trapped ions concentration on the dynamics of dust-acoustic (DA) periodic travelling waves in dusty plasmas, which consist of dust fluid, trapped ions, and Maxwellian electrons. The physical nature of DA solitary and periodic travelling waves in the plasma model at hand is governed by a Korteweg-de Vries (KdV) type equation. Bifurcation analysis of the Hamiltonian system is applied to demonstrate the probability of the presence of DA solitary and periodic travelling waves for the KdV type equation. A careful discussion illustrates that non-linear phenomena of DA solitary and periodic travelling waves are modified due to the presence of the trapped ions concentration. It is found numerically that the negative amplitude of DA periodic travelling waves is amplified, as the trapping parameter that determines the number of the trapped ions is increased. The numerical simulation gives rise to significant highlights on the dynamics of solitary and periodic travelling waves in astrophysical environments such as Saturn's moon Enceladus.
The problem of dust-ion-acoustic (DIA) solitons is addressed in an electronegative collisionless dusty plasma with the presence of trapped-nonthermal electrons. To do this, we revisited the Cairns-Gurevich distribution that describes, concurrently, the evolution of the energetic electrons and those trapped in the plasma potential and analytically expressed its corresponding density. The modifications, due to the presence of this trapped nonthermal density, arising in small-amplitude DIA solitons propagation associated with both space and experimental plasmas are analyzed. In particular, we have found that, in both cases, a proportion of trapped nonthermal electrons relatively significant makes the solitary structure less spiky. Besides, we have shown that the amplitude and width of the experimental plasma soliton seem more affected due to the low ionic mass ratio.
The Landau damping of the dust ion‐acoustic wave (DIAW) in a dusty plasma with non‐extensive distributed components is analysed relying on the kinetic approach. The electron, ion, and dust particles are effectively modelled by the non‐extensive distribution function of the Tsallis statistics. For a collisionless plasma with different values of plasma components indices, the general dispersion relation is achieved, and the non‐extensivity effects on the frequency, as well as the Landau damping of the DIAW, are studied. We show that for , the preliminary results of the Maxwellian plasma are obtained. The decrease of wave damping is achieved by increasing the coefficient q index and the ion‐to‐electron density ratio. The damping rate also increases with an increasing ion‐to‐electron temperature ratio.
The head-on collision between two ion-acoustic solitons (IASs) is studied in pair ions plasmas with hybrid Cairns–Tsallis-distributed electrons. The chosen model is inspired from the experimental studies of Ichiki et al. [Phys. Plasmas 8, 4275 (2001)]. The extended Poincaré–Lighthill–Kuo (PLK) method is employed to obtain the phase shift due to the IASs collision. Both analytical and numerical results reveal that the magnitude of the phase shift is significantly affected by the nonthermal and nonextensive parameters ( α and q ), the number density ratios ( μ and υ ) as well as the mass ratio σ . For a given mass ratio σ ≃ 0.27 (Ar ⁺ , SF 6 − ), the magnitude of the phase shift Δ Q ( 0 ) decreases slightly (increases) with the increase of q ( α ). The effect of α on Δ Q ( 0 ) is more noticeable in the superextensive distribution case ( q < 1). As σ increases [ σ ≃ 0.89 (Xe ⁺ , SF 6 − )], the phase shift becomes wider. In other terms, the phase shift was found to be larger under the effect of higher densities of the negative ions. Our findings should be useful for understanding the dynamics of IA solitons’ head-on collision in space environments [namely, D -regions ( H + , O 2 − ) and F-regions (H ⁺ , H ⁻ ) of the Earth’s ionosphere] and in laboratory double pair plasmas [namely, fullerene (C ⁺ , C ⁻ ) and laboratory experiment (Ar ⁺ , F ⁻ )].
The nonlinear propagation of ion–acoustic (IA) waves in a magneto–rotating plasma is studied by considering the Kappa-Cairns electron distribution. Employing the perturbation scheme, Korteweg–de Vries equation is derived. It is seen that both positive and negative potential solitons can be supported in the present plasma model. The numerical results reveal that the Kappa-Cairns distributed electrons modify features of the electrostatic waves significantly. The effects of non–thermal parameters, plasma rotation frequency, ion temperature, and the wave propagation angle on electrostatic solitary wave structures are also discussed here. It is found that the plasma parameters considerably influence the propagation of IA waves in rotating plasmas. Furthermore, using the bifurcation theory of planar dynamical systems to the K-dV equation, we have presented the existence of solitary and periodic traveling waves. Our study may be helpful to understand the behavior of ion–acoustic wave in the rotating plasma.
The effect of trapped non-thermal polarization force on low-frequency solitary waves (dust acoustic solitons) is addressed in a collisionless complex plasma. For this purpose, we have firstly redefined the three-dimensional Cairns–Gurevich distribution that describes, simultaneously, the evolution of the energetic ions and those trapped in the plasma potential well, and derived the associated both density and polarization force expressions. The polarization force effects are remarkably modified due to the presence of the trapped non-thermal ions. In particular, we have found that the polarization force magnitude decreases with the increase of ion non-thermality character. Next, the modifications arising in dust-acoustic (DA) solitons and its energy due to the presence of these trapped non-thermal density and polarization force are analyzed. In particular, we have found that the presence of the polarization force leads to an increase in the amplitude and width of DA solitary wave can be propagated in different complex plasma media. Finally, a complementary study on the polarization force effects on the DA energy, associated with both space and experimental dusty plasmas, is also carried out. Our numerical results have shown that more is strong the polarization force more is higher the energy transported by DA solitons.
A first theoretical attempt is made to investigate the effects of the trapped nonthermal polarization force on dust-acoustic solitons in collisionless dusty plasmas. For this purpose, a three-dimensional Cairns–Gurevich distribution, that describes, simultaneously, the evolution of the energetic ions and those trapped in the plasma potential well, is revisited, and, consequently, both density and polarization force expressions are derived. The effects of the polarization force are sensibly modified due to the presence of the trapped nonthermal ions. In particular, we have found that, for both experimental and space dusty plasmas, the polarization force magnitude decreases as the relative fraction of the nonthermal ions increases. The changes arising in both linear and nonlinear dust-acoustic (DA) solitons propagation due to the presence of these trapped nonthermal density and polarization force are also analyzed. We have shown that due to the polarization force acting on the dust grains, the effect of the ion nonthermality is boosted. Besides, we have found that an increase in polarization force magnitude leads to an increase in the amplitude and width of DA solitons.
The linear and nonlinear properties of dust‐electron acoustic waves (DEAWs) propagating in magnetized, collisionless, dusty plasma system containing inertial cold electrons, Maxwellian hot electrons, nonthermal ions, and arbitrarily (positively or negatively) charged stationary dust are investigated. The reductive perturbation technique is employed to reduce the basic set of fluid equations to the modified Korteweg‐de Vries equation or Ostrovsky's equation, which governs the dynamics of small amplitude DEAWs in a weakly magnetized dusty nonthermal plasma. The approximate analytical as well as numerical solutions reveal that the basic characteristics of DEA nonlinear structures are found to be significantly modified by the key plasma configuration parameters. It is found that the leading compressive or rarefactive solitary wave structure separates from a trailing wave packet during a considerable time under the influence of magnetic field‐induced Lorentz force.
The properties of obliquely propagating electron-acoustic solitary waves (OPEASWs) have been investigated in a magnetized superthermal plasma (containing inertial cold electron species, inertialess electrons following Vasyliunas-Schamel distribution function, and static ions) via the fluid dynamical approach. The reductive perturbation technique is employed to derive the Schamel equation and the solitary wave solution of the Schamel equation is used to examine the basic features of small but finite amplitude OPEASWs in such a magnetized plasma in the presence of trapped k-superthermal hot electron population. The basic features (width, amplitude, speed, etc.) of OPEASWs are found to be significantly modified by the different plasma configuration parameters, such as plasma superthermality, obliqueness, the particle trapping effect, and external magnetic field effect. The nature of electrostatic disturbances, that may propagate in different realistic space and laboratory plasma systems (e.g., in Saturn ring), is briefly discussed.
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The effect of the exchange-correlation potential (ECP) on the profile of dust-acoustic solitary waves (DASWs) in dusty plasma containing degenerate electrons and ions was investigated. The present study focuses on the linear and nonlinear long-wavelength dust-acoustic waves in non-relativistic plasmas. The linear acoustic waves are analyzed using the multi-fluid model, whereas the reductive perturbation theory is applied to derive a modified Korteweg-de Vries (KdV) equation for small- but finite-amplitude waves. The effects of dust concentration and the exchange-correlation potential were investigated. These parameters remarkably change the amplitude and the width of the solitary waves. The findings from this investigation may lead to further understanding of the characteristics of nonlinear electrostatic waves exciting in the dense Fermi gas which is present in metallic nano structures or in near-supernovae environment in the presence of massive charge defects.
In the present work, employing the nonlinear field equations of a hot dusty plasma in the presence of nonthermal electrons and trapped ions, we studied the amplitude modulation of nonlinear waves in such a plasma medium by use of the reductive perturbation method and obtained the modified nonlinear Schrödinger equation. The modulational instability (MI) was investigated and the effects of the proportion of the fast electrons (α), the trapping parameter (b) and the plasma parameters such as the dust-ion temperature ratio (σd), the partial unperturbed electron to dust density (δ), and the ion-electron temperature ratio (σi) on it was discussed. For the investigation of modulational instability problems three parameters P/Q, Kmax and Γmax play the central role. The variations of these parameters with the wave number k and the other physical parameters are discussed and the possibility of occurence of modulational instability is indicated.
In this paper, dressed solitons in dusty plasma in the presence of polarization forces acting on the dust grains are investigated. Including fourth-order nonlinearities of electric potential and integrating the resulting energy equation, an exact soliton solution is obtained for dust acoustic waves in dusty plasma. This exact solution reduces to the dressed soliton solution obtained for the system using renormalization procedure in the reductive perturbation method, when Mach number is expanded in terms of soliton velocity. An equation including higher-order corrections is also derived, and the polarization effect is show to have a significant role in the characteristics of these corrections.
Effects of plasma nonextensivity on the nonlinear cnoidal ion-acoustic wave in unmagnetized electron-positron-ion plasma have been investigated theoretically. Plasma positrons are taken to be Maxwellian, while the nonextensivity distribution function was used to describe the plasma electrons. The known reductive perturbation method was employed to extract the KdV equation from the basic equations of the model. Sagdeev potential, as well as the cnoidal wave solution of the KdV equation, has been discussed in detail. We have shown that the ion-acoustic periodic (cnoidal) wave is formed only for values of the strength of nonextensivity (q). The q allowable range is shifted by changing the positron concentration (p) and the temperature ratio of electron to positron (σ). For all of the acceptable values of q, the cnoidal ion-acoustic wave is compressive. Results show that ion-acoustic wave is strongly influenced by the electron nonextensivity, the positron concentration, and the temperature ratio of electron to positron. In this work, we have investigated the effects of q, p, and σ on the characteristics of the ion-acoustic periodic (cnoidal) wave, such as the amplitude, wavelength, and frequency.
Alternative localized dust-ion acoustic waves are investigated in a magnetized charge varying dusty plasma with nonthermal electrons having a vortex-like velocity distribution. The correct non-Maxwellian charging currents are obtained based on the well-known orbit limited motion theory. Following the standard reductive perturbation technique, a Schamel–Zakharov Kuznetsov Burgers (S-ZKB) equation is derived. It is shown that due to an interplay between trapping and nonthermality, our dusty plasma model may support solitary as well as shock waves the main quantities (phase velocity, amplitude and width) of which are drastically influenced by trapping, nonthermality and charge variation. Due to the flexibility provided by the outlined distribution function (two concepts of non isothermality), we stress that our model should provide a good fit of the space observations.
It is now well known that space plasmas frequently contain particle components that exhibit high, or superthermal, energy tails with approximate power law distributions in velocity space. Such nonthermal distributions, with overabundances of fast particles, can be better fitted, for supra- and superthermal velocities, by generalized Lorentzian or kappa distributions, than by Maxwellians or one of their variants. Employing the kappa distribution, with real values of the spectral index κ, in place of the Maxwellian we introduce a new plasma dispersion function expected to be of significant importance in kinetic theoretical studies of waves in space plasmas. It is demonstrated that this function is proportional to Gauss’ hypergeometric function 2F1[1,2κ+2;κ+2;z] enabling the well-established theory of the hypergeometric function to be used to manipulate dispersion relations. The reduction, for integer values of κ, to the less general so-called modified plasma dispersion function [Phys. Fluids B 3, 1835 (1991)] is demonstrated. An example illustrating the use of the function is presented.
A numerical investigation is presented to show the existence, formation, and possible realization of large-amplitude dust acoustic (DA) solitary waves in a charge varying dusty plasma with nonthermal ions. These nonlinear localized structures are self-consistent solutions of the collisionless Vlasov equation with a population of fast particles. The spatial patterns of the variable charge DA solitary wave are significantly modified by the nonthermal effects. The results complement and provide new insights into previously published results on this problem.
The problem of a one-dimensional stationary nonlinear electrostatic wave in a plasma free from interparticle collisions is solved exactly by elementary means. It is demonstrated that, by adding appropriate numbers of particles trapped in the potential-energy troughs, essentially arbitrary traveling wave solutions can be constructed. When one passes to the limit of small-amplitude waves it turns out that the distribution function does not possess an expansion whose first term is linear in the amplitude, as is conventionally assumed. This disparity is associated with the trapped particles. It is possible, however, to salvage the usual linearized theory by admitting singular distribution functions. These, of course, do not exhibit Landau damping, which is associated with the restriction to well-behaved distribution functions. The possible existence of such waves in an actual plasma will depend on factors ignored in this paper, as in most previous works, namely interparticle collisions, and the stability of the solutions against various types of perturbations.
It is naturally important question for us to ask under what physical situation should the κ-deformed statistics be suitable for the statistical description of a system and what should the κ parameter stand for. In this Letter, a formula expression of κ parameter is derived on the basis of the κ–H theorem, the κ-velocity distribution and the generalized Boltzmann equation in the framework of κ-deformed statistics. We thus obtain a physical interpretation for the parameter κ≠0 with regard to the temperature gradient and the external force field. We show, as the q-statistics based on Tsallis entropy, the κ-deformed statistics may also be the candidate one suitable for the statistical description of the systems in external fields when being in the nonequilibrium stationary state, but has different physical characteristics. Namely, the κ-distribution is found to describe the nonequilibrium stationary state of the system where the external force should be vertical to the temperature gradient.
It is shown that the nonlinear propagation of dust-acoustic waves in a strongly coupled dusty plasma is governed by a modified Korteweg–de-Vries–Burgers (KdV–Burgers) equation. The latter is derived from a set of generalized hydrodynamic equations for strongly correlated dust grains in a liquid-like state, a Boltzmann electron distribution, and a non-isothermal vortex-like ion distribution. The numerical solutions of the modified KdV–Burgers equation are presented in order to provide some salient features of dust-acoustic solitary and shock structures that may exist in laboratory dusty plasmas where the dust grains are in a strongly coupled liquid phase.
In the present effort we consider the most general nonlinear particle kinetics within the framework of the Fokker–Planck picture. We show that the kinetics imposes the form of the generalized entropy and subsequently we demonstrate the H-theorem. The particle statistical distribution is obtained, both as stationary solution of the nonlinear evolution equation and as the state which maximizes the generalized entropy. The present approach allows to treat the statistical distributions already known in the literature in a unifying scheme. As a working example we consider the kinetics, constructed by using the κ-exponential exp{κ}(x)=(1+κ2x2+κx)1/κ recently proposed, which reduces to the standard exponential as the deformation parameter κ approaches to zero and presents the relevant power law asymptotic behaviour exp{κ}(x)∼x→±∞|2κx|±1/|κ|. The κ-kinetics obeys the H-theorem and in the case of Brownian particles, admits as stationary state the distribution f=Z−1exp{κ}[−(βmv2/2−μ)] which can be obtained also by maximizing the entropy Sκ=∫dnv[c(κ)f1+κ+c(−κ)f1−κ] with c(κ)=−Zκ/[2κ(1+κ)] after properly constrained.
The book Introduction to Plasma Physics by Shukla and Mamun deals with various aspects of collective processes in dusty plasmas. The first introductory chapters review dust charging and the forces on dust grains in the plasma. The next two chapters give an elaborate description of the various waves and instabilities present in plasmas. In our opinion this makes the book a must for scientists involved in dusty plasma research as for the first time these phenomena are clearly explained and catalogued in a single work. Magnetic as well as non-magnetic plasmas are treated and where applicable examples from laboratory or space plasmas are given.
The text is suitable for graduate level teaching as well as referencing purposes. The authors state in the preface: `This book has grown out of research work on topics on which the authors have spent a considerable amount of time and thought.' This explains the final chapters of the book, where `hot topics' on respectively elongated grains, non-linear waves and dust crystals are discussed. Since these chapters deal with state-of-the-art research, the results are inevitably not presented in a systematic way, but rather as a compilation of recent papers.
Throughout the book the subject is treated using a theoretical approach. This makes it complementary to the book Dusty Plasmas: Physics, Chemistry and Technological Impacts in Plasma Processing edited by A Bouchoule which takes an applied approach. The research on dusty plasmas is a relatively new and rapidly expanding area of science. This book will serve as a handbook on waves and instabilities dusty plasmas in the coming years. But the character of the last chapters shows that more is to come in this exciting field of research.
E Stoffels and W W Stoffels
The detection of a new LF electrostatic wave in an unmagnetized collisionless dusty plasma is reported. This new wave should have relevance to LF noise in the F-ring of Saturn. Due to the special feature of the dusty plasma in this ring, the present dust acoustic wave can exist as a normal mode of the system.
An attempt is made to study variable charge dust acoustic (DA) solitons within the theoretical framework of the Tsallis statistical mechanics. The correct nonextensive ion charging current is presented for the first time based on the orbit motion limited (OML) approach. The variable dust charge is then expressed in terms of the Lambert function and we take advantage of this transcendental function to investigate nonlinear localized DA waves in a charge varying dusty plasma with nonextensive ions more rigorously. Our results reveal that the ion nonextensivity makes the dust acoustic solitary structure more spiky. As the ions deviate from their thermodynamic equilibrium, the dust grain charge becomes least negative and the dust accumulation more effective. In addition, the nonextensive character of the ions contributes to the electron depletion. The latter is more pronounced as the ions evolve far away from their thermal equilibrium. Our results should help in providing a good fit between theoretical and experimental results.
The existence of arbitrary amplitude dust acoustic solitary waves is investigated in an unmagnetized dusty plasma comprising a negatively charged dust fluid, superthermal electrons, and nonthermal ions having a κ-vortex-like velocity distribution function. It is shown that due to electron and ion superthermality, the present dusty plasma model may support subsonic as well as supersonic electrostatic solitary waves involving cusped potential humps. The interplay between different concepts of nonisothermality is highlighted. Our results should help to understand the role that phase-space holes may play in reconnection by scattering and energizing particles and to provide an explanation of the intense solitary structures observed by the FAST mission.
The effects of dust temperature and trapped ions are incorporated in the study of dust-acoustic solitary waves. An energy integral equation involving the Sagdeev potential is derived, and the basic properties of large amplitude solitary structures are investigated. It is shown that the effects of dust temperature, resonant ions and equilibrium free electron density significantly change the regions of the existence of large amplitude solitary waves. Expanding the Sagdeev potential to include higher-order nonlinearities of electric potential, an exact steady state solution is also obtained which confirms the possibility of dust-acoustic soliton in the small amplitude limit. Furthermore, two asymptotic cases of the stationary solution are found which are related to the contribution of trapped ions.
The problem of nonlinear variable charge dust acoustic waves in a dusty plasma with trapped ions is revisited. The correct non-isothermal ion charging current is presented for the first time based on the orbit motion limited (OML) approach. The variable dust charge is then expressed in terms of the Lambert function and we take advantage of this new transcendental function to investigate nonlinear localized dust acoustic waves in a charge varying dusty plasma with trapped ions more rigorously. (c) 2008 Elsevier B.V. All rights reserved.
The nonextensivity in a non-isothermal plasma system with the Coulombian
long-range interactions is studied in the framework of Tsallis statistics. We
present for first time a mathematical expression of the nonextensive parameter
q based on the mathematical theory about the generalized Boltzmann equation and
the q-H theorem and the Maxwellian q-velocity distribution. We obtain a new
physical explanation for q concerning the nature of non-isothermal
configurations in plasma systems with Coulombian long-range interactions. We
also provide one illustration for Almeida theorem (Physica A 300(2001)424) from
the kinetic analyses of plasma systems, which means that Tsallis statistics
might be a suitable statistics for the description of a nonequilibrium system
with a temperature gradient in it.
New acoustic waves originating from a balance of dust particle inertia and plasma pressure are investigated. It is shown that these waves can propagate linearly as a normal mode in a dusty plasma, and non-linearly as supersonic solitons of either positive or negative electrostatic potential.