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In this tutorial, three-dimensional (3D) Cairns and Kappa-Cairns distribution functions are re-examined both analytically and numerically. The difference between one-dimensional (1D) and (3D) Cairns distribution functions (CDF) has been explained by deriving reduced 1D Cairns distribution function. It is noted that expressions of 1D and reduced 1D distributions such as Maxwellian and Kappa distributions are similar to each other, while the plots of 1D and reduced 1D CDF are significantly different from each other. The effect of non-thermality parameter on the 3D CDF is also studied by plotting it as a function of magnitude of the 3D velocity. It shows that the peak of the distribution function shifts toward higher speeds with an increase in the value of non-thermality parameter. The longitudinal dielectric response function is derived by using the 3D CDF for studying kinetic behavior of electrostatic waves in an unmagnetized isotropic plasma. The dielectric function is written in terms of plasma dispersion function and is then used to derive the dispersion relations and Landau damping increments for electron plasma waves, ion acoustic waves, and dust acoustic waves in a Cairns distributed plasma. The expressions of the dispersion relation and Landau damping rate of Cairns distributed plasma change into the corresponding expressions of the Maxwellian distributed plasma when the nonthermality parameter is taken equal to zero. Mathematical manipulations have been done for 3D Kappa-Cairns distribution function (KCDF) to find the correct normalization factor. An appropriate and valid range of values of the spectral index κ is obtained by calculating second moment of the velocity by integrating KCDF over 3D velocity space. The effect of non-thermality parameter and the spectral index κ on the 3D KCDF is studied by plotting it as a function of magnitude of the 3D velocity. It is also seen that the velocities where the tails of KCDF exists are much higher compared to the velocities where the tail of Kappa distribution occurs. For both 3D CDF and KCDF, it has been observed that the difference in the distribution functions becomes negligible when the value of the non-thermality parameter becomes more than 0.5.

The semiclassical hydrodynamic model is used to study the effect of electron exchange-correlation potential, quantum Bohm term, and degenerate pressure on the dynamics of dust ion acoustic waves by following the two-fluid theory in collisional, unmagnetized dusty plasma. For linear analysis, the dispersion relation modified by the exchange-correlation coefficient is derived. For nonlinear analysis, the standard perturbative approach is used to derive a deformed Korteweg–deVries equation with a linear damping term for finite amplitude waves. The analytical and numerical investigations in the presence of low collisional frequencies reveal the existence of compressive dissipative solitons. Considering the dense astrophysical objects, the dissipative compressive solitons are numerically investigated with the effect of different plasma parameters including collisions and exchange-correlation potential.

Using kinetic formulation with a quantum plasma dynamic approach, dust acoustic waves (DAWs) in three component non-relativistic, isotropic, unmagnetized, and collisionless quantum dusty plasma are analytically analyzed. Electrons are considered as quantum mechanical, following the Fermi–Dirac distribution, while ions and dust are considered as classical, obeying Maxwell–Boltzmann statistics. The longitudinal response function, in terms of susceptibility tensors, is derived and divided into two parts, i.e., dispersion relation (real part) and Landau damping (imaginary part). The dispersion relations and Landau damping rates are analyzed with the effect of the arbitrary degree of temperature degeneracy and quantum recoil. Two cases of the DAWs are discussed regarding the arbitrary degeneracy, which are nearly non-degenerate and nearly degenerate quantum plasmas. The dispersion relations and Landau damping rates for DAWs in both cases are examined analytically and numerically, using the data of semiconductor quantum wells and dense astrophysical plasmas.

An effective single fluid quantum magnetohydrodynamic model is used for multicomponent dusty plasma composed of degenerate electrons, ions, and nondegenerate dust species for the investigation of nonlinear circularly polarized dust Alfven waves in the presence of quantum corrected terms like Fermi and radiation pressures of electrons and ions as well as Jeans term that originates in a self-gravitating, dusty plasma. A set of Zakharovlike equations are derived and are used independently for soliton solutions, incorporating quantum terms and the Jeans term. Using the Sagdeev potential approach, it is observed that ordinary solitons may result in the absence of the Jeans term, whereas cusp solitons are obtained in the presence of the Jeans term. The variation of Sagdeev potential and its associated solitary structure for different parameters are discussed for both right and left handed circularly polarized dust Alfven waves. It is also observed that increasing the values of the magnetic field and decreasing the dust mass enhance the width of the cusp soliton, while the amplitude remains the same. The modulational instability of the system in view of both quantum and Jeans terms is also looked into, and different conditions of stability and instability for both right and left handed circularly polarized quantum dust Alfven waves are discussed.An effective single fluid quantum magnetohydrodynamic model is used for multicomponent dusty plasma composed of degenerate electrons, ions, and nondegenerate dust species for the investigation of nonlinear circularly polarized dust Alfven waves in the presence of quantum corrected terms like Fermi and radiation pressures of electrons and ions as well as Jeans term that originates in a self-gravitating, dusty plasma. A set of Zakharovlike equations are derived and are used independently for soliton solutions, incorporating quantum terms and the Jeans term. Using the Sagdeev potential approach, it is observed that ordinary solitons may result in the absence of the Jeans term, whereas cusp solitons are obtained in the presence of the Jeans term. The variation of Sagdeev potential and its associated solitary structure for different parameters are discussed for both right and left handed circularly polarized dust Alfven waves. It is also observed that increasing the values of the magnetic field and decreasing ...

The hybrid Cairns-Tsallis distribution is applied to study the linear and nonlinear characteristics of ion coupled drift waves in electron-positron-ion plasma with the effect of slow rotation. Using the parametric range approximation, a dispersion relation for ion acoustic waves is derived in the presence of density inhomogeneity. In the spatially limited region, the Zakharov–Kuznetsov-type equation in the nonlinear regime is derived for oblique drift solitary waves, which are then numerically investigated under the influence of the hybrid-distribution. It has been observed that the amplitude and width of spatially limited solitary waves strongly depend on both nonextensive-q and nonthermal parameter α. However, there exists a restricted range in the q and α parameters, above/below which the nonthermality and nonextensivity effects are considered to be ceased. The influence of the rotational frequency on the dispersion properties of the oblique drift solitary wave is also discussed. This study may provide a deep insight into the applicability of the Cairns-Tsallis distribution in plasmas having an excess of energetic particles in the presence of density inhomogeneity greatly observed in both space and laboratory environments.

Inertial Alfven solitary waves are investigated in an intermediate β ( α ≡ ( β 2 ) / Q < 1 ) quantum plasma, taking into account electron temperature degeneracy correction. We discuss the general coupling parameter Λ to find out the validity of the proposed model in nearly non-degenerate ξ ( = e υ k B T e ) ≪ 1 and nearly degenerate δ ( = T e T F ) ≪ 1 limiting cases. Employing the reductive perturbation technique, a Korteweg–de Vries (KdV) equation is derived for weakly nonlinear inertial Alfven solitons in the presence of nearly non-degenerate and nearly degenerate pressure corrections. It is found that in the presence of electron inertia and thermal pressure, the inertial Alfven solitary waves are accompanied by both hump and dip solitons. Moreover, the effects of the degeneracy parameter G ( = L i 3 2 ( − ξ ) L i 1 2 ( − ξ ) ) are reflected in the graphical analysis for both limiting cases. The significance of work to astrophysical plasmas is highlighted.

Nonlinear Alfven waves are studied in a fluid model for nonrelativistic, magnetized spin-1/2 quantum plasmas with an arbitrary degeneracy effect. Following a local Fermi-Dirac distribution function, a modified equation of state is utilized which is applicable to both classical and degenerate limits. Using the fluid equations for Hall magnetohydrodynamics with quantum corrections due to statistical effects, Bohm potential, spin magnetization energy, and temperature degeneracy, a set of modified Zakharov equations are derived for circularly polarized nonlinear Alfven waves. Ions are assumed to be cold, and the spin effects of electrons are incorporated through spin force along with spin magnetization current. A linear dispersion relation for finite amplitude Alfven waves duly modified by spin magnetization and arbitrary temperature degeneracy effects is also obtained. Employing the Sagdeev potential approach, the properties of Alfven solitary profiles in quantum plasmas with arbitrary degeneracy effects of electrons are analyzed. The amplitude of Sagdeev potential and of the associated soliton structure for both right and left-hand circularly polarized Alfven waves is observed to decrease with the decrease in the value of the arbitrary temperature degeneracy factor G for the case of the nearly degenerate limit. Similarly, it is found that the amplitude of Sagdeev potential and of the related solitary profile increases for both kinds of circular polarized Alfven waves with the increasing value of G in the case of the nearly non-degenerate limit.Nonlinear Alfven waves are studied in a fluid model for nonrelativistic, magnetized spin-1/2 quantum plasmas with an arbitrary degeneracy effect. Following a local Fermi-Dirac distribution function, a modified equation of state is utilized which is applicable to both classical and degenerate limits. Using the fluid equations for Hall magnetohydrodynamics with quantum corrections due to statistical effects, Bohm potential, spin magnetization energy, and temperature degeneracy, a set of modified Zakharov equations are derived for circularly polarized nonlinear Alfven waves. Ions are assumed to be cold, and the spin effects of electrons are incorporated through spin force along with spin magnetization current. A linear dispersion relation for finite amplitude Alfven waves duly modified by spin magnetization and arbitrary temperature degeneracy effects is also obtained. Employing the Sagdeev potential approach, the properties of Alfven solitary profiles in quantum plasmas with arbitrary degeneracy effects of ...

Linear and nonlinear kinetic Alfven waves (KAWs) are studied in collisionless, non-relativistic two fluid quantum magneto-plasmas by considering arbitrary temperature degeneracy. A general coupling parameter is applied to discuss the range of validity of the proposed model in nearly degenerate and nearly non-degenerate plasma limits. Linear analysis of KAWs shows an increase (decrease) in frequency with the increase in parameter ζ ( δ ) for the nearly non-degenerate (nearly degenerate) plasma limit. The energy integral equation in the form of Sagdeev potential is obtained by using the approach of the Lorentz transformation. The analysis reveals that the amplitude of the Sagdeev potential curves and soliton structures remains the same, but the potential depth and width of soliton structure change for both the limiting cases. It is further observed that only density hump structures are formed in the sub-alfvenic region for value K z 2 > 1. The effects of parameters ζ, δ on the nonlinear properties of KAWs a...

Low frequency electrostatic dust drift acoustic (DDA) waves are studied in an inhomogeneous dust magnetoplasma comprised of dust components of opposite polarity, Boltzmannian ions, and nonextensive distributed electrons. The magnetic-field-aligned dust sheared flow drives the electrostatic drift waves in the presence of ions and electrons. The sheared flow decreases or increases the frequency of the DDA wave, mostly depending on its polarity. The conditions of instability for this mode, with nonextensivity and dust streaming effects, are discussed. The nonlinear dynamics is then investigated for the DDA wave by deriving the Koeteweg-deVries (KdV) nonlinear equation. The KdV equation yields an electrostatic structure in the form of a DDA soliton. The relevancy of the work to laboratory four component dusty plasmas is illustrated.

Nonlinear circularly polarized Alfven waves are studied in Fermionic spin-1/2 quantum plasmas. Using the basic equations for Hall magnetohydrodynamics including quantum corrections, the set of Zakharov-like equations are obtained for circularly polarized nonlinear Alfven waves. In order to investigate the properties of the Alfven solitary structure in the presence of spin magnetization and quantum plasma beta, the Sagdeev potential approach is employed. For the case of right-handed circularly polarized Alfven waves, the amplitude of the Sagdeev potential and the associated solitary profile is observed to enhance with the increase of quantum plasma beta and magnetization energy due to electron spin-1/2 effects. However, it is found that the amplitude of the Sagdeev potential and the related solitary profile decrease with the increasing values of quantum plasma beta and magnetization energy for the case of left-handed circularly polarized Alfven waves. An increase in the width of the solitary structure is a...

By applying the kinetic theory of the Valsove-Poisson model and the reductive perturbation technique, a Korteweg-de Vries (KdV) equation is derived for small but finite amplitude ion acoustic waves in multi-ion plasma composed of positive and negative ions along with the fraction of electrons. A correspondent equation is also derived from the basic set of fluid equations of adiabatic ions and isothermal electrons. Both kinetic and fluid KdV equations are stationary solved with different nature of coefficients. Their differences are discussed both analytically and numerically. The criteria of the fluid approach as a limiting case of kinetic theory are also discussed. The presence of negative ion makes some modification in the solitary structure that has also been discussed with its implication at the laboratory level.

Low frequency electrostatic coupled dust drift dust acoustic waves are studied in an inhomogeneous, collisional four component dust magnetoplasma composed of dust components of opposite polarity, along with Boltzmannian ions and electrons. The nonlinear evolution equation in the form of an ordinary differential equation and its limiting cases are derived and solved using the Tanh-method. The numerical analysis of the obtained solutions is studied for both laboratory and cosmic plasma systems. It is observed that, depending on the values of the plasma parameters like ion and electron temperatures, and charge number, both rarefactive and compressive shock and solitary waves may exist. It is shown that the concepts of a critical ion and electron temperatures/density in the nonlinear equations treatment, and of a changeover from compressive to rarefactive shock and soliton characters, correspond to the formation of rarefactive regimes, at which the electric stresses maximize and density minimizes.

Ion-acoustic solitary waves (IASWs) are investigated in a plasma having a cold positron beam fluid, electrons following a vortex-like distribution with entropic index q, and dynamic ions. Using a standard procedure, a pseudo-potential energy equation is derived. The presence of nonextensive q - distributed trapped electrons and cold positron beam has been shown to influence the small amplitude soliton structure quite significantly. From the analysis of our results, it is shown that compressive IASWs are supported in this plasma model. As the real plasma situations are observed with plasma species having a relative flow, our present analysis should be beneficial for comprehending the electrostatic solitary structures observed in fusion plasma devices and positron winds observed in astrophysical plasmas.

A theoretical investigation is carried out for understanding the properties of electron-acoustic solitary waves (EASWs) in a beam plasma whose constituents are a cold beam electron fluid, hot nonextensive electrons obeying a vortex-like distribution with nonextensive factor q, and stationary ions. An energy integral (Schamel KdV) equation is derived by employing pseudo-potential (reductive perturbation) approach. The presence of nonextensive q-distributed hot trapped electrons and cold electron beam has been shown to influence the soliton structure quite significantly. The nonlinear dispersion relation is derived to analyze the dependency of the electron acoustic solitary wave quantities. From the analysis of our results, it is shown that the present plasma model supports the compressive EASWs. As the real plasma situations are observed with plasma species having a relative flow, so our present analysis should be useful for understanding the electrostatic solitary structures observed in the dayside auroral zone and other regions of the magnetosphere.

Dust magnetosonic waves are studied in degenerate dusty plasmas with spin and exchange correlation effects. Using the fluid equations of magnetoplasma with quantum corrections due to the Bohm potential, temperature degeneracy, spin magnetization energy, and exchange correlation, a generalized dispersion relation is derived. Spin effects are incorporated via spin force and macroscopic spin magnetization current. The exchange-correlation potentials are used, based on the adiabatic local-density approximation, and can be described as a function of the electron density. For three different values of angle, the dispersion relation is reduced to three different modes under the low frequency magnetohydrodynamic assumptions. It is found that the effects of quantum corrections in the presence of dust concentration significantly modify the dispersive properties of these modes. The results are useful for understanding numerous collective phenomena in quantum plasmas, such as those in compact astrophysical objects (e...

Linear properties of high and low frequency waves are studied in an electron-positron-ion (e-p-i) dense plasma with spin and relativity effects. In a low frequency regime, the magnetohydrodynamic (MHD) waves, namely, the magnetoacoustic and Alfven waves are presented in a magnetized plasma, in which the inertial ions are taken as spinless and non-degenerate, whereas the electrons and positrons are treated quantum mechanically due to their smaller mass. Quantum corrections associated with the spin magnetization and density correlations for electrons and positrons are re-considered and a generalized dispersion relation for the low frequency MHD waves is derived to account for relativistic degeneracy effects. On the basis of angles of propagation, the dispersion relations of different modes are discussed analytically in a degenerate relativistic plasma. Numerical results reveal that electron and positron relativistic degeneracy effects significantly modify the dispersive properties of MHD waves. Our present analysis should be useful for understanding the collective interactions in dense astrophysical compact objects, like, the white dwarfs and in atmosphere of neutron stars.

The propagation of ion-acoustic (IA) solitary and rogue waves is investigated in a two-dimensional ultrarelativistic degenerate warm dense plasma. By using the reductive perturbation technique, the cylindrical Kadomtsev–Petviashvili (KP) equation is derived, which can be further transformed into a Korteweg–de Vries (KdV) equation. The latter admits a solitary wave solution. However, when the frequency of the carrier wave is much smaller than the ion plasma frequency, the KdV equation can be transferred to a nonlinear Schrodinger equation to study the nonlinear evolution of modulationally unstable modified IA wavepackets. The propagation characteristics of the IA solitary and rogue waves are strongly influenced by the variation of different plasma parameters in an ultrarelativistic degenerate dense plasma. The present results might be helpful to understand the nonlinear electrostatic excitations in astrophysical degenerate dense plasmas.

Dust acoustic solitary waves in a dusty plasma containing dust of opposite polarity (adiabatic positive and negative dust), non-isothermal electrons and ions (following vortex like distribution) are theoretically investigated by employing pseudo-potential approach, which is valid for arbitrary amplitude structures. The propagation of small but finite amplitude solitary structures is also examined by using the reductive perturbation method. The basic properties of large (small) amplitude solitary structures are investigated by analyzing the energy integral (modified Korteweg-de Vries equation). It is shown that the effects of dust polarity, trapping of plasma particles (electrons and ions), and temperatures of dust fluids significantly modify the basic features of the dust-acoustic solitary structures that are found to exist in such an opposite polarity dust-plasma medium. The relevance of the work in opposite polarity dust-plasma, which may occur in cometary tails, upper mesosphere, Jupiter's magnetosphere, is briefly discussed.

Low frequency magnetosonic waves are studied in magnetized degenerate electron-positron-ion plasmas with spin effects. Using the fluid equations of magnetoplasma with quantum corrections due to the Bohm potential, temperature degeneracy, and spin magnetization energy, a generalized dispersion relation for oblique magnetosonic waves is derived. Spin effects are incorporated via spin force and macroscopic spin magnetization current. For three different values of angle θ, the generalized dispersion relation is reduced to three different relations under the low frequency magnetohydrodynamic assumptions. It is found that the effect of quantum corrections in the presence of positron concentration significantly modifies the dispersive properties of these modes. The importance of the work relevant to compact astrophysical bodies is pointed out.

Dust ion acoustic (DIA) solitons in an unmagnetized pair-ion (PI) plasmas with adiabatic pair-ions, non-isothermal electrons, and negatively charged background dust are investigated, using both small and arbitrary amplitude techniques. An energy integral equation involving the Sagdeev potential is derived, and basic properties of the large amplitude solitary structures are investigated. The effects of dust concentration, resonant electrons, and ion temperatures on the profiles of the Sagdeev potential and corresponding solitary waves are studied. The related Schamel-Korteweg-de Vries (S-KdV) equation with mixed-nonlinearity is derived by expanding the Sagdeev potential. Asymptotic solutions for different orders of nonlinearity are discussed for DIA solitary waves. The present work is applicable to understand the wave phenomena and associated nonlinear electrostatic perturbations in the doped pair ion plasmas, not completely filtered e.g., pair ion-electron plasmas, enriched with an extra massive charged component (e.g., dust defects), which may be academic for the moment but might be of interest for forthcoming experiments in laboratory (space) plasmas.

Linear and nonlinear coupled electrostatic drift and ion acoustic waves are studied in inhomogeneous, collisional pair ion–electron plasma. The Korteweg–de Vries–Burgers (KdVB) equation for a medium where both dispersion and dissipation are present is derived. An attempt is made to obtain exact solution of KdVB equation by using modified tanh–coth method for arbitrary velocity of nonlinear drift wave. Another exact solution for KdVB is obtained, which gives a structure of shock wave. Korteweg–de Vries (KdV) and Burgers equations are derived in limiting cases with solitary and monotonic shock solutions, respectively. Effects of species density, magnetic field, obliqueness, and the acoustic to drift velocity ratio on the solitary and shock solutions are investigated. The results discussed are useful in understanding of low frequency electrostatic waves at laboratory pair ion plasmas.