Reductive Perturbation Technique Research Articles

Evolution of Subsonic Shock Waves Associated with Reconnection Jets in Earth’s Magnetotail

Motivated by the signatures of nonlinear electrostatic waves observed by the Magnetospheric Multiscale spacecraft mission in reconnection jet regions of Earth's magnetotail, we have explored the dynamical features of ion-acoustic shock waves in the magnetotail. In this investigation, we have examined the dynamics and characteristics of ion-acoustic subsonic shock waves in non-Maxwellian space plasma comprising of two counterstreaming ion beams with suprathermal electrons, assumed to follow a kappa (κ) distribution. A reductive perturbation technique has been adopted to establish an evolution equation for small amplitude electrostatic shock structures. Importantly, subsonic waves only exist when the beam velocity exceeds a certain threshold, beyond which supersonic and subsonic waves may coexist. The combined effects of the beam velocity and the non-Maxwellian electron statistics have been analyzed to examine the characteristics of subsonic shock waves. Both symmetric and asymmetric (in relative beam density) models have been considered, leading to distinct possibilities in the evolution of subsonic shock waves. The findings of the investigation will help unfold the relatively unexplored dynamical characteristics of subsonic shock waves that may form and propagate in the magnetosphere.

Modified Korteweg–de Vries solitons with quartic nonlinearity in a dusty plasma

The present multicomponent dusty plasma with ions, Cairns distributed electrons and immobile dusts has been investigated first time through modified Korteweg–de Vries (mKdV) equation of quartic nonlinearity derived by reductive perturbation technique. In this new investigation, it is found that the dust ion-acoustic (DIA) solitary waves have smaller amplitudes compared to the amplitudes of mKdV-DIA compressive solitons of our previous investigation []. Roles of non-thermal parameter (β), dust to ion density ratio (σ), number of dust charge (Z d ) and initial streaming speed of ion (u i0) in the growth of amplitudes and widths of this new mKdV-DIA solitons are investigated.

Dust acoustic solitons in a positively charged dust plasma with regularized-κ distributed electrons in the presence of generalized polarization force

The propagation features of dust acoustic waves in a three-component plasma system composed of regularized Kappa distributed electrons, Maxwellian ions, and dust grains carrying positive charges are investigated. The reductive perturbation technique is employed to derive the KdV equation. A generalized expression for the polarization force is derived and the effect of the polarization force is taken into consideration as well. The bifurcation analysis is used, and the solitary wave solution was investigated. The critical value of the superthermal spectral index κ is introduced at which the solitonic structure turns up from rarefactive to compressive. It is found that in the range 0<κ<2.23, a rarefactive structure is obtained while the compressive structure appears for κ>2.23. In addition, it is found that by increasing the value of cutoff parameter α, the polarization strength increases too. All the obtained results are helpful to investigate the characteristics of the nonlinear wave propagating in the mesosphere region.

تخامد لانداو للأمواج السوليتونية الصوتية الغبارية في بلازما غبارية غير شاملة

تم دراسة الأمواج السوليتونية الصوتية الغبارية تحت تأثير تخامد لانداو للبلازما الغبارية الفضائية تحوي أيونات خاضعة لتوزع سرعة غير شامل. قمنا بالتحقيق في تأثير تخامد لانداو والبارمتر غير الشامل على البنى السوليتونية الصوتية الغبارية من خلال الحل العددي لمعادلة كورديفيك- دي فريس المعدلة بحد تخامد لانداو. باستخدام طريقة الاضطراب الاختزالية قمنا باستنتاج معادلة كورتيفيك – دي فريس مع إضافة حد تخامد لانداو. أظهرت هذه الدراسة أنً كثافة حبيبات الغبار تلعب دوراَ أساسياً في ظهور أو عدم ظهور الأمواج السوليتونية DA في البلازما الغبارية، والتأثير الواضح للخاصية غير الشاملة للأيونات على ظاهرة تخامد لانداو للبنى اللاخطية المتشكلة في البلازما الغبارية. تفيد نتائج هذه الدراسة في محاولة فهم الآلية الفيزيائية للانتشار الأمواج السوليتونية تحت تأثير تخامد لانداو في بيئات البلازما المغبرة غير الشاملة كالمناطق الكوكبية، والأغلفة المغناطيسية، وبيئات البلازما الفضائية.

Landau damping effects on dust ion-acoustic solitary waves in a nonthermal pair-ion plasma

The Landau damping effects on the nonlinear propagation of dust ion-acoustic solitary waves (DIASWs) are studied in an electron-free unmagnetized collisionless dusty pair-ion plasma. The two species of nonthermal ions (positive and negative) are described by the kinetic Vlasov equations, whereas the massive charged dust grains form only the background plasma. Using the multi-scale reductive perturbation technique generalized for the applications to the Vlasov equation, we derive a modified Korteweg–de Vries (KdV) equation that governs the evolution of DIASWs with the effects of linear resonance, in particular, linear Landau damping. The properties of the phase velocity and the Landau damping rate of DIASWs are studied with the effects of positive and negative-ion nonthermality parameters ( α p , α n ), the ratios of the positive to negative ion masses (m) and temperatures (T) as well as the dust to negative-ion number density ratio (δ). It is found that, depend on the degree of positive to negative ion mass ratios (m), both of damped and growing oscillations occur in a collisionless dusty pair-ion plasma due to the resonance phenomena between the wave and the nonthermal particles of the system. The properties of the decay rates of the amplitude of the KdV soliton with a small effect of Landau damping are also studied with the above system of parameters. The results may be useful for understanding the localization of solitary pulses and associated resonance damping and/or growing oscillations of the wave in laboratory and space plasmas, in which the number density of free electrons is much smaller than that of ions and highly charged heavy, micron seized dust grains.

Effect of trapping of electrons and positrons on the evolution of shock wave in magnetized plasma: A complex trapped K-dV burgers’ equation

Our research demonstrates that positive and negative ions, along with trapped electrons and positrons, propagate as small-amplitude electrostatic shock waves in a magnetized collisions-less plasma. The trapped Korteweg-de Vries Burger's (T-KdVB) equation was derived by the widely recognized standard reductive perturbation technique (RPT), employing two separate scales of standard coordinates. The study examines the impact of temperature ratio on the shock wave characteristics of positive and negative ions, as well as positrons with electrons. The analysis and discussion focus on the impacts of negative ion number densities, as well as the number densities of electrons and positrons. The T-KdVB equations are responsible for governing rarefactive shock waves. Additionally, it is important to acknowledge that nonlinearity and dissipation play a key role in controlling the amplitude of the shock.

Nonlinear behavior of dust acoustic periodic soliton structures of nonlinear damped modified Korteweg–de Vries equation in dusty plasma

In this article, the nonlinear plasma model known the damped modified Korteweg–de Vries equation under consideration on the base of analytical technique to know the novel behavior of this nonlinear model. We formulated the nonlinear plasma model by utilizing the RP (reductive perturbation) technique with basic set of fluid system which consisted on cold negative charge dust fluid, immobile background neutral charges, positive charge ion fluid, fast (ionized) electrons and thermally electrons. We also constructed various kinds of novel soliton solutions including kink wave solitons, periodic solitary waves, anti-kink wave solitons for the damped modified KdV equation. The determined soliton solutions are novel, interested and did not determined in the previous research. The physical interpretation of demonstrated solutions represent in contour, two-dim and three-dim with computational software. The study prove that our proposed approach is powerful, efficient and can also be fruitful for study of other nonlinear models.

Dynamic behavior of positron acoustic multiple-solitons in an electron–positron-ion plasma

We explore the dynamic characteristics of positron acoustic multiple-solitons in an unmagnetized plasma containing mobile cold positrons, Kappa-distributed superthermal hot electrons and positrons, and stationary positive ions. This study investigates the overtaking collisional effects, various parametric impacts, and phase shifts in the electron–positron-ion (e-p-i) plasma. Through the reductive perturbation technique, we derived the Korteweg-de Vries (KdV) equation and the modified Korteweg-de Vries (mKdV) equation. The multiple-soliton solutions (MSS) are then obtained using the simplified Hirota method and Cole-Holf transformation. The investigation shows that the amplitudes and widths of multi-solitons decrease with the increasing hot positron concentration but increase with the increasing hot electron index parameter. The results are expected to help us understand the dynamics of waves propagation in the pulsar magnetosphere, the active galactic nuclei, the quasar’s relativistic jet, the inner region of the accretion disk surrounding a black hole, etc.

On the dynamics of soliton interactions in the stellar environments

The effects of trapping of relativistically degenerate electrons are studied on the formation and interaction of nonlinear ion-acoustic solitary waves (IASWs) in quantum plasmas. These plasmas are detected in high-density astrophysical entities and can be created in the laboratory by interacting powerful lasers with matter. The formula for the number density of electrons in a state of relativistic degeneracy is provided, along with an analysis of the non-relativistic and ultra-relativistic scenarios. While previous studies have delved into specific aspects of relativistic effects, there needs to be a more detailed and systematic examination of the fully relativistic limit, which is essential for gaining a holistic perspective on the behavior of solitons in these extreme conditions. The aim of this work is to comprehensively investigate the fully relativistic limit of the system to fill this gap. The reductive perturbation technique is utilized to deduce the Korteweg–de Vries (KdV) equation, which is used to analyze the properties of the IASWs. Hirota bilinear formalism is applied to obtain single- and multi-soliton solutions for the KdV equation. The numerical analysis is focused on the plasma properties of the white dwarf in the ongoing investigation. The amplitude of the IASWs is found to be maximum for the non-relativistic, intermediate for the ultra-relativistic, and minimum for the fully relativistic limit. Most importantly, it is found that the fastest interaction occurs in the non-relativistic limit and the slowest in the fully relativistic limit.

Nonlinear structure and growth rate of solitary waves for extended Zakharov-Kuznetsov equation in plasma having (r, q) distribution electrons

The Reductive Perturbation Technique (RPT) is used to investigate ion acoustic waves (IAWs) in magnetized plasmas made up of electrons obeying the generalized form of ( r,q ) distribution. The analysis makes use of the Extended Zakharov-Kuznetsov (EZK) equation and the hydrodynamic theory of plasmas. The study investigates the effects of electron flatness ( r ) at low energy and superthermal ( q ) at high energy on the characteristics of IAWs, and the solutions to the EZK equation are plotted for different values of the plasma characteristics, such as the total ion cyclotron frequency, the ion to electron temperature ratio, and the ion-specific heat. The research goes into great detail to estimate the instability threshold and growth rate of these waves by applying a small-k perturbation technique. Overall, this investigation contributes to the better understanding of ion acoustic structures in space plasmas.

The role of κ‐deformed Kaniadakis distributed electrons on the dust ion‐acoustic waves in charge‐varying dusty plasma

AbstractThe one‐dimensional dynamics of nonlinear electrostatic dust‐ion acoustic waves (DIAWs), in a plasma containing Kaniadakis‐distributed electrons, fluid ions, and variable‐charge dust grains has been investigated. The κ−generalized electron charging current is derived based on the orbit‐motion limited approach. The reductive perturbation technique has been used to derive a KdV Burger equation of propagation. We have highlighted the dependence between the values of the dust particles charge Qd0 and the validity domain of the Kaniadakis parameter κ, where the Burger term C has positive. By integrating the KdV Burger equation numerically and assuming a laboratory dusty plasma, a transition from oscillatory to monotonic shock waves is observed with increasing values of κ. We have also found that the phase velocity of DIA shock waves decreases, and the dissipation can prevail over that of dispersion as the electrons move further away from thermal equilibrium (κ is increased). The coexistence of compressive and rarefactive of monotonic shock profile is observed. It is found that an increase in κ leads to an increase of the compressive monotonic shock wave and decrease the rarefactive monotonic shock wave.

Effect of trapping and reflection on dust acoustic solitary waves in nonthermal opposite polarity dust plasmas

The study of resonant wave-particle interactions (WPIs) is crucial in plasma systems where charged plasma particles interact via long-range electromagnetic waves. Our research focuses on exploring the impact of trapping and reflection, along with the superthermality of Kappa resonance electrons and ions, on the characteristics of dust acoustic waves (DAWs) in opposite polarity dust plasma (OPDP). Both linear and non-linear analyses were conducted. Two distinct types of dust acoustic modes, namely fast and slow, have been observed in the linear regime of two different instances of WPIs. Moving on to the non-linear regime, the Schamel KdV (SKdV) equation has been derived using the reductive perturbation technique. In both cases, a stationary solution in the form of a dust acoustic double-layer wave (DADLW) has been successfully obtained. Our findings are highly relevant to astrophysical plasma environments with non-thermal trapped and reflected particles.

Fast and slow magnetosonic shock waves in non-Maxwellian plasmas

Obliquely propagating nonlinear fast and slow magnetosonic wave modes in a hot non-Maxwellian dissipative plasma are investigated in the current work. Modified temperatures have been derived for the non-extensive Q- and (r,q)-distributions that correspond to the physical properties of such non-Maxwellian plasmas. The reductive perturbation technique has been employed to derive the linear dispersion relation (LDR) and Kadomstev-Petvashvilli-Burgers (KPB) equation for slow and fast magnetosonic wave modes in two dimensions. We then investigated the effect of non-extensive parameter Q, spectral indices (r,q) and kinematic viscosity ν on the LDR and nonlinear propagation of KPB shock profiles for both the slow and fast mode magnetosonic waves. We found that linear and nonlinear propagation of fast and slow magnetosonic wave modes have been considerably modified in such non-Maxwellian plasmas. The results presented here would depict a realistic picture of the propagation of linear and nonlinear fast and slow magnetosonic wave modes in non-Maxwellian plasmas.

Low-frequency nonlinear ion-acoustic cnoidal waves in a superthermal plasma with a monoenergetic electron beam

This study analyzes the behavior of nonlinear electrostatic ion-acoustic cnoidal waves (IACWs) in a magnetoplasma characterized by two distinct temperatures of superthermal electrons and a monoenergetic electron beam. For this purpose and based on fluid theory, the reductive perturbation technique is applied to reduce the basic equations to a third-order Korteweg-de Vries (KdV) equation. Under certain conditions, the KdV equation can be used for modeling symmetric CWs; when these conditions are not met, it is replaced by a Kawahara equation to describe these waves correctly. The symmetric IACWs features are examined in detail to determine the effect of pertinent plasma parameters. This study may help model nonlinear structures in astrophysical and space plasmas and understand the mechanism of CWs in the plasma sheath region.

Ion-acoustic cnoidal waves in a non-Maxwellian plasma with regularized κ-distributed electrons

This study examines how regularized kappa distributed (RKD) electrons affect the characteristics of the ion-acoustic cnoidal waves (IACWs) in normal plasma. The RKD is a generalized form of the standard kappa distribution that accounts for the impacts of thermal broadening and finite size effects on the plasma particles. By employing the reductive perturbation technique, the Korteweg–de Vries equation is derived, and its nonlinear cnoidal wave (CW) solution is obtained and analyzed both analytically and numerically. It is found that the amplitude of compressive IACWs increases with an enhanced cutoff parameter α and decreases with increasing superthermality κ. Rarefactive IACWs yield opposing results as compared to compressive IACWs under the impact of κ and α. The results provide insight into the behavior of CWs in normal plasmas with non-Maxwellian distributions and contribute to the understanding of wave-particle interactions in laboratory and space plasmas.

Effect of κ-deformed Kaniadakis distribution on the modulational instability of electron-acoustic waves in a non-Maxwellian plasma

In this paper, the modulational instability (MI) of the high-frequency electron-acoustic waves (EAWs) is reported in a non-Maxwellian plasma composed of two distinct types of electrons and stationary ions. One type of electrons is treated as a cold inertial fluid, whereas the other type is considered as inertialess species following κ-deformed Kaniadakis distribution. The fluid equations to the current model are reduced via a reductive perturbation technique to a nonlinear Schrödinger equation, which is then used to compute the MI and the growth rate of the EAWs. It is instructive to note that the deformation parameter (which develops the Kaniadakis entropy) and the hot-to-cold electron density ratio (hot electron concentration) significantly affect the conditions for MI. The modulated envelope black (dark and gray) solitons are investigated. The current results are beneficial in analyzing the spectrum of the cosmic rays, which violates manifestly the Boltzmann–Gibbs statistics. Moreover, the obtained results can be used to understand the mystery of many observations in stars where the presence of non-Maxwellian particles dominates.

Dust-ion-acoustic shock waves in the presence of (r, q) distributed electrons

A three-component dusty plasma system, with non-Maxwellian electrons, is investigated. The ions are considered to be mobile, the (spherical) dust is assumed to be stationary with charge- fluctuations, and the electrons follow the generalized velocity distribution function (VDF). The respective Burger’s equation, which manifests a shock structure, is derived by employing a reductive perturbation technique (RPT). It is deduced that dust charge fluctuations induce shock potential in a collisionless plasma system. The resultant perturbed potential is evaluated as affected by different plasma parameters, e.g. the ratios of electron to ion temperature (σ i ) and equilibrium densities (μ), and the non-thermal indices r and q. The shock amplitude is seen to decrease when the value of σ i is increased, and vice versa, whereas an opposite trend is observed for μ. It is found that for r = 0 and increasing values of spectral index q, amplitude of the potential distribution enhances while its width decreases. It is further noted that non-Maxwellian VDF significantly modifies the potential distribution. For the limiting case, namely, r = 0 and q → ∞ , the corresponding Maxwellian results are retrieved which benchmarks our findings. Present work will be useful in understanding the nonlinear propagation of dust-ion-acoustic shock waves in system where non-thermal population—as caused by various physical processes—of electrons are observed.

Archipelagos, islands, necklaces, and other exotic structures in external force-driven chaotic dusty plasmas

In this paper, the modified Kadomtsev Petviashvili (mKP) equation is derived by considering the dynamics of the dust-acoustic waves (DAWs) with kappa-distributed hot and cold ions and Boltzmannian electrons. The reductive perturbation technique is applied for deriving mKP. The bifurcation theory of the planar dynamical system is used to obtain the phase portrait of the DAWs in the framework of mKP equation. The formation of dust-acoustic solitary waves (DASWs) is analyzed for different plasma parameters in the Saturn’s magnetosphere. In the current plasma model, the external periodic force is introduced to study the quasiperiodic and chaotic behavior of the DAWs. It is noted that periodic initial conditions lead to the emergence of many outlandish features in the system by comparison with the solitary initial conditions. The frequency of the external periodic force ω plays an important role in the transition from quasiperiodic to chaotic behavior. Moreover, the strength of the external periodic force, the value of the nonlinear coefficient of the mKP equation, and the nonlinear acoustic speed are also found to have a significant effect on the chaotic behavior of the system.

The separation of one-soliton-shock to multi-soliton-shock of dust-ion acoustic wave using Lax pair and Darboux transformation of Burgers’ equation

Properties of dust-ion acoustic waves (DIAWs) in an unmagnetized dusty plasma consisting of mobile ions, superthermal electrons, and negatively charged dust clouds with charge fluctuation are investigated. Burgers’ equation is derived from the fluid equation of the plasmas by considering the reductive perturbation technique. The Lax pair of the evolution equation is obtained, and the characteristics of the DIAWs are investigated by employing the Darboux transformation. Some new waves (one-soliton, two-soliton-shocks, etc.) other than usual shocks are observed as the Darboux transformation is used through the Lax pair. It is also shown that all the physical plasma parameters have a significant effect on the structure of the nonlinear waves.

Obliquely propagating nonlinear magnetosonic waves in non-Maxwellian plasmas

In this paper, propagation characteristics of obliquely propagating nonlinear magnetosonic waves in hot nonthermal plasmas have been studied. The expressions of modified temperatures have been derived for non-Maxwellian Q-nonextensive and (r, q) distributions and then incorporated into the one-fluid magnetohydrodynamic model. By employing the reductive perturbation technique, we derived the linear dispersion relation (LDR) and nonlinear Kadomstev–Petvashvilli (KP) equation for slow and fast magnetosonic wave modes in two dimensions. We then investigated the LDR and nonlinear propagation of KP solitons for both the slow and fast mode magnetosonic waves and found that propagation characteristics are significantly altered by considering the effect of modified temperature. The results presented here would depict a realistic picture of the propagation of nonlinear magnetosonic waves in non-Maxwellian plasmas.