Reductive Perturbation Method Research Articles

Ion acoustic soliton with thermal ions and non-thermal electrons in a high-relativistic electron-positron-ion plasma

In this investigation, both compressive and rarefactive solitons are shown to exist in electron-positron-ion plasma consisting of thermal positrons, non-thermal electrons, and high relativistic thermal ions. The reductive perturbation method (RPM) is used to derive the nonlinear Korteweg-de Vries (KdV) equation from the governing normalized basic set of equations. It is shown that only the fast ion-acoustic mode may produce the small amplitude rarefactive and compressive KdV solitons. It has been observed that the addition of variable ion species temperature dramatically alters the fundamental characteristics of amplitude and width. Moreover, the amplitude decreases as the ion to electron temperature ratio rises. The soliton's amplitude is also increased by the relativistic factor. High-relativistic electron-positron-ion plasmas are found in extreme astrophysical environments like pulsar magnetospheres, gamma-ray bursts, and active galactic nuclei. These plasmas’ wave and soliton activity can shed light on the dynamics of these astrophysical systems.

Consequence of head‐on collision and double‐layer soliton with linear analysis in multi‐component dusty plasmas

AbstractThe unmagnetized multicomponent collisional dusty plasmas have been considered to investigate the consequence of head‐on collision of dust‐ion‐acoustic waves (DIAWs) with nonlinear characteristic analysis of DIAWs and collisionless plasmas have been considered for studying the production of double layer (DL). The plasma system consists of positively charged ion fluid, negatively charged cold dust fluid, thermal electrons, q‐nonextensive electrons, and immobile background neutral particles. The extended Poincaré‐Lighthill‐Kuo (ePLK) method has been used to derive two‐sided damped Korteweg–De Vries (dKdV) equations and the reductive perturbation method has been used to derive modified KdV (mKdV) equation and the standard Gardner equation (SGE). Linear dispersion relation is derived using linear perturbation technique. The nonlinear propagation of DIA Gardner solitons (DIAGSs) and double layers (DLs) as well as phase shifts have been investigated for the effects of the concerned parameters. It is found that the DIA soliton structures are significantly modified by the species densities, dust mass, and superextensive parameter. The rarefactive dip shape solitary Gardner solitons become produced for the effects of concerned parameters and the positive (negative) DLs become produced in the range of () for the effect of temperatures on the species. The superextensive and the subextensive parameters of electrons significantly affect the DL soliton structures.

Effect of Cairns-Tsallis distribution on ion acoustic waves in interstellar medium

Ion acoustic waves (IAWs) in an electron–positron–ion (EPI) plasma are studied as affected by various plasma parameters, e.g., electron density, ion temperature, cold ion cyclotron frequency, nonextensive and nonthermal parameters. Plasma fluid equations have been applied to study the plasma system for which reductive perturbation method (RPM) and small-k expansion method are used to derive the relevant ZK and modified ZK (mZK) equations and the instability growth rate of IAWs, respectively. The effects of nonextensive and nonthermal parameters, external magnetic field, ion–electron temperature ratio and electron density on the dispersion relation, amplitude and width, energy and multi-dimensional instability of IAWs have been shown. It is found that the dispersion relation increases with the increasing nonthermality parameter, but decreases with the increasing nonextensivity parameter. Increasing nonextensive parameter leads to the decreasing soliton energy and the increasing growth rate. Increasing nonthermal parameter leads to the decreasing growth rate. These results will be helpful in the determination of plasma dynamics for electron–positron–ion plasma systems containing Cairns-Tsallis distributed electrons and positrons in interstellar.

Nonlinear dissipative wave trains in a system of self-propelled particles

The paper addresses the existence of modulated nonlinear periodic wave trains in a system of self-propelled particles (SPPs). The reductive perturbation method reduces the model hydrodynamics equations to a one-dimensional (1D) complex Ginzburg-Landau (CGL) equation. The modulational instability (MI) phenomenon is studied, where an expression for the instability growth rate is proposed. The latter is used to discuss regions of parameters where trains of solitonic waves are likely to be obtained. This is highly influenced by the values of the variances of Gaussian noise in self-diffusion and binary collision. Solutions for the CGL equations are also studied via the Porubov technique, using a combination of Jacobi and Weierstrass elliptic functions. Wave propagation in the self-propelled particles flock includes modulated nonlinear wave trains, nonlinear spatially localized periodic patterns, and continuous waves.

Bifurcation Analysis of Driver's Characteristics in Car-Following Model

Abstract According to traffic flow theory, traffic is affected not only by road conditions such as bottlenecks, the environment, interruptions, and so on but also by the driver's behavior. To control and manage increasingly complex traffic networks, it also becomes necessary to study the effects of driver characteristics significantly. In this research, a novel car-following model is proposed which considers both the driver's cautious and aggressive instincts for optimal and relative velocity integrals. To analyze the stability of the new model, a small perturbation method was used. Further, the modified Korteweg–de-Vries equations were established with the help of a reductive perturbation method. In bifurcation analysis, we examine the existence and stability of Hopf bifurcation in various systems. This helps to gain deeper insight into the behavior of these dynamical systems and can be used to develop more efficient control strategies. Numerical simulations and theoretical analyses both show that the aspects of the enhanced model related to driver characteristics have a major affect on traffic flow stability. Additionally, the model can adeptly handle traffic congestion and quickly return to its normal state if any disruption occurs.

(3+1)-dimensional cylindrical dust ion-acoustic solitary waves in dusty plasma

The basic properties of (3+1)-dimensional cylindrical dust ion-acoustic (DIA) solitary waves have been studied in a multi-component plasma medium consisting of inertial ions, Maxwellian electrons, and stationary negative dust particles. The reductive perturbation method has been used to derive a cylindrical (3+1)-dimensional Kadomtsev–Petviashvili equation for small but finite amplitude of dust ion-acoustic waves. The impacts of nonplanar geometry and presence of static negative dust particles on the dust ion-acoustic solitary structures have been investigated. DIA solitary waves with both positive and negative polarity is observed to generate for this existing multi-component and complex plasma medium. The changes in the DIA wave structure in radial, azimuthal, and axial coordinates over time have been presented analytically. The results of this present investigation may contribute to a better understanding of laboratory plasma events as well as nonlinear electrostatic excitations in astrophysical environments.

(2+1)-dimensional modulated nonlinear ion-acoustic soliton packets in magnetized electronegative plasma

The dynamics of propagation of (2+1)-dimensional ion-acoustic wave (IAW) packets in magnetized electronegative plasma made of negative ions with electrons, both in Boltzmann distribution and cold mobile positive ions is studied. For this purpose, the standard reductive perturbation method is used to derive the (2+1)-dimensional nonlinear Schrödinger (NLS) equation, which governs the modulation of positive IAW packets. The coefficients of the obtained NLS equation do not only depend on the system parameters specially the electronegative parameters such as the negative ion concentration ratio and the electron-to-negative ion temperature ratio (α and σn, respectively), but also on the magnetic field (ωp). The modulational instability is studied to analyze the areas of the emergence of waves. In order to describe the main characteristics of IAWs, the variational method is performed involving trial function solution of NLS equation and the exact (input profile) and numerical (output profile) solutions are also discussed.

Modulational instability of ion-acoustic waves in a dense quantum plasma

Modulational instability of ion-acoustic (IA) waves is investigated in an unmagnetized quantum plasma by adopting an equation of state for Fermi electron gas. To this end, a basic set of one-dimensional quantum hydrodynamic equations is reduced to a nonlinear Schrödinger equation using the reductive perturbation method. Quantum corrections are described using a nondimensional quantum parameter H proportional to the quantum diffraction parameter. Based on normalized electron temperature and chemical potential, new quantum criteria of modulational instability for IA waves are obtained and analyzed. Moreover, formation of rogue waves due to the modulational instability of IA waves and propagation of modulated IA packets in the form of bright and dark envelope solitons are examined for two different types of astrophysical plasma regions: white dwarf stars and warm dense matter.

Rogue waves dynamics of cubic–quintic nonlinear Schrödinger equation with an external linear potential through a modified Noguchi electrical transmission network

In this paper, we have considered a modified Noguchi nonlinear transmission network with a dispersive element. Using the reductive perturbation method in the semi-discrete limit, we show how wave dynamics in the network can be governed by a cubic–quintic nonlinear Schrödinger equation (CQ-NLSE) with an external linear potential. Through the modified Darboux transformation, we constructed the first, second and third order rogue wave solutions. Although the linear potential function Γ(τ) does not influence the instability criterion of waves propagating in the network, it has a significant effect on the dynamics of the amplitude, phase and shirp of rogue waves. Our results also reveal that it is possible both to predict the position where the amplitude of rogue waves reaches its maximum value and to reduce their spatial width.

Higher-order dust kinetic Alfvén wave solitons and quasi-periodic waves in a polarized dusty plasma

This investigation presents the study of dust kinetic Alfvén wave solitons (DKAWSs) and quasi-periodic waves in a polarized dusty plasma comprising inertial dust, Cairns–Tsallis (CT) distributed ions and electrons. Using the reductive perturbation method, the KdV equation is derived and the Tanh-method is employed to derive its solution to study the dust kinetic Alfvén solitary structures. Furthermore, by considering the contribution of higher-order effects, the KdV-type inhomogeneous equation is derived and its solution results in the formation of dressed DKAWSs. The combined effects of polarization force and other plasma parameters on the characteristics of different kinds of DKAWSs are analyzed. Most importantly, from the conservation of KdV equations in the presence of external periodic perturbations, the study of quasi-periodic wave has also been illustrated. The analysis shows that the periodic solitonic behavior is transformed to quasi-periodic and also approaches chaotic oscillations. By changing different physical parameters, the numerical analysis of nonlinear DKAWs from periodic to quasi-periodic and to chaotic behavior has also been presented with external perturbation. The findings of this investigation are useful to understand the insight into physics of nonlinear phenomena for studying the dynamics of DKAWSs in space and astrophysical plasma environments.

Homotopy time-fractional enhancement soliton solution for six component complex cometary plasma with thermal heavy ions

A dusty plasma system composed of three inertial species and three Kappa-distributed inertial-less species is considered. The hydrodynamic fluid equation used to describe investigated plasma system. The plasma system under consideration is a very rich system, so in this paper the effective Debye length and effective plasma frequency are considered to normalize all physical quantities. The Kortweg de-Veris equation is derived using the reductive perturbation method. Two envelope solitons (bright and dark) are propagating, and their appearance depends mainly on the values thermal parameter of heavy ions (oxygen ions) as well as the mass ratio for dust plasma. The homotopy perturbed transform method (HPTM) is used to solve the reduced time-fractional Korteweg-de Vries (TFKdV) equation. The effects of fractional parameters on the solution solutions are examined. Our results show that the shape of the solitary pulse is highly controlled by the fractional order, as well as the plasma parameters in addition to that the HPTM is very useful method, where few terms approximately three terms are enough to get successful approximate solution for fractional type equations.

Effect of Variable Dust Charge on Overtaking Collision of Multi-Solitons in Dusty Plasma

Abstract: The propagation and overtaking collision of dust acoustic multi-soliton in a dusty plasma consisting of kappa-distributed electrons, Boltzmann-distributed ions, and negatively charged dust grains has been investigated. In the Theoretical study, we use the reductive perturbation method to derive the Korteweg–De Vries (KdV) equation. Using the Hirota bilinear method, we have found four-soliton and five-soliton solutions of the KdV equation. It is observed that the dust charge variation coefficients are significantly affected by the spectral indices of the electrons. We discovered that plasma systems with variable-charged dust grains support the formation of solitons with greater amplitude. Our results help understanding the nonlinear phenomena that form in the magnetospheres of Saturn and Comet Halley as well as in interstellar dusty plasma. Keywords: Dusty plasma, Overtaking collision, Dust charge variation, Korteweg–De Vries (KdV) equation, Hirota bilinear method.

Dust‐ion‐acoustic shock wave excitations at super‐critical points with quartic nonlinearity

AbstractProgress in understanding the propagation of dust ion acoustic shock waves for the super critical values of any specific parameter which accompany an unmagnetized collisionless four‐component dusty multi‐ion nonextensive plasma is presented. To accomplish this goal, the formation of modified Burgers‐type equation having quartic nonlinearity via the reductive perturbation method and its analytical solution is yielded for the first time. It is found that the compressive electrostatic shocks are supported not only around the super critical values but also at the super critical values of the specific parameters. The presented results would be applicable to comprehend wave propagation in interplanetary plasmas and laboratory plasmas.

Existence of both compressional and rarefactive nonlinear solitary waves in a chain of dust particles

Not only the compressional dust acoustic solitary wave, but also the rarefactive one in a chain of the dust particles are verified by using the molecular dynamics simulation method. The compressional dust acoustic solitary wave is also obtained by using the traditional reductive perturbation method, while the rarefactive dust acoustic solitary wave solution is fitted by the numerical simulation results. The application scope of the traditional reductive perturbation method to derive compressional dust acoustic solitary waves in a chain of the dust particles is given. The application scope of the fitted results of the rarefactive dust acoustic solitary waves is also given.

Small amplitude dust acoustic solitary waves in a magnetized dusty plasma with nonthermal distribution for both electrons and ions

The propagation of dust acoustic solitary waves (DASWs) has been investigated in a magnetized dusty plasma consisting of negatively charged dust particles, nonthermal ions and nonthermal electrons. Using reductive perturbation method, the Korteweg-de-Vries (K-dV) equation is derived. It is found that the basic structures (such as polarity, amplitude, width) of DASWs are influenced by nonthermal ions and nonthermal electrons. The results show that both compressive and rarefactive DASWs exist in such dusty plasma. It is observed that by increasing the nonthermality parameter of ions and electrons, the nonlinear wave phase velocity increases. Furthermore, nonthermal distributed ions have more effect on the nonlinear wave phase velocity of the DASWs than nonthermal distributed electrons. The dependence of the angle that the external magnetic field makes with the propagation direction of wave on the amplitude and width of DASWs has been also investigated. It was concluded that this angle strongly affects the DASWs. Finally, the effect of the magnetic field appears only in the width and not in the amplitude of the DASWs. The findings of this study can be helpful in understanding the structures of DASWs in the dusty plasma in which both nonthermal ions and nonthermal electrons exist. Such structures can be found in different regions of space and astrophysics mediums such as the upper ionosphere and Saturn’s E-ring.

Nonlinear density waves on graphene electron fluids

The hydrodynamic behavior of charged carriers leads to nonlinear phenomena such as solitary waves and shocks, among others. As an application, such waves might be exploited on plasmonic devices either for modulation or signal propagation along graphene waveguides. We study the nature of nonlinear perturbations following an approach similar to Sagdeev potential analysis and also by performing the reductive perturbation method on the hydrodynamic description of graphene electrons, taking into consideration the effect of Bohm quantum potential and odd viscosity. Thus, deriving a dissipative Kadomtsev-Petviashvili-Burgers (KPB) equation for the bidimensional flow as well as its unidimensional limit in the form of Korteweg--de Vries--Burgers (KdVB). The stability analysis of these equations unveils the existence of unstable modes that can be excited and launched through graphene plasmonic devices.

Solitary wave solution of (3+1)-dimensional cylindrical Kadomstev–Petviashvili equation in warm opposite polarity dusty plasma

A theoretical investigation is done to study the propagation of dust-acoustic (DA) solitary waves in a dusty plasma system containing dynamic positively as well as negatively charged warm dust species and nonthermally distributed ion and electron species. The standard reductive perturbation method is employed to derive the (3+1)-dimensional cylindrical Kadomstev–Petviashvili (KP) equation (also known as cylindrical Korteweg–de Vries equation), which admits solitary wave solution for small but finite amplitudes. The parametric regimes for the existence of DA solitary waves are obtained. Two modes, namely, slow and fast are observed corresponding to different phase velocities. The slow mode has negative solitary pulses, whereas the fast mode contains both positive and negative DA solitary waves. It is shown that the ratio of positively charged dust to negatively charged dust number density significantly modifies the basic features of the DA solitary waves. The noteworthy effects of other parameters (viz. the nonthermal parameter and ion-to-electron temperature ratio) on the basic properties (namely, phase speed, polarity, amplitude, and width) of the DA solitary waves are also mentioned. The fundamental features and the underlying physics of the electrostatic solitary structures that are relevant to cometary tails, upper mesosphere, Jupiter’s magnetosphere, etc., are briefly discussed.

The jamming transition of multi-lane lattice hydrodynamic model with passing effect

Multi-lane environments are not uncommon on the highway. In multi-lane scenarios, vehicles can not only switch freely between lanes but also overtake each other. However, as basic driving behavior, the passing effect under multi-lane environments has been neglected in traffic flow research. To fill this gap, we present a modified multi-lane lattice hydrodynamic model considering the passing effect. Based on the reduction perturbation method, the corresponding stability norm of the proposed model is obtained, which reveals that the total number of lanes is positively correlated with the stability of traffic flow. When the stability condition does not hold, to further examine the formation and transmission process of traffic jams near the neutral stability curve, we carry out a nonlinear stability analysis of the model. The modified Korteweg-de Vries (mKdV) equation and the existence of the above mKdV equation are deduced, respectively. When the passing ratio is sufficiently small, i.e., the above existence holds, we observe that the jamming transition emerges between uniform flow and kink jam; once the passing ratio exceeds the threshold, the jamming transition occurs among uniform flow and kink-Bando traffic wave through the chaotic phase. We also conduct numerical simulations to verify the theoretical derivation.

Three-Dimensional Cylindrical Dust-Acoustic Solitary Pulses in Warm Nonthermal Plasma

The nonlinear propagation of 3-D cylindrical dust-acoustic (DA) solitary waves in an unmagnetized dusty plasma system consisting of mobile adiabatic positively charged dust (PCD) grains, nonthermal ions, and electrons is investigated. The basic properties of nonlinear DA solitary structures, which exist in such a warm dusty plasma medium, are studied. The ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3+1$</tex-math> </inline-formula> )-dimensional cylindrical Kadomstev–Petviashvili equation (cKPE) (also known as cylindrical Korteweg-de Vries (KdV) equation) is derived by using the reductive perturbation method (RPM). The possible region of the formation of solitons is shown. The effects of different plasma parameters (viz., the nonthermal parameter, the dust-to-ion temperature ratio, the ion-to-dust number density ratio, etc.) on the characteristics of the DA solitary waves are studied, and also the graphical representation of the cylindrical DA solitary structures is presented. The polarity of DA solitary waves changes with different plasma parameters (viz., the ion-to-electron temperature ratio, the nonthermality of ions and electrons, the ion-to-dust number density, etc.). These results may improve our understanding of DA solitary structures in space and laboratory plasma devices where nonthermal dusty plasma occurs.

On the propagation and interaction of ion-acoustic solitary, periodic, shock, and breather waves in a non-Maxwellian electron–positron–ion magnetoplasma

This investigation analyzes the propagation of nonlinear ion-acoustic waves (IAWs) in an unmagnetized, collisionless plasma composed of inertial positive ions and inertialess Maxwellian positrons as well as the inertialess non-Maxwellian electrons that obey (r, q)-distribution. To observe the impact of particle trapping on the nonlinear IAWs in an electron–positron–ion plasma, the Korteweg–De Vries (KdV) and modified KdV (mKdV) equations are derived using a reductive perturbation method. In the distribution function, the spectral parameters (r, q) put up their contribution to the flatness and high-energy tails, respectively. An important aspect of this investigation is the determination of well-known quasi-periodic solutions, multi-soliton solutions, breathers, and shocks under the variation of different physical parameters, especially spectral indices (r, q). Finally, the interaction of solitons is also presented for discussion of the complete profile. In addition, a detailed comparison, especially in a periodic wave, is made between the generalized (r, q)-distribution and the limiting cases of Kappa and Maxwellian distributions. The results presented in this study contribute to a better understanding of the characteristics of both high- and low-energy parts of the electron distribution function as well as the formation of periodic, soliton, multi-soliton, breathers, and shocks in space and astrophysical plasmas.