KdV Burgers Research Articles

Magnetoacoustic shocks with geometrical effects in spin−12 dense plasmas

The propagation characteristics of magnetoacoustic shock waves are investigated in an electron-ion dense magnetoplasma that accounts for spin−12 electrons and geometrical effects within the framework of a two-fluid quantum magnetohydrodynamic model. For this purpose, we have employed the reductive perturbation technique and derived small-amplitude planar Korteweg de Vries Burgers (KdVB) and cylindrical KdVB equations. Numerically, cylindrical KdVB equations are analyzed by choosing the plasma parameters consistent with compact astrophysical systems. It is observed that the density, magnetic field, and viscosity are the parameters that ascertain significant modifications in the structure and propagation of magnetoacoustic shock waves. The amplitude of the shock wave becomes larger in the case of cylindrical geometry and propagates faster than that of planar shock waves. Furthermore, the results are compared with analytical solutions in the limit of earlier times to show an excellent agreement of the results. However, the magnetization energy is found to mitigate the amplitude of shock structures in a dense magnetoplasma where quantum spin effects cannot be ignored.

Electron acoustic waves in a 2-electron, dissipative, quantum magneto plasma

A theoretical study is presented for the oblique propagation of linear and nonlinear electron acoustic waves (EAWs) in a two-electron population, dissipative, quantum magnetoplasma. The linear dispersion relation yields a complex plasma wave frequency $\omega$ , denoting a decaying wave, with the amount of damping directly related to the strength of the dissipative force. The small amplitude analysis for the nonlinear structures using the reductive perturbation technique, gives the Korteweg-de Vries-Burgers equation. The numerical solutions depict a soliton for negligible dissipation and a monotonic shock for relatively large dissipation. For intermediate values, the solutions are damped oscillations. The dispersion of the EAWs gets enhanced with increase in strength of the magnetic field and a higher value of the ratio of hot to cold electron density. The stability analysis is also carried out, and the corresponding phase portraits are plotted to observe the trajectories. Of the two critical points, one is a saddle point, hence always unstable, while the other is a stable focus or a stable node, depending on the relative strengths of the plasma parameters.

On the well-posedness and asymptotic behaviour of the generalized Korteweg–de Vries–Burgers equation

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p < 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.

Exact solutions of the Rosenau–Hyman equation, coupled KdV system and Burgers–Huxley equation using modified transformed rational function method

In this research, we study the exact solutions of the Rosenau–Hyman equation, the coupled KdV system and the Burgers–Huxley equation using modified transformed rational function method. In this paper, the simplest equation is the Bernoulli equation. We are not only obtain the exact solutions of the aforementioned equations and system but also give some geometric descriptions of obtained solutions. All can be illustrated vividly by the given graphs.

On the numerical evaluation for studying the fractional KdV, KdV-Burgers and Burgers equations

This paper is devoted to present an accurate numerical procedure to solve fractional (Caputo sense) Korteweg-de Vries, Korteweg-de Vries-Burgers and Burgers equations by using the spectral Chebyshev collocation method and finite difference method (FDM). The proposed problem is reduced to a system of ODEs with the help of the properties of Chebyshev polynomials of the third kind. This system is solved by using the FDM. Some theorems about the convergence analysis are stated and proved. A numerical simulation and a comparison with the previous work are presented.

KdV–Burgers equation in the modified continuum model considering the effect of friction and radius on a curved road

Considering the effect of friction and radius on a curved road, we propose an extended continuum model based on the full velocity difference model (FVDM, for short). Through the linear stability theory, the linear stability criterion of the continuum model is gained. By applying the nonlinear analysis, the KdV–Burgers equation is derived to describe the propagating behavior of traffic density wave near the neutral stability line. Numerical simulations show that traffic flow evolves into multiple local cluster, which corresponding to some particular traffic phenomenon with distinctive curved road factors. Numerical results indicate that the effect of friction and radius can suppress the traffic congestion and improve the stability of traffic flow efficiently.

Effect of Superthermal Polarization Force on Dust Acoustic Nonlinear Structures

Abstract An investigation of the dust acoustic shock waves as well as solitary waves in an unmagnetized dusty plasma consisting of fluid of negatively charged dust grains, superthermal ions, and Maxwellian electrons under the influence of superthermally modified polarization force is presented. The polarization force is significantly influenced by superthermal ions. Reductive perturbation technique has been used to derive the Korteweg-de Vries-Burgers equation. It is illustrated that the superthermal polarization force significantly alters the characteristics of the negative polarity shock and solitary waves. It is also examined that the soliton energy gets depleted by the influence of superthermal polarization force.

Analytical description of the structure of special discontinuities described by a generalized KdV–Burgers equation

Traveling wave solutions of a generalized KdV–Burgers equation are studied. The nonlinearity is specified as a piecewise linear flux function consisting of four parts. A boundary value shock-structure problem is solved. An analytical solution describing the structures of special discontinuities (undercompressive shocks) is obtained, and the behavior of these solutions is examined for various parameters of the problem.

Dust acoustic shock waves in magnetized dusty plasma

We have presented a theoretical study of the dust acoustic (DA) shock structures in a magnetized, electron depleted dusty plasma in the presence of two temperature superthermal ions. By deriving a Korteweg–de Vries–Burgers equation and studying its shock solution, we aim to highlight the effects of magnetic field and obliqueness on various properties of the DA shock structures in the presence of kappa-distributed two temperature ion population. The present model is motivated by the observations of Geotail spacecraft in the Earth's magnetotail and it is seen that the different physical parameters such as superthermality of the cold and hot ions, the cold to hot ion temperature ratio, the magnetic field strength, obliqueness and the dust kinematic viscosity greatly influence the dynamics of the shock structures so formed. The results suggest that the variation of superthermalities of the cold and hot ions have contrasting effects on both positive and negative polarity shock structures. Moreover, it is noted that the presence of the ambient magnetic field affects the dispersive properties of the medium and tends to make the shock structures less wide and more abrupt. The findings of present investigation may be useful in understanding the dynamics of shock waves in dusty plasma environments containing two temperature ions where the electrons are significantly depleted.

Nonlinear analysis for a modified continuum model considering driver’s memory and backward looking effect

Account for the “integral form of optimal velocity changes with memory” and “backward looking” effect, an extended car-following model is proposed in this paper. Through the relation between macro and micro variables, the extended car-following is transformed to a new continuum model. According to the linear stability theory, the linear stability’s condition of the presented model is obtained. The modified KdV–Burgers equation is established through non-linear analysis to describe the spreading behavior of the traffic density wave near the neutral stability line. The numerical simulation results show that the “backward looking” effect can improve the traffic flow’s stability and the “integral form of optimal velocity changes with memory” can relieve the traffic congestion.

Stationary Shock Waves with Oscillating Front in Dislocation Systems of Semiconductors

The paper presents a study of weakly nonlinear wave processes in the cylindrical region of a hole gas surrounding a negatively charged dislocation in an n-type semiconductor crystal. It is shown that shock waves propagating along the dislocation are the solutions of the Korteweg–de Vries–Burgers equation when the dispersion and dissipation of medium are taken into account. Estimates are obtained for the basic physical parameters characterizing the shock wave and the region inside the Reed cylinder.

Effect of Defects on the Spatial Localization of Nonlinear Acoustic Waves

A self-consistent mathematical model that includes equations of elasticity theory and kinetic equations for the density of different types of point defects is reduced to a nonlinear equation of evolution that combines the familiar Korteweg–de Vries–Burgers and Klein–Gordon equations of wave dynamics. Exact analytical solutions for this equation are found and analyzed.

New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations

In this paper, we extend the model of the Korteweg–de Vries (KDV) and Korteweg–de Vries–Burger’s (KDVB) to new model time fractional Korteweg–de Vries (TFKDV) and time fractional Korteweg–de Vries-Burger’s (TFKDVB) with Liouville–Caputo (LC), Caputo–Fabrizio (CF), and Atangana-Baleanu (AB) fractional time derivative equations, respectively. We utilize the q-homotopy analysis transform method (q-HATM) to compute the approximate solutions of TFKDV and TFKDVB using LC, CF and AB in Liouville–Caputo sense. We study the convergence analysis of q-HATM by computing the Residual Error Function (REF) and finding the interval of the convergence through the h-curves. Also, we find the optimal values of h so that, we assure the convergence of the approximate solutions. The results are very effective and accurate in solving the TFKDV and TFKDVB.

An extended macro model accounting for acceleration changes with memory and numerical tests

Considering effect of acceleration changes with memory, an improved continuum model of traffic flow is proposed in this paper. By applying the linear stability theory, we derived the new model’s linear stability condition. Through nonlinear analysis, the KdV–Burgers equation is derived to describe the propagating behavior of traffic density wave near the neutral stability line. Numerical simulation is carried out to study the extended traffic flow model, which explores how acceleration changes with memory affected each car’s velocity, density and fuel consumption and exhaust emissions. Numerical results demonstrate that acceleration changes with memory have significant negative effect on dynamic characteristic of traffic flow. Furthermore, research results verify that the effect of acceleration changes with memory will deteriorate the stability of traffic flow and increase cars’ total fuel consumptions and emissions during the whole evolution of small perturbation.

The nonlinear analysis for a new continuum model considering anticipation and traffic jerk effect

Considering the anticipation and traffic jerk simultaneously, a new macro model of traffic flow is put forwarded in this paper. By means of linear stability theory, the new model's linear stability with the effect of traffic jerk and anticipation term is obtained.Through nonlinear analysis, the KdV–Burgers equation is derived to describe the propagating behavior of traffic density wave near the neutral stability line. Numerical simulation is carried out to study the influence about the traffic jerk and anticipation effect. The evolution of traffic jam and corresponding energy consumption are then explored and compared each other. The outputs of these findings demonstrate the extended model can suppress traffic jam and reduce energy consumption in real traffic condition.

Nonlinear waves in layered media: Solutions of the KdV–Burgers equation

Abstract We use the KdV–Burgers equation to model a behaviour of a soliton which, while moving in non-dissipative medium encounters a barrier with dissipation. The modelling included the case of a finite width dissipative layer as well as a wave passing from a non-dissipative layer into a dissipative one. The dissipation results in reducing the soliton amplitude/velocity, and a reflection and refraction occur at the boundary(s) of a dissipative layer. In the case of a finite width barrier on the soliton path, after the wave leaves the dissipative barrier it retains a soliton form and a reflection wave arises as small and quasi-harmonic oscillations (a breather). The first order approximation in the expansion by the small dissipation parameter is studied.

Bifurcation Analysis for Dust-Acoustic Waves in a Four-Component Plasma Including Warm Ions

Investigation of linear dust-acoustic (DA) waves was analyzed in a system of collisionless, dissipative, and unmagnetized four-component dusty plasma consisting of cold positively and negatively charged particles of dust, with ions and electrons are of the Boltzmann distribution. The compatibility condition is obtained by using the normal mode analysis. The Korteweg de Vries–Burgers (KdV–Burgers) equation is obtained when the plasma system is nonlinearly analyzed by the reductive perturbation method. To study the nonlinear waves, we obtain some interesting physical solutions. These solutions are in the form of soliton, a combination between a shock and a soliton, and finally monotonic and oscillatory shock waves. Graphical illustration for these solutions was presented. The characteristics of the DA solitary and shock waves are significantly modified by the presence of positive and negative dust ions, the ratio of the ion to electron temperatures, and it is also found that the basic features are affected by the dust kinematic viscosity. The planar dynamical systems bifurcation theory, which was used to establish the existence of solitary wave solutions and periodic traveling wave solutions, has been established. Accordingly, the phase portrait topology and the potential diagram are illustrated for the KdV–Burgers equation. As a result of different phase orbits, there is an advantage to predict different classes of traveling wave solutions. The electric field is determined. Generalization of the obtained results in this paper can be used to investigate the nature of plasma waves in both laboratory and space.

Existence regimes for shocks in inhomogeneous magneto-plasmas having entropy

The finding of connection of plasma density and temperature with entropy gives an incitement to study different plasma models with respect to entropy. Nonlinear dissipative one- and two-dimensional structures (shocks) are investigated in nonuniform magnetized plasma with respect to entropy. The dissipation comes in the medium through ion-neutral collisions. The linear dispersion relation is derived. The Korteweg–deVries-Burgers and Kadomtsev-Petviashvili-Burgers equations are derived for nonlinear drift waves in 1-D and 2-D by employing the drift approximation. It is found that vd/u (vd is the diamagnetic drift velocity and u is the velocity of nonlinear structure) plays a significant role in the shock formation. It is also found that entropy has a significant effect on the strength of shocks. It is noticed that vd/u determines the rarefactive and compressive nature of the shocks. It is observed that upper and lower bounds exist for the shock velocity. It is also observed that the existing regimes for both one- and two-dimensional shocks for kappa distributed electrons are different from shocks with Cairns distributed electrons. Both rarefactive and compressive shocks are found for the 1-D drift waves with kappa distributed electrons. Interestingly, it is noticed that entropy enhances the strength of one- and two-dimensional shocks.

Bifurcations in the Generalized Korteweg–de Vries Equation

We study the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with periodic in the spatial variable boundary conditions. For various values of parameters, in a sufficiently small neighborhood of the zero equilibrium state we construct asymptotics of periodic solutions and invariant tori. Separately we consider the case when the stability spectrum of the zero solution contains a countable number of roots of the characteristic equation. In this case we state a special nonlinear boundary-value problem which plays the role of a normal form and determines the dynamics of the initial problem.

Whistler precursor and intrinsic variability of quasi-perpendicular shocks

The structure of a whistler precursor in a quasi-perpendicular shock is studied within two-fluid approach in one-dimensional case. The complete set of equations is reduced to the KdV equation, if no dissipation is included. With a phenomenological resistive dissipation the structure is described with the KdV–Burgers equation. The shock profile is intrinsically time dependent. For sufficiently strong dissipation, temporal evolution of a steepening profile results in generation of a stationary decaying whistler ahead of the shock front. With the decrease of the dissipation parameter, whistler wave trains begin to detach and propagate toward the upstream and the ramp is weakly time dependent. In the weakly dissipative regime the shock front experiences reformation.