Magnetosonic Shocks in Ultra-Relativistic Dissipative Degenerate Plasmas

Magnetosonic shock structures in dissipative magnetized degenerate electron ion plasma are studied.The two fluid quantum magnetohydrodynamic equations for non-degenerate ions and ultra-relativistic degenerate electron fluids with the Maxwell equations are presented.Using the reductive perturbation technique the Korteweg de Vries Burgers(KdVB)equation is derived and its solution is presented with the tanh method.Astrophysical plasma parameters are used to study the effects of variation of plasma density,magnetic held intensity and kinematic viscosity on the propagation characteristics of nonlinear shock structures in such plasma systems.     

Magnetosonic shock waves in dense astrophysical electron–positron–ion plasmas

Multifluid quantum magnetohydrodynamic model (QMHD) is used to investigate small but finite amplitude magnetosonic shock waves in dense) electron–positron–ion (e–p–i) plasmas. The Korteweg–de Vries–Burgers (KdVB) equation is derived by using reductive perturbation method. It is noticed that variations in the positron density modify the profile of magnetosonic shocks in dense e–p–i plasmas significantly. The numerical results are also presented by taking into account the dense plasma parameters from published literature of astrophysical conditions, in compact stars.     

Ion-acoustic shock waves in multi-electron temperature collisional plasma

The nonlinear propagation of ion-acoustic waves in a collision-dominated double electron temperature plasma is considered. Accounting for the ion viscosity and the ion heat conductivity, it is shown by means of two-warm fluid equations that the nonlinear evolution of the ion-acoustic waves is governed by the Korteweg—de Vries—Burgers equation. Stationary shock solution of the KdV—Burgers equation is presented.     

New exact solutions of Burgers’ type equations with conformable derivative

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.     

Analysis of the stability and density waves for traffic flow

In this paper, the optimal velocity model of traffic is extended to take into account the relative velocity. The stability and density waves for traffic flow are investigated analytically with the perturbation method. The stability criterion is derived by the linear stability analysis. It is shown that the triangular shock wave, soliton wave and kink wave appear respectively in our model for density waves in the three regions: stable, metastable and unstable regions. These correspond to the solutions of the Burgers equation, Korteweg－de Vries equation and modified Korteweg－de Vries equation. The analytical results are confirmed to be in good agreement with those of numerical simulation. All the results indicate that the interaction of a car with relative velocity can affect the stability of the traffic flow and raise critical density.     

Modified (2+1)-dimensional displacement shallow water wave system and its approximate similarity solutions

Recently, a new (2+1)-dimensional shallow water wave system, the (2+1)-dimensional displacement shallow water wave system (2DDSWWS), was constructed by applying the variational principle of the analytic mechanics in the Lagrange coordinates. The disadvantage is that fluid viscidity is not considered in the 2DDSWWS, which is the same as the famous Kadomtsev—Petviashvili equation and Korteweg—de Vries equation. Applying dimensional analysis, we modify the 2DDSWWS and add the term related to the fluid viscidity to the 2DDSWWS. The approximate similarity solutions of the modified 2DDSWWS (M2DDSWWS) is studied and four similarity solutions are obtained. For the perfect fluids, the coefficient of kinematic viscosity is zero, then the M2DDSWWS will degenerate to the 2DDSWWS.     

The element-free Galerkin method of numerically solving a regularized long-wave equation

The element-free Galerkin (EFG) method is used in this paper to find the numerical solution to a regularized long-wave (RLW) equation. The Galerkin weak form is adopted to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. The effectiveness of the EFG method of solving the RLW equation is investigated by two numerical examples in this paper.     

Effects of double spectral electron distribution and polarization force on dust acoustic waves in a negative dusty plasma

The nonlinear features of dust acoustic waves (DAWs) propagating in a multicomponent dusty plasma with negative dust grains, Maxwellian ions, and double spectral electron distribution (DSED) are investigated. A Korteweg de Vries Burgers equation (KdVB) is derived in the presence of the polarization force using the reductive perturbation technique (RPT). In the absence of the dissipation effect, the bifurcation analysis is introduced and various types of solutions are obtained. One of these solutions is the rarefactive solitary wave solution. Additionally, in the presence of the dissipation effects, the tanh method is employed to find out the solution of KdVB equation. Both of the monotonic and the oscillatory shock structures are numerically investigated. It is found that the correlation between dissipation and dispersion terms participates strongly in creating the dust acoustic shock wave. The limit of the DSED to the Maxwell distribution is examined. The distortional effects in the profile of the shock wave that result by increasing the values of the flatness parameter, r, and the tail parameter, q, are investigated. In addition, it has been shown that the proportional increase in the value of the polarization parameter R enhances in both of the strength of the monotonic shock wave and the amplitude of the oscillatory shock wave. The effectiveness of non-Maxwellian distributions, like DSED, in several of plasma situations is discussed as well.     

Radially ingoing and outgoing dispersive and dissipative structures in electron‐ion plasmas in the presence of electron trapping

In this article, non‐linear propagation of ingoing and outgoing electrostatic waves on the ion time scale in an unmagnetized, non‐relativistic electron‐ion (ei) plasma in the presence of warm ions, ion kinematic viscosity, and trapped Maxwellian electrons was examined in a non‐planar geometry. In the weak non‐linearity limit, modified soliton and shock equations were derived with the inclusion of electron trapping in cylindrical and spherical geometries. The finite difference method was used to solve all these equations in the non‐planar geometries using the planar versions of these equations as an initial input. The results were compared with their counterparts with quadratic non‐linearity and the main differences were expounded. It was shown that the spatio‐temporal scales over which the shocks form for the non‐planar trapped Burgers equation are much shorter by comparison with the shocks admitted by the non‐planar trapped Korteweg de Vries Burgers equation. It was also found that unlike their non‐linear shock counterparts, the solitary structures admitted by the non‐planar trapped Korteweg de Vries equation exhibit a phase shift.     

An improved element-free Galerkin method for solving generalized fifth-order Korteweg-de Vries equation

In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.     

An improved element-free Galerkin method for solving the generalized fifth-order Korteweg–de Vries equation

In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.     

A study of solitary waves by He's semi-inverse variational principle

The dynamics of solitary waves in the presence of perturbation terms is studied in this paper with the aid of the semi-inverse variational principle. In this paper, shallow water waves as well as internal gravity waves in a density-stratified ocean are considered. These are respectively modeled by the Korteweg–de Vries equation as well as the compound Korteweg–de Vries equation. An analytical solution of the solitary wave is found in each case.     

扰动KdV方程孤子的同伦映射解

利用同伦映射方法研究了一类非线性KdV(Korteweg de Vries)方程. 首先引入一个同伦变换,使相应的方程求孤子解问题转化为映射变换问题.然后利用映射特性得到了原方程孤子的近似解. 关键词： 孤子 扰动 同伦映射     

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.     

Modified Burgers equation by local discontinuous Galerkin method

In this paper, we present the local discontinuous Galerkin method for solving Burgers’ equation and the modified Burgers’ equation. We describe the algorithm formulation and practical implementation of the local discontinuous Galerkin method in detail. The method is applied to the solution of the one-dimensional viscous Burgers’ equation and two forms of the modified Burgers’ equation. The numerical results indicate that the method is very accurate and efficient.     

Periodic and chaotic behaviour in a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations

The dynamical behaviour of a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations (extension of the Korteweg–de Vries equation to two space variables) are studied in this paper. Harmonic solutions of non-resonance and primary resonance are obtained using the perturbation method. Chaotic motion under harmonic excitations is studied using the Melnikov method.A wide range of solutions for the reduced perturbed generalized Korteweg–de Vries equations, in which non-linear phenomena appearing within transition from regular harmonic response (periodic solutions) to chaotic motion, are obtained using the time integration Runge–Kutta method. When chaos is found, it is detected by examining the phase plane, the Poincaré map, the sensitivity solution of the solution to initial conditions, and by calculating the largest Lyapunov exponent.     

Bifurcation analysis for ion acoustic waves in a strongly coupled plasma including trapped electrons

The nonlinear ion acoustic wave propagation in a strongly coupled plasma composed of ions and trapped electrons has been investigated. The reductive perturbation method is employed to derive a modified Korteweg–de Vries–Burgers (mKdV–Burgers) equation. To solve this equation in case of dissipative system, the tangent hyperbolic method is used, and a shock wave solution is obtained. Numerical investigations show that, the ion acoustic waves are significantly modified by the effect of polarization force, the trapped electrons and the viscosity coefficients. Applying the bifurcation theory to the dynamical system of the derived mKdV–Burgers equation, the phase portraits of the traveling wave solutions of both of dissipative and non-dissipative systems are analyzed. The present results could be helpful for a better understanding of the waves nonlinear propagation in a strongly coupled plasma, which can be produced by photoionizing laser-cooled and trapped electrons [1], and also in neutron stars or white dwarfs interior.     

Exact solutions of stochastic fractional Korteweg de–Vries equation with conformable derivatives

We deal with the Wick-type stochastic fractional Korteweg de–Vries(KdV) equation with conformable derivatives.With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.     

An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)

A generalized Fisher equation(GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance.The exact mathematical result of the GFE has been widely used in population dynamics and genetics,where it originated.Many researchers have studied the numerical solutions of the GFE,up to now.In this paper,we introduce an element-free Galerkin(EFG) method based on the moving least-square approximation to approximate positive solutions of the GFE from population dynamics.Compared with other numerical methods,the EFG method for the GFE needs only scattered nodes instead of meshing the domain of the problem.The Galerkin weak form is used to obtain the discrete equations,and the essential boundary conditions are enforced by the penalty method.In comparison with the traditional method,numerical solutions show that the new method has higher accuracy and better convergence.Several numerical examples are presented to demonstrate the effectiveness of the method.     

Element-free Galerkin （EFG） method for a kind of two-dimensional linear hyperbolic equation

The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin (EFG) method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.