Modified Burgers Equation Research Articles

Cylindrical and Spherical Electron-Acoustic Shock Waves in Electron-Positron-Ion Plasmas with Nonextensive Electrons and Positrons

Electron-acoustic shock waves (EASWs) in an unmagnetized four-component plasma (containing hot electrons and positrons following the q-nonextensive distribution, cold mobile viscous electron fluid, and immobile positive ions) are studied in nonplanar (cylindrical and spherical) geometry. With the help of the reductive perturbation method, the modified Burgers equation is derived. Analytically, the effects of nonplanar geometry, nonextensivity, relative number density and temperature ratios, and cold electron kinematic viscosity on the basic properties (viz. amplitude, width, speed, etc.) of EASWs are discussed. It is examined that the EASWs in nonplanar geometry significantly differ from those in planar geometry. The results of this investigation can be helpful in understanding the nonlinear features of EASWs in various astrophysical plasmas.

Cylindrical and Spherical Ion-Acoustic Shock Waves in Nonextensive Electron-Positron–Ion Plasma

The properties of cylindrical and spherical ion-acoustic shock waves in an unmagnetized plasma system consisting of inertial ions, nonextensive electrons, and nonextensive positrons are investigated. A modified Burgers equation is derived using the reductive perturbation technique and its numerical solution is obtained. It is found that the properties of the cylindrical and spherical ion-acoustic shock waves significantly differ from those of 1-D planar shock waves. The findings of this investigation may be used in understanding the wave propagation in laboratory and space plasmas.

Ion-Scale Electrostatic Nonplanar Shock Waves in Dusty Plasmas with Two-Temperature Superthermal Electrons

The basic properties of nonplanar (viz. cylindri- cal and spherical) dust-ion-acoustic (DIA) shock waves in an unmagnetized dusty plasma system (consisting of inertial ions, negatively charged immobile dust, and superthermal electrons with two distinct temperatures) are investigated by employing the reductive perturbation method. The modified Burgers equation is derived and is numerically analyzed in order to examine the basic properties of DIA shock struc- tures. The effects of nonplanar geometry, electron superther- mality, and ion kinematic viscosity on the basic features of DIA shock waves are discussed. It is found that the prop- erties of the cylindrical and spherical DIA shock waves in dusty plasmas with two-temperature superthermal elec- trons significantly differ from those of one-dimensional planar shocks. The implications of our results in space plasmas (viz. star formation, supernovae explosion, solar wind, pulsar magnetosphere, Saturn's outer magnetosphere (R ∼ 13−18 RS ,w hereRS is the radius of Saturn), Saturn's inner magnetosphere (R <9 RS , etc.)) and laboratory plas- mas (viz. laser-induced implosion, capsule implosion, shock tube, etc.), where superthermal electrons with two distinct temperatures occurs, are briefly discussed.

An Approach with HaarWavelet Collocation Method forNumerical Simulations of Modified KdV and ModifiedBurgers Equations

An Approach with HaarWavelet Collocation Method forNumerical Simulations of Modified KdV and ModifiedBurgers Equations

Cylindrical and Spherical Ion-Acoustic Shock Waves in a Relativistic Degenerate Multi-Ion Plasma

A rigorous theoretical investigation has been made to study the existence and basic features of the ion-acoustic (IA) shock structures in an unmagnetized, collisionless multi-ion plasma system (containing degenerate electron fluids, inertial positively as well as negatively charged ions, and arbitrarily charged static heavy ions). This investigation is valid for both non-relativistic and ultra-relativistic limits. The reductive perturbation technique has been employed to derive the modified Burgers equation. The solution of this equation has been numerically examined to study the basic properties of shock structures. The basic features (speed, amplitude, width, etc.) of these electrostatic shock structures have been briefly discussed. The basic properties of the IA shock waves are found to be significantly modified by the effects of arbitrarily charged static heavy ions and the plasma particle number densities. The implications of our results in space and interstellar compact objects like white dwarfs, neutron stars, black holes, and so on have been briefly discussed.

Dissipative shock waves in all-normal-dispersion mode-locked fiber lasers

We propose an interpretation of the pronounced "M" spectral shape that is a recurrent feature in all-normal-dispersion mode-locked fiber laser dynamics. Our interpretation involves shock wave formation regularized by dissipation, modeled by a modified Burgers equation. The large fringes appearing at the edges of the spectrum result from discontinuities in the spectral phase.

Two different methods for numerical solution of the modified Burgers' equation.

A numerical solution of the modified Burgers' equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computing L 2 and L ∞ error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM.

Time-dependent non-planar DIA shock waves in non-extensive dusty plasma

AbstractThe propagation of dust ion-acoustic shock waves (DIASHWs) in an unmagnetized dissipative dusty plasma system consisting of inertial ions, non-inertial, non-extensive q-distributed electrons, and negatively charged stationary dust is investigated in bounded non-planar (cylindrical and spherical) geometry. A modified Burgers equation is derived and its numerical solution is obtained. It is found that the basic features of DIASHWs are significantly modified by the effects of electron non-extensivity and ion kinematic viscosity in bounded geometry. It is also shown that the propagation characteristics of non-planar DIASHWs in a non-extensive plasma are qualitatively different from those of planar ones.

A weakly non-linear theory for spiral density waves excited by accretion disc turbulence

We develop an analytic theory to describe spiral density waves propagating in a shearing disc in the weakly non-linear regime. Such waves are generically found to be excited in simulations of turbulent accretion discs, in particular if the said turbulence arises from the magneto-rotational instability (MRI). We derive a modified Burgers equation governing their dynamics, which includes the effects of non-linear steepening, dispersion and a bulk viscosity to support shocks. We solve this equation approximately to obtain non-linear sawtooth solutions that are asymptotically valid at late times. In this limit, the presence of shocks is found to cause the wave amplitude to decrease with time as t−2. The validity of the analytic description is confirmed by direct numerical solution of the full non-linear equations of motion. The asymptotic forms of the wave profiles of the state variables are also found to occur in MRI simulations, indicating that dissipation due to shocks plays a significant role apart from any effects arising from direct coupling to the turbulence.

Towards nonlinear stability of sources via a modified Burgers equation

Coherent structures are solutions to reaction–diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at x = ± ∞ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a Burgers-type equation as a model problem that captures some of the essential features of sources, we show how this phenomenon can be analysed and asymptotic nonlinear stability be established in this simpler context.

Numerical solutions of the modified Burgers’ equation by Petrov–Galerkin method

The modified Burgers’ equation (MBE) is solved numerically by the Petrov–Galerkin method using a linear hat function as the trial function and a cubic B-spline function as the test function. Product approximation has been used in this method. A linear stability analysis of the scheme shows it to be unconditionally stable. The accuracy of the presented method is demonstrated by two test problems. The numerical results are found in good agreement with the exact solutions.

A fourth-order numerical scheme for solving the modified Burgers equation

A finite-difference scheme based on fourth-order rational approximants to the matrix–exponential term in a two-time level recurrence relation is proposed for the numerical solution of the modified Burgers equation. The resulting nonlinear system, which is analyzed for stability, is solved using an already known modified predictor–corrector scheme. The results arising from the experiments are compared with the corresponding ones known from the available literature.

Propagation of nonlinear acoustic plane waves in an elastic gas-filled tube

This paper deals with modeling of nonlinear plane acoustic waves propagating through an elastic tube filled with thermoviscous gas. A description of the interactions between gas and an elastic tube wall is carried out by the continuity equation of a wall velocity. Simplification on the basis of the local reaction assumption enables to model an acoustic treatment on the tube wall by using a wall impedance. Because there are considerable losses due to wall friction, the influences of the acoustic boundary layer were also considered. Using certain assumptions a special form of the Burgers equation was derived which enables to describe the propagation of nonlinear waves in the elastic tube. This model equation takes into account nonlinear, dissipative, and dispersion effects which compete each other. Characteristic lengths of the supposed effects and numerical results with respect to the source frequency were used for a qualitative analysis of the model equation. Applicability of this model equation was demonstrated by series of measurements. By application of the long-wave approximation the Korteweg-de Vries-Burgers and Kuramoto-Sivashinsky equations were derived from the modified Burgers equation.

Cylindrical and spherical dust-acoustic shock waves in a strongly coupled dusty plasma

Cylindrical and spherical dust-acoustic (DA) shock waves propagating in a strongly coupled dusty plasma are theoretically investigated. The generalized hydrodynamic model, in which highly charged dust are treated as strongly coupled, but electrons and ions are treated as weakly coupled, is employed. The modified Burgers equation, which is derived by using the reductive perturbation technique, is numerically solved to examine the effects of nonplanar cylindrical and spherical geometries on the basic features of the DA shock waves that are formed due to a strong correlation among highly negatively charged dust. It is shown that the effects of nonplanar cylindrical and spherical geometries significantly modify the basic properties of the DA shock structures. The implications of our results in laboratory dusty plasma experiments are briefly discussed.

Nonlinear spherically divergent shock waves propagating in a relaxing medium

The propagation of nonlinear spherically diverging N-waves in atmosphere was studied experimentally and theoretically. The relative effects of nonlinear, dissipation, and relaxation phenomena on the N-wave duration and amplitude were investigated based on the numerical solutions of the modified Burgers equation. It is shown that, under the experimental conditions, the duration of a pulse increases mainly due to nonlinear propagation, whereas the amplitude depends on the combined effects of nonlinearity, dissipation, and relaxation. The frequency response of the measuring system is obtained. The calibration of the amplitude and duration of the experimental waveforms is performed based on the nonlinear lengthening of the propagating pulse. The results of numerical modeling show good agreement with experimental data.

Nonlinear shear wave interaction in soft solids

This paper describes nonlinear shear wave experiments conducted in soft solids with transient elastography technique. The nonlinear solutions that theoretically account for plane and nonplane shear wave propagation are compared with experimental results. It is observed that the cubic nonlinearity implied in high amplitude transverse waves at f(0)=100 Hz results in the generation of odd harmonics 3f(0), 5f(0). In the case of the nonlinear interaction between two transverse waves at frequencies f(1) and f(2), the resulting harmonics are f(i)+/-2f(j)(i,j=1,2). Experimental data are compared to numerical solutions of the modified Burgers equation, allowing an estimation of the nonlinear parameter relative to shear waves. The definition of this combination of elastic moduli (up to fourth order) can be obtained using an energy development adapted to soft solid. In the more complex situation of nonplane shear waves, the quadratic nonlinearity gives rise to more usual harmonics, at sum and difference frequencies, f(i)+/-f(j). All components of the field have to be taken into account.

Asymptotic Solution for a Kind of Boundary Layer Problem

This paper studies the partial differential equation with a small coefficient in the highest-order item. This kind of equation is also named as boundary layer problem. The Burgers equation and modified Burgers equation are analyzed in this approach. First, these equations are transferred into the strong nonlinear ones, and then the corresponding strong nonlinear equations are solved based on the perturbation method. The results from the asymptotic method are comparable with those obtained from numerical computation.

Numerical treatment for the modified burgers equation

In this paper, we consider the solution of the modified Burger's equation by using the collocation method with quintic splines. Applying the Von-Neumann stability analysis method we show that the proposed method is unconditionally stable. By conducting a comparison between the absolute error for our numerical results and the analytic solution of the modified Burger's equation we will test the accuracy of the proposed method.

Large‐Time Asymptotics for Periodic Solutions of the Modified Burgers Equation

In this paper, we construct large‐time asymptotic solution of the modified Burgers equation with sinusoidal initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study.

Conte Truncated Expansion and Applications

In the special Conte truncated expansion approach one obtains different solutions of the Prigogine–Lefever equation by use of various solutions of a type of Riccati equation, including the periodic soliton solutions and singular soliton solutions. In order to acquire conveniently the soliton solutions of the Boussinesq equation, a proper transformation is applied. Using the special Conte truncated expansion approach yields the known bell-shape solutions and some new soliton solutions like cot2 × sec2, tan2 × c sec2, tanh2 × sech2, etc. We also study the soliton solutions of the modified Burgers equation (MBE). Using leading term analysis, we find the exponent is a fraction, i.e., −\( \frac{1}{2} \). Therefore, the special Conte truncated expansion approach cannot be used directly. A transformation is first made to them another form of the MBE. Various soliton solutions of MBE are then presented, including the periodic solutions and singular soliton solutions.