KdV-Burgers Equation Research Articles

Generalized Sagdeev potential theory for shock waves modeling

In this paper, we develop an innovative approach to study the shock wave propagation using the Sagdeev potential method. We also present an analytical solution for Korteweg de Vries Burgers (KdVB) and modified KdVB equation families with a generalized form of the nonlinearity term which agrees well with the numerical one. The novelty of the current approach is that it is based on a simple analogy of the particle in a classical potential with the variable particle energy providing one with a deeper physical insight into the problem and can easily be extended to more complex physical situations. We find that the current method well describes both monotonic and oscillatory natures of the dispersive-diffusive shock structures in different viscous fluid configurations. It is particularly important that all essential parameters of the shock structure can be deduced directly from the Sagdeev potential in small and large potential approximation regimes. Using the new method, we find that supercnoidal waves can decay into either compressive or rarefactive shock waves depending on the initial wave amplitude. Current investigation provides a general platform to study a wide range of phenomena related to nonlinear wave damping and interactions in diverse fluids including plasmas.

A Multi-Domain Bivariate Pseudospectral Method for Evolution Equations

In this paper, we present a new general approach for solving nonlinear evolution partial differential equations. The novelty of the approach is in the combination of spectral collocation and Lagrange interpolation polynomials with Legendre–Gauss–Lobatto grid points to descritize and solve equations in piece-wise defined intervals. The method is used to solve several nonlinear evolution partial differential equations, namely, the modified KdV–Burgers equation, modified KdV equation, Fisher’s equation, Burgers–Fisher equation, Burgers–Huxley equation and the Fitzhugh–Nagumo equation. The results are compared with known analytic solutions to confirm accuracy, convergence and to get a general understanding of the performance of the method. In all the numerical experiments, we report a high degree of accuracy of the numerical solutions. Strategies for implementing various boundary conditions are discussed.

KdV–Burgers equation in a new continuum model based on full velocity difference model considering anticipation effect

In this paper, a new continuum model based on full velocity difference car following model is developed with the consideration of driver’s anticipation effect. By applying the linear stability theory, the new model’s linear stability 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 shows that the new model possesses the local cluster, and it is capable of explaining some particular traffic phenomena Numerical results show that when considering the effects of anticipation, the traffic jams can be suppressed efficiently. The key improvement of this new model is that the anticipation effect can improve the stability of traffic flow.

A new continuum model based on full velocity difference model considering traffic jerk effect

In this paper, a new continuum model is developed based on full velocity difference car-following model, which takes the traffic jerk effect into account. The critical condition for traffic flow is derived, and density waves occur in traffic flow because of the small disturbance. Near the neutral stability line, nonlinear analysis is taken to derive the KdV–Burgers equation for describing the density wave, and one of the solutions is given. Numerical simulation is carried out to show the local traffic described by the model.

On nonlinear superposition of the KdV–Burgers shock waves and the behavior of solitons in a layered medium

Superposition of explicit (analytic) monotone non-increasing shock waves for the KdV–Burgers equation is studied. Initial profile chosen as a sum of two such shock waves gradually transforms into a single shock wave of a somewhat complex yet qualitatively predictable structure. The layered media consist of layers with both dispersion and dissipation and layers with dispersion but without dissipation. In the latter case the waves are described by the KdV equation, while in the former — by the Kdv–Burgers one. A soliton solution of the KdV equation meeting a layer with dissipation transforms somewhat similarly to a ray of light in air crossing semi-transparent plate. Both nonlinear superposition and the behavior of solitons are demonstrated in detail, modelled numerically and graphically presented.

Potential Symmetries, Lie Transformation Groups and Exact Solutions of Kdv-Burgers Equation

Potential Symmetries, Lie Transformation Groups and Exact Solutions of Kdv-Burgers Equation

An extended continuum model accounting for the driver's timid and aggressive attributions

Considering the driver's timid and aggressive behaviors simultaneously, a new continuum model is put forwarded in this paper. By applying the linear stability theory, we presented the analysis of new model's linear stability. Through nonlinear analysis, the KdV–Burgers equation is derived to describe density wave near the neutral stability line. Numerical results verify that aggressive driving is better than timid act because the aggressive driver will adjust his speed timely according to the leading car's speed. The key improvement of this new model is that the timid driving deteriorates traffic stability while the aggressive driving will enhance traffic stability. The relationship of energy consumption between the aggressive and timid driving is also studied. Numerical results show that aggressive driver behavior can not only suppress the traffic congestion but also reduce the energy consumption.

Effect of secondary electron emission on nonlinear dust acoustic wave propagation in a complex plasma with negative equilibrium dust charge

In this paper, we have investigated the effect of secondary electron emission on nonlinear propagation of dust acoustic waves in a complex plasma where equilibrium dust charge is negative. The primary electrons, secondary electrons, and ions are Boltzmann distributed, and only dust grains are inertial. Electron-neutral and ion-neutral collisions have been neglected with the assumption that electron and ion mean free paths are very large compared to the plasma Debye length. Both adiabatic and nonadiabatic dust charge variations have been separately taken into account. In the case of adiabatic dust charge variation, nonlinear propagation of dust acoustic waves is governed by the KdV (Korteweg-de Vries) equation, whereas for nonadiabatic dust charge variation, it is governed by the KdV-Burger equation. The solution of the KdV equation gives a dust acoustic soliton, whose amplitude and width depend on the secondary electron yield. Similarly, the KdV-Burger equation provides a dust acoustic shock wave. This dust acoustic shock wave may be monotonic or oscillatory in nature depending on the fact that whether it is dissipation dominated or dispersion dominated. Our analysis shows that secondary electron emission increases nonadiabaticity induced dissipation and consequently increases the monotonicity of the dust acoustic shock wave. Such a dust acoustic shock wave may accelerate charge particles and cause bremsstrahlung radiation in space plasmas whose physical process may be affected by secondary electron emission from dust grains. The effect of the secondary electron emission on the stability of the equilibrium points of the KdV-Burger equation has also been investigated. This equation has two equilibrium points. The trivial equilibrium point with zero potential is a saddle and hence unstable in nature. The nontrivial equilibrium point with constant nonzero potential is a stable node up to a critical value of the wave velocity and a stable focus above it. This critical value increases with increasing secondary electron emission. Thus, in the presence of secondary electron emission, higher velocity shock waves are monotonic. The results have been numerically explained considering the presence of Al2O3 dust grains in the plasma.

Sparse Optimal Control of the KdV-Burgers Equation on a Bounded Domain

In this work we consider measure-valued optimal control problems involving the KdV-Burgers (KdVB) equation as a state equation. These optimal control problems are motivated by an inverse problem and a control problem involving the flow of water in a channel over topography. Well-posedness of the optimal control problem is established which involves the investigation of the \KdVB equation for irregular source terms. Moreover optimality conditions for the control problem are derived. Efficient numerical schemes based on spectral methods are proposed for the state and adjoint equation, as well as adequate optimization methods. The theoretical results are illustrated by two numerical examples.

Survey and Review

The Survey and Review paper of this issue, “Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws,” by G. A. El, M. A. Hoefer, and M. Shearer, connects two well-known areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves, and uses a raft of applied mathematics tools: phase portrait analysis, multiple scales, averaging, etc. The theory of nonlinear dispersive waves has its roots in nineteenth century studies of water waves and gained an enormous prominence after the computer discovery of the soliton by Zabusky and Kruskal in 1965. This discovery led to the theory of integrable Hamiltonian partial differential equations with ramifications in many areas of mathematics, and also in mathematical physics, nonlinear optics, etc. The evolution of the theory of hyperbolic conservation laws has followed a similar pattern; while it started with Riemann's work on gas dynamics, it only came of age in the 1950s and 1960s with the contributions of Lax, Oleinik, and others. Regardless of the smoothness of the initial data, solutions of nonlinear conservation laws typically develop discontinuities and therefore have to be understood in a weak sense. To make matters more complicated, weak solutions are often not unique, and the solution with physical relevance has to be identified with the help of an entropy condition. A way to avoid these difficulties is to regularize the differential equations. Regularization is of course not only a matter of mathematical convenience: it brings to the equation “higher order” physical effects ignored in the conservation law formulation. The dissipative regularization of the inviscid Burgers equation \( u_t+(u^2/2)_x = 0\) results in the Burgers equation \( u_t+(u^2/2)_x = \epsilon u_xx\), \(\epsilon >0\). The inviscid shock that jumps from the upstream state \( u^-\) to the downstream state \(u^+\), \(u^->u^+\), is regularized to become a viscous, smooth traveling wave solution that shares the shock speed and upstream and downstream values. In situations without dissipation, regularization may arise from dispersion, as in the Korteweg--de Vries (KdV) equation \( u_t+(u^2/2)_x = \mu u_xxx\), and in that case the regularization of shock solutions leads to so-called dispersive shock waves, entities with a structure much richer than that of the dissipative traveling wave. The paper focuses on the modified KdV Burgers equation \( u_t+(u^3)_x = \epsilon u_xx+\mu u_xxx\), a universal asymptotic model where the corresponding conservation law \( u_t+(u^3)_x = 0\) has a nonconvex flow \( u^3\). As the paper shows, many remarkable phenomena appear due to the competition among nonconvexity, dissipation, and dispersion; these include undercompressive shock waves and double-wave complexes such as shock-rarefactions.

Well-posedness of Korteweg-de Vries-Burgers equation on a finite domain

In this paper,we consider the Korteweg-de Vries-Burgers equation on a finite domain with initial value and nonhomogeneous boundary conditions. This particular problem arises in the theory of ferroelectricity. We first get the local well-posedness of the problem, and then under the help of the local result, we use nonlinear interpolation to have the global well-posedness of the problem.

Korteweg-de Vries-Burgers equation in a multi-component magnetized plasma with nuclei of heavy elements

The nonlinear properties of ion-acoustic (IA) waves are investigated in a relativistically degenerate magnetized quantum plasma, whose constituents are non-degenerate inertial ions, degenerate electrons and immobile positively-charged heavy elements. For nonlinear studies, the well-known reductive perturbation technique is employed to derive the Korteweg-de Vries-Burger equation in the presence of relativistically degenerate electrons. Numerically, the amplitude, width, and phase speed are shown to be associated with the localized IA solitons, and shocks are shown to be significantly influenced by the various intrinsic parameters relevant to our model. The solitary and the shock wave properties have been to be influenced in the non-relativistic, as well as the ultrarelativistic, limits. The effects of the external magnetic field and the obliqueness are found to change the basic properties of IA waves significantly. The present analysis can be useful in understanding the collective process in dense astrophysical environments, like there of non-rotating white dwarfs, neutron stars, etc.

Characteristics of nonlinear dust acoustic waves in a Lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation

In this paper, characteristics of small amplitude nonlinear dust acoustic wave have been investigated in a unmagnetized, collisionless, Lorentzian dusty plasma where electrons and ions are inertialess and modeled by generalized Lorentzian Kappa distribution. Dust grains are inertial and equilibrium dust charge is negative. Both adiabatic and nonadiabatic fluctuation of charges on dust grains have been taken under consideration. For adiabatic dust charge variation reductive perturbation analysis gives rise to a KdV equation that governs the nonlinear propagation of dust acoustic waves having soliton solutions. For nonadiabatic dust charge variation nonlinear propagation of dust acoustic wave obeys KdV-Burger equation and gives rise to dust acoustic shock waves. Numerical estimation for adiabatic grain charge variation shows the existence of rarefied soliton whose amplitude and width varies with grain charges. Amplitude and width of the soliton have been plotted for different electron Kappa indices keeping ion velocity distribution Maxwellian. For non adiabatic dust charge variation, ratio of the coefficients of Burger term and dispersion term have been plotted against charge fluctuation for different kappa indices. All these results approach to the results of Maxwellian plasma if both electron and ion kappa tends to infinity.

Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs

By using a counterexample, we proved the fractional chain rule appeared in many references does not hold under Riemann–Liouville definition and Caputo definition of fractional derivative. It shows that this chain rule is invalid in investigating exact solutions of nonlinear fractional partial differential equations (PDEs). In this paper, based on the homogenous balanced principle, the function-expansion method of separation variable type are introduced. By using this method, a series of nonlinear time fractional PDEs such as time fractional KdV equation and Burgers equation, time fractional diffusion–convection equations are studied from mathematical viewpoint. The dynamical properties of these exact solutions are discussed and the profiles of several representative exact solutions are illustrated.

Spectral stability theory of heteroclinic solutions to the Korteweg-de Vries-Burgers equation with an arbitrary potential

The analysis of stability of heteroclinic solutions to the Korteweg–de Vries–Burgers equation is generalized to the case of an arbitrary potential that gives rise to heteroclinic states. An example of a specific nonconvex potential is given for which there exists a wide set of heteroclinic solutions of different types. Stability of the corresponding solutions in the context of uniqueness of a solution to the problem of decay of an arbitrary discontinuity is discussed.

Addendum to “Travelling waves for a non-local Korteweg–de Vries–Burgers equation” [J. Differential Equations 257 (3) (2014) 720–758

We add a theorem to F. Achleitner, C.M. Cuesta and S. Hittmeir (2014) [1]. In that paper we studied travelling wave solutions of a Korteweg–de Vries–Burgers type equation with a non-local diffusion term. In particular, the proof of existence and uniqueness of these waves relies on the assumption that the exponentially decaying functions are the only bounded solutions of the linearised equation. In this addendum we prove this assumption and thus close the existence and uniqueness proof of travelling wave solutions.

A new approach for some NLDEs with variable coefficients

This study is related to new soliton solutions compound KdV-Burgers equation, generalized KdV equation and (2+1) dimensional Broer–Kaup equations with variable coefficients. The generalized Kudryashov method (GKM) has been tackled for procuring exact solutions of these equations. By using this method, dark and bright soliton solutions of these equations have been found. Also, some graphical representations have been given to observe the structure of these solutions.

Nonlinear ion acoustic dissipative shock structure with exchange-correlation effects in quantum semiconductor plasmas

Ion acoustic shocks in the electron-hole-ion semiconductor plasmas have been studied. The quantum recoil effects, exchange-correlation effects and degenerate pressure of electrons and holes are included. The ion species are considered classical and their dissipation is taken into account via the dynamic viscosity. The Korteweg de Vries Burgers equation is derived by using reductive perturbation approach. The excitation of shock waves in different semiconductor plasmas is pointed out. For numerical analyses, the plasma parameters of different semiconductors are considered. The impact of variation of the plasma parameters on the strength of the shock wave in the semiconductor plasmas is discussed.

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 field intensity and kinematic viscosity on the propagation characteristics of nonlinear shock structures in such plasma systems.

An effective analytic approach for solving nonlinear fractional partial differential equations

Nonlinear fractional differential equations are widely used for modelling problems in applied mathematics. A new analytic approach with two parameters c1 and c2 is first proposed for solving nonlinear fractional partial differential equations. These parameters are used to improve the accuracy of the resulting series approximations. It turns out that much more accurate series approximations are obtained by choosing proper values of c1 and c2. To demonstrate the applicability and effectiveness of the new method, two typical fractional partial differential equations, the nonlinear gas dynamics equation and the nonlinear KdV-Burgers equation, are solved.