KdV-Burgers Equation Research Articles

Stability of nonstationary solutions of the generalized KdV-Burgers equation

The stability of nonstationary solutions to the Cauchy problem for a model equation with a complex nonlinearity, dispersion, and dissipation is analyzed. The equation describes the propagation of nonlinear longitudinal waves in rods. Previously, complex behavior of traveling waves was found, which can be treated as discontinuity structures in solutions of the same equation without dissipation and dispersion. As a result, the solutions of standard self-similar problems constructed as a sequence of Riemann waves and shocks with a stationary structure become multivalued. The multivaluedness of the solutions is attributed to special discontinuities caused by the large effect of dispersion in conjunction with viscosity. The stability of special discontinuities in the case of varying dispersion and dissipation parameters is analyzed numerically. The computations performed concern the stability analysis of a special discontinuity propagating through a layer with varying dispersion and dissipation parameters.

Effect of nonthermality fraction on dust acoustic growth rate in inhomogeneous viscous dusty plasmas

The properties of linear dust acoustic (DA) waves in inhomogeneous viscous dusty plasmas with non-thermal electrons and ions have been investigated. A linear dispersion relation is obtained with the non-adiabatic dust charge fluctuation and fraction of nonthermality of electron-ion. The dependence of frequency and the damping rate of waves on the nonthermality fraction and dust kinematic viscosity coefficient are discussed. To study the dust acoustic shock waves, KdV-Burgers (KdV-B) equation for homogeneous dissipative dusty plasma has been considered and solved by means of tanh method. The obtained solution is a particular combination of a solitary wave with a Burgers shock wave. The present results are useful in the context of space plasma.

Soliton Solutions of Fractional Order KdV-Burger's Equation

In this article, the new exact travelling wave solutions of the time-and space-fractional KdV-Burgers equation has been found. For this the fractional complex transformation have been implemented to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations, in the sense of the Jumarie's modified Riemann-Liouville derivative. Afterwards, the improved (G'/G)-expansion method can be implemented to celebrate the soliton solutions of KdV-Burger's equation of fractional order.

Ion acoustic shocks in magneto rotating Lorentzian plasmas

Ion acoustic shock structures in magnetized homogeneous dissipative Lorentzian plasma under the effects of Coriolis force are investigated. The dissipation in the plasma system is introduced via dynamic viscosity of inertial ions. The electrons are following the kappa distribution function. Korteweg-de Vries Burger (KdVB) equation is derived by using reductive perturbation technique. It is shown that spectral index, magnetic field, kinematic viscosity of ions, rotational frequency, and effective frequency have significant impact on the propagation characteristic of ion acoustic shocks in such plasma system. The numerical solution of KdVB equation is also discussed and transition from oscillatory profile to monotonic shock for different plasma parameters is investigated.

New Exact Solutions of some (2 + 1)-Dimensional Nonlinear Evolution Equations Via Extended Kudryashov Method

In this work, we propose an extended Kudryashov method to present new exact solutions of some nonlinear partial differential equations. The key idea of this method is to take full advantages of the Bernoulli and the Riccati equations involving parameters. We choose the (2 + 1)-dimensional Painleve integrable Burgers equations and the (2 + 1)-dimensional Korteweg-de Vries-Burgers equation to illustrate the validity and advantages of the method. By means of this method many new and general exact solutions have been found.

Nonlinear features of ion acoustic shock waves in dissipative magnetized dusty plasma

The nonlinear propagation of small as well as arbitrary amplitude shocks is investigated in a magnetized dusty plasma consisting of inertia-less Boltzmann distributed electrons, inertial viscous cold ions, and stationary dust grains without dust-charge fluctuations. The effects of dissipation due to viscosity of ions and external magnetic field, on the properties of ion acoustic shock structure, are investigated. It is found that for small amplitude waves, the Korteweg-de Vries-Burgers (KdVB) equation, derived using Reductive Perturbation Method, gives a qualitative behaviour of the transition from oscillatory wave to shock structure. The exact numerical solution for arbitrary amplitude wave differs somehow in the details from the results obtained from KdVB equation. However, the qualitative nature of the two solutions is similar in the sense that a gradual transition from KdV oscillation to shock structure is observed with the increase of the dissipative parameter.

Head-on collision of dust-acoustic shock waves in strongly coupled dusty plasmas

A theoretical investigation is carried out to study the propagation and the head-on collision of dust-acoustic (DA) shock waves in a strongly coupled dusty plasma consisting of negative dust fluid, Maxwellian distributed electrons and ions. Applying the extended Poincaré–Lighthill–Kuo method, a couple of Korteweg–deVries–Burgers equations for describing DA shock waves are derived. This study is a first attempt to deduce the analytical phase shifts of DA shock waves after collision. The impacts of physical parameters such as the kinematic viscosity, the unperturbed electron-to-dust density ratio, parameter determining the effect of polarization force, the ion-to-electron temperature ratio, and the effective dust temperature-to-ion temperature ratio on the structure and the collision of DA shock waves are examined. In addition, the results reveal the increase of the strength and the steepness of DA shock waves as the above mentioned parameters increase, which in turn leads to the increase of the phase shifts of DA shock waves after collision. The present model may be useful to describe the structure and the collision of DA shock waves in space and laboratory dusty plasmas.

Approximate analytical solution of the nonlinear fractional KdV–Burgers equation: A new iterative algorithm

In this paper, explicit and approximate solutions of the nonlinear fractional KdV–Burgers equation with time–space-fractional derivatives are presented and discussed. The solutions of our equation are calculated in the form of rabidly convergent series with easily computable components. The utilized method is a numerical technique based on the generalized Taylor series formula which constructs an analytical solution in the form of a convergent series. Five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method. Graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution. The results reveal that the method is very effective and simple in determination of solution of the fractional KdV–Burgers equation.

Computable analysis of a boundary-value problem for the generalized KdV-Burgers equation

In this paper, we investigate the computability of the solution operator of the generalized KdV–Burgers equation with initial-boundary value problem. Here, the solution operator is a nonlinear map H 3m� 1 .R C / � H m .0,T/ ! C.Œ0,T� ;H 3m� 1 .R C //, from the initial-boundary value data to the solution of the equation. By a technique that is widely used for the study of nonlinear dispersive equation, and using the type 2 theory of effectivity as computable model, we prove that the solution map is Turing computable, for any integerm � 2, and computable real numberT > 0. Copyright © 2014 John Wiley & Sons, Ltd.

Finite time blow-up and breaking of solitary wind waves.

The evolution of surface water waves in finite depth under wind forcing is reduced to an antidissipative Korteweg-de Vries-Burgers equation. We exhibit its solitary wave solution. Antidissipation accelerates and increases the amplitude of the solitary wave and leads to blow-up and breaking. Blow-up occurs in finite time for infinitely large asymptotic space so it is a nonlinear, dispersive, and antidissipative equivalent of the linear instability which occurs for infinite time. Due to antidissipation two given arbitrary and adjacent planes of constant phases of the solitary wave acquire different velocities and accelerations inducing breaking. Soliton breaking occurs in finite space in a time prior to the blow-up. We show that the theoretical growth in amplitude and the time of breaking are both testable in an existing experimental facility.

Subharmonic bifurcations and chaos for the traveling wave solutions of the compound Kdv–Burgers equation with external and parametrical excitations

The subharmonic bifurcations and chaotic motions are investigated both analytically and numerically for the traveling solutions of compound Kdv–Burgers equation with external and parametrical excitations. The critical curves separating the chaotic and non-chaotic regions are obtained. The chaotic feature on the system parameters are discussed in detail. Some new dynamical phenomena including the “controllable frequency” are presented. The conditions for subharmonic bifurcations are also obtained. Numerical results are given, which verify the analytical ones.

Space—time fractional KdV—Burgers equation for dust acoustic shock waves in dusty plasma with non-thermal ions

The KdV—Burgers equation for dust acoustic waves in unmagnetized plasma having electrons, singly charged nonthermal ions, and hot and cold dust species is derived using the reductive perturbation method. The Boltzmann distribution is used for electrons in the presence of the cold (hot) dust viscosity coefficients. The semi-inverse method and Agrawal variational technique are applied to formulate the space—time fractional KdV—Burgers equation which is solved using the fractional sub-equation method. The effect of the fractional parameter on the behavior of the dust acoustic shock waves in the dusty plasma is investigated.

Competition between geometric dispersion and viscous dissipation in wave propagation of KdV-Burgers equation

In this paper the competitive relationship between the geometric dispersion and the viscous dissipation in the wave propagation of the KdV-Burgers equation is investigated by the generalized multi-symplectic method. Firstly, the generalized multi-symplectic formulations for the KdV-Burgers equation are presented in Hamiltonian space. Then, focusing on the inherent geometric properties of the generalized multi-symplectic formulations, a 12-point difference scheme is constructed. Finally, numerical experiments are performed with fixed step-sizes to obtain the maximum damping coefficient that insures that the scheme constructed is generalized multi-symplectic, and to study the competition between the geometric dispersion and the viscous dissipation in the wave propagation of the KdV-Burgers equation. The competition phenomena are comprehensively illustrated in the wave forms as well as in the phase diagrams: for the KdV equation (a particular case of the KdV-Burgers equation without dissipation), there is a closed orbit in the phase diagram; and the closed orbit is substituted by a heteroclinic one with the appearance of the viscous dissipation; moreover, the heteroclinic orbit changes from the saddle-node type to the saddle-focus type with an increase of the damping coefficient.

Travelling waves for a non-local Korteweg–de Vries–Burgers equation

We study travelling wave solutions of a Korteweg–de Vries–Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors. In contrast, travelling waves of the non-local KdV–Burgers equation are not in general monotone, as is the case for the corresponding classical KdV–Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV–Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone.

The [formula omitted]-expansion method using modified Riemann–Liouville derivative for some space-time fractional differential equations

In this paper, the fractional partial differential equations are defined by modified Riemann–Liouville fractional derivative. With the help of fractional derivative and traveling wave transformation, these equations can be converted into the nonlinear nonfractional ordinary differential equations. Then G′G-expansion method is applied to obtain exact solutions of the space-time fractional Burgers equation, the space-time fractional KdV-Burgers equation and the space-time fractional coupled Burgers’ equations. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. These results reveal that the proposed method is very effective and simple in performing a solution to the fractional partial differential equation.

Dust-acoustic shock waves in a magnetized non-thermal dusty plasma

A theoretical investigation is carried out to study the basic properties of dust-acoustic (DA) shock waves propagating in a magnetized non-thermal dusty plasma (containing cold viscous dust fluid, non-thermal ions, and non-thermal electrons). The reductive perturbation method is used to derive the Korteweg–de Vries–Burgers equation. It is found that the basic properties of DA shock waves are significantly modified by the combined effects of dust fluid viscosity, external magnetic field, and obliqueness (angle between external magnetic field and DA wave propagation direction). It is shown that the dust fluid viscosity acts as a source of dissipation, and is responsible for the formation of DA shock structures in the dusty plasma system under consideration. The implications of our results in some space and laboratory plasma situations are briefly discussed.

Dust acoustic shock waves in dusty plasma of opposite polarity with non-extensive electron and ion distributions

A theoretical investigation has been made of obliquely propagating nonlinear electrostatic shock structures. The reductive perturbation method has been used to derive the Korteweg-de Vries-Burger (KdV-Burger) equation for dust acoustic shock waves in a homogeneous system of a magnetized collisionless plasma comprising a four-component dusty plasma with massive, micron-sized, positively, negatively dust grains and non-extensive electrons and ions. The effect of dust viscosity coefficients of charged dusty plasma of opposite polarity and the non-extensive parameters of electrons and ions have been studied. The behavior of the oscillatory and monotonic shock waves in dusty plasma has been investigated. It has been found that the presence of non-extensive parameters significantly modified the basic properties of shock structures in space environments.

Extended models of non-linear waves in liquid with gas bubbles

In this work we generalize the models for non-linear waves in a gas–liquid mixture taking into account an interphase heat transfer, a surface tension and a weak liquid compressibility simultaneously at the derivation of the equations for non-linear waves. We also take into consideration high order terms with respect to the small parameter. Two new non-linear differential equations are derived for long weakly non-linear waves in a liquid with gas bubbles by the reductive perturbation method considering both high order terms with respect to the small parameter and the above-mentioned physical properties. One of these equations is the perturbation of the Burgers equation and corresponds to main influence of dissipation on non-linear waves propagation. The other equation is the perturbation of the Burgers–Korteweg–de Vries equation and corresponds to main influence of dispersion on non-linear waves propagation.

Lie symmetry reductions and exact solutions of a coupled KdV–Burgers equation

In this paper, based on classical Lie group method, with the help of Maple software, we study a coupled KdV–Burgers equation, and get its infinitesimal generator, commutation table of Lie algebra, symmetry group and similarity reductions. In particular, solitary wave solutions and similarity solutions of the coupled KdV–Burgers equation are derived from the reduction equations.

Modified approximation for the KdV–Burgers equation

In this paper, a new finite difference called the Restrictive Taylor approximation (RTA) is implemented to find numerical solution of KdV-Burgers, this method is a new explicit method. The accuracy of the method is assessed in terms of the absolute error which is very close to zero and the error norms L2, L∞.