mKdV Burgers Equation Research Articles

Bifurcation analysis of electron-acoustic waves in electron beam-plasma with suprathermal electrons

It has investigated the properties of nonlinear electron-acoustic waves propagating in a four-component collisionless, unmagnetized plasma. The system consists of cool inertial background electrons of temperature T c , the cool inertial electron beam of temperature T b , hot inertialess suprathermal electrons modeled by a κ-distribution of temperature T h , and ions. The nonlinear modified Korteweg-de Vries-Burgers (mKdV-Burgers) equation that characterizes the electron acoustic waves in such a plasma model is derived. The wave solutions of the resultant mKdV-Burgers equation are investigated using planar dynamical system bifurcation theory. The unperturbed hot, cool, and beam electron densities and the superthermal electron parameter are revealed to have a considerable influence on the nonlinear electron-acoustic traveling waves and the associated electric fields. The system evolves also monotonic shock waves when the dissipation term dominates over the dispersion term. The current model might be useful for understanding the generation of nonlinear electron-acoustic waves in a variety of astrophysical plasma settings, including the Earth's magnetosphere.

The evolution equation of non-linear waves and its exact solutions by subsidiary ordinary differential equation method

In this work, we investigate the dynamics of the equatorial Rossby waves by including the complete Coriolis force, external source and dissipation. The amplitude evolution of equatorial Rossby waves is described as an extended non-linear mKdV–Burgers equation from a potential vorticity equation and it is unlike the standard mKdV–Burgers equation. Built on the obtained model, the corresponding physical phenomena related to the non-linear Rossby waves are analyzed. Also, the subsidiary ordinary differential equation method is employed to solve the solitary solution of the mKdV equation. By analyzing the solution, we find that the horizontal component of Coriolis parameter works on the amplitude of the Rossby waves. Meanwhile, we use the Adomian decomposition method to obtain the approximate soliton solution of the model.

Dynamics of ion-acoustic waves in nonrelativistic magnetized multi-ion quantum plasma: the role of trapped electrons

The presented work provides a new perspective of quantum nonlinear propagation of ion-acoustic (IA) waves in a magnetized collisionless and dissipative multi-ion plasma system, composed of cold positive and negative charged nonrelativistic ions, negatively charged immobile heavy ions and trapped electrons; all existed in a quantizing magnetic field. We address the reductive perturbation method to derive the Korteweg-de Vries- Burgers (KdV-Burgers) and modified KdV-Burgers equations. The bifurcation theory of planar dynamical systems is applied to investigate the existence of the solitary wave and the periodic traveling wave solutions of the resulting mKdV-Burgers equation. Accordingly, the phase portrait topology is illustrated for this equation. To study the nonlinear waves in case of mKdV- Burgers, we obtain some interesting physical solutions for mKdV- Burgers. These solutions are in the form of soliton, a combination between the shock and the soliton, and finally monotonic and oscillatory shock waves. Only rarefactive IA waves are obtained for the slow mode of phase velocity. It is found that the IA solitons, IA shocks are affected by the plasma system parameters. The presented theoretical research can be successfully implemented to understanding the solitary and shocks structures in dense magnetized quantum plasmas such as those existing in white dwarfs and in the intense-laser plasma interactions.

Generalized (2 + 1)-dimensional mKdV-Burgers equation and its solution by modified hyperbolic function expansion method

Abstract We investigate a new higher-dimensional nonlinear dynamics model to describe the generation and evolution of Rossby waves. We derive a generalized (2 + 1)-dimensional modified Korteweg-de Vries (mKdV)-Burgers equation by considering the quasi-geostrophic potential vorticity equation under the generalized β approximation with dissipation and external source in barotropic fluids, and by utilizing multiple scales and the perturbation expansion method. Qualitative analysis yields that the generalized β and shear basic flow can induce the nonlinear Rossby solitary waves, and dissipation has effects on the propagation of Rossby waves as well. Then, we analyze the conservation laws for the mass and energy of Rossby solitary waves. Moreover, the asymptotic kink-shaped solitary wave solution for the generalized (2 + 1)-dimensional mKdV-Burgers equation is explored by modified hyperbolic function expansion method. The solitary wave solutions and figures are analyzed, the results show that the dissipation effect causes the speed and amplitude of Rossby solitary waves to decrease exponentially and the width of Rossby solitary waves to increase exponentially with time.

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.

Characteristics of dust acoustic waves in dissipative dusty plasma in the presence of trapped electrons

The formation and propagation of nonlinear dust acoustic waves (DAWs) as solitary and solitary/shock waves in an unmagnetized, homogeneous, dissipative and collisionless dusty plasma comprising negatively charged micron sized dust grains in the presence of free and trapped electrons with singly charged non-thermal positive ions is discussed in detail. The evolution characteristics of the solitary and shock waves are studied by deriving a modified Korteweg–de Vries–Burgers (mKdV–Burgers) equation using the reductive perturbation method. The mKdV–Burgers equation is solved considering the presence (absence) of dissipation. In the absence of dissipation the system admits a solitary wave solution, whereas in the presence of dissipation the system admits shock waves (both monotonic and oscillatory) as well as a combination of solitary and shock wave solutions. Standard methods of solving the evolution equation of shock (solitary) waves are used. The results are discussed numerically using standard values of plasma parameters. The findings may be useful for better understanding of formation and propagation of waves in astrophysical plasma.

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.

Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient

This paper studies the bounded traveling wave solutions of MKdV-Burgers equation with the negative dispersive coefficient by the theory of planar dynamical systems, undetermined coefficients method. The global phase portraits under the different parameter conditions, as well as the existent number and conditions of the bounded traveling wave solutions are obtained for the dynamical system corresponding to the traveling wave solutions of MKdV-Burgers equation. The relation is investigated between the profiles of the bounded traveling wave solutions and dissipative coefficient. And a critical value characterizing the scale of dissipative effect, is given, which is different from the one proposed by R.F. Bikbaev in his article. Focusing on the open issue proposed by R.F. Bikbaev, based on the bell and kink profile solitary wave solutions of MKdV-Burgers equation we presented, approximate damped oscillatory solutions of MKdV-Burgers equation are obtained according to the evolution relation of orbits corresponding to the approximate damped oscillatory solutions in the global phase portraits.

Nonlinear Alfvén wave dynamics in plasmas

Nonlinear Alfvén wave dynamics is presented using Lagrangian fluid approach in a compressible collisional magnetized plasma. In the framework of two fluid dynamics, finite electron inertia is shown to serve as a dispersive effect acting against the convective nonlinearity. In a moving frame, the Alfvén wave can, therefore, form an arbitrarily strong amplitude solitary wave structure due to the balance between nonlinearity and dispersion. Weak amplitude Alfvén waves are shown to be governed by a modified KdV equation, which extends for finite dissipation to a mKdV-Burgers equation. These equations have well known solutions. Next, we have analyzed the fourth order nonlinear Alfvén wave system of equations both numerically and by approximation method. The results indicate a collapse of the density and magnetic field irrespective of the presence of dispersion. The wave magnetic field, however, appears to be less singular showing collapse only when the dispersive effects are negligible. These results may contribute to our understanding of the generation of strongly localized magnetic fields (and currents) in plasmas and are expected to be of special importance in the astrophysical context of magnetic star formation.

Linear and nonlinear analysis of dust acoustic waves in dissipative space dusty plasmas with trapped ions

The propagation of linear and nonlinear dust acoustic waves in a homogeneous unmagnetized, collisionless and dissipative dusty plasma consisted of extremely massive, micron-sized, negative dust grains has been investigated. The Boltzmann distribution is suggested for electrons whereas vortex-like distribution for ions. In the linear analysis, the dispersion relation is obtained, and the dependence of damping rate of the waves on the carrier wave number \(k\), the dust kinematic viscosity coefficient \(\eta _{d}\) and the ratio of the ions to the electrons temperatures \(\sigma _{i}\) is discussed. In the nonlinear analysis, the modified Korteweg–de Vries–Burgers (mKdV–Burgers) equation is derived via the reductive perturbation method. Bifurcation analysis is discussed for non-dissipative system in the absence of Burgers term. In the case of dissipative system, the tangent hyperbolic method is used to solve mKdV–Burgers equation, and yield the shock wave solution. The obtained results may be helpful in better understanding of waves propagation in the astrophysical plasmas as well as in inertial confinement fusion laboratory plasmas.

Beta effect and slowly changing topography Rossby waves in a shear flow

In barotropic fluids, based on the quasi-geostrophic potential vorticity equation, an inhomogeneous mKdV-Burgers equation including slowly changing topography and an external source is derived by employing the perturbation method and stretching transforms of time and space. With the inspection of the evolution of the amplitude of Rossby waves, it is found that beta effect, topography effect, slowly changing topography and an external source are all the important factors, and that the solitary Rossby wave is induced thought the basic stream function has a shear flow . On the assumption that the nonlinear and topographic effects are in balance, an inhomogeneous mKdV-Burgers equation is derived, the results show that the topography and Rossby wave interact in the barotropic flows. The inhomogeneous mKdV-Burgers equation describing the evolution of the amplitude of solitary Rossby wave as the change of Rossby parameter β(y) with latitude y, topographic forcing, slowly changing topography and the external source is obtained.

Infinite series symmetry reduction solutions to the modified KdV–Burgers equation

From the point of view of approximate symmetry, the modified Korteweg-de Vries–Burgers (mKdV–Burgers) equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV–Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV–Burgers equation satisfies the Painlevé II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.

New exact solutions to MKDV-Burgers equation and (2 + 1)-dimensional dispersive long wave equation via extended Riccati equation method

In this paper, with the aid of symbolic computation and a general ansätz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansätz. The method can also be applied to other nonlinear partial differential equations.

Boundary control of MKdV-Burgers equation

The boundary control of MKdV-Burgers equation was considered by feedback control on the domain [0,1]. The existence of the solution of MKdV-Burgers equation with the feedback control law was proved. On the base, priori estimates for the solution was given. At last, the existence of the weak solution of MKdV-Burgers equation was proved and the global-exponential and asymptotic stability of the solution of MKdV-Burgers equation was given.

Exploratory Approach to Explicit Solution ofNonlinear Evolution Equations

By introducing a concept of “rank,” an exploratory approach is presented forfinding the explicit traveling wave solutions to some physically interestingnonlinear evolution equations in physics and other fields. To show the efficiencyof this approach, the explicit traveling wave solutions to some well-knownequations, such as the MKdV-Burgers equation, the generalizedFitzhugh-Nagumo equation, the generalized Burgers-Fisher equation, and some coupledones in fluid mechanics, are given.

Exact and explicit solitary wave solutions to some nonlinear equations

Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative Φ4-model equation, the generalized Fisher equation, and the elastic-medium wave equation.