Zakharov Kuznetsov Burgers Equation Research Articles

Optimal decay estimate and asymptotic profile for solutions to the generalized Zakharov–Kuznetsov–Burgers equation in 2D

We consider the Cauchy problem for the generalized Zakharov–Kuznetsov–Burgers equation in 2D. This is one of the nonlinear dispersive–dissipative equations, which has a spatial anisotropic dissipative term −μuxx. In this paper, we prove that the solution to this problem decays at the rate of t−34 in the L∞-sense, provided that the initial data u0(x,y) satisfies u0∈L1(R2) and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the L∞-norm of the solution. As a result, we prove that the given decay rate t−34 of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schrödinger equation, we derive the explicit asymptotic profile for the solution.

Generalized variational structures of the (3+1)-dimensional Zakharov–Kuznetsov–Burgers equation in dusty plasma

The center of this paper is to establish the generalized variational structure (GVS) of the [Formula: see text]-dimensional Zakharov–Kuznetsov–Burgers equation (ZKBe) by taking advantage of the Semi-inverse method (SIM). Two different GVSs are extracted and the derivation process is presented in detail. The extracted GVSs reveal the energy conservation law and can offer some new insights on the study of the variational method.

Negative energy dust acoustic waves evolution in a dense magnetized quantum Thomas–Fermi plasma

Propagation of nonlinear waves in the magnetized quantum Thomas–Fermi dense plasma is analyzed. The Zakharov–Kuznetsov–Burgers equation is derived by using the theory of reductive perturbation. The exact solution contains both solitary and shock terms. Also, it is shown that rarefactive waves propagate in most cases. Both the associated electric field and the wave energy have been derived. The effects of dust and electrons temperature, dust density, magnetic field magnitude, and direction besides the effect of the kinematic viscosity on the amplitude, width, and energy of the formed waves are discussed. It is shown that the negative energy wave is formed and its value is enhanced due to the increase of the kinematic viscosity and the ambient magnetic field which lead to an increase in the instability. The present results are helpful in controlling the stabilization of confined Thomas–Fermi dense magnetoplasma that are found in white dwarfs and in the high-intensity laser-solid matter interaction experiments.

Nonlinear self-adjointness, conserved quantities and Lie symmetry of dust size distribution on a shock wave in quantum dusty plasma

The current exploration explains new traveling wave profiles by the Lie symmetry approach Here, we derive the (3+1)-dimensional Zakharov–Kuznetsov burgers equation for the shock wave phenomena in plasma with dust-charged particles having quantum effects. The structures of this model are made by Abelian algebra. The considered equation is reduced to an ordinary differential equation by the use of this Abelian algebra. The complete derivation of the classical symmetries of the considered model and reduction corresponding to the first four symmetries has been added. The new auxiliary equation method is implemented for the formation of the new traveling wave structures. The graphical explanation of some obtained results are elaborated in detail. The nonlinear self-adjointness of the considered model is also part of this work. By this concept, we have evaluated the conservation laws of the model. This work is new and does not exist in literature.

Analytical study of nonlinear water wave equations for their fractional solution structures

This paper examines the three-dimensional nonlinear time-fractional water wave equations for their analytical wave solutions. These are the equations of the names [Formula: see text]-Zakharov–Kuznetsov–Burgers equation and [Formula: see text]-Yu–Toda–Sasa–Fukuyama equation. The obtained wave solutions are in the form of kink, periodic and singular waves by utilizing the [Formula: see text]-expansion approach. The aforesaid solutions are verified and demonstrated graphically via symbolic soft computations.

Symmetry analysis of a generalized 2D Zakharov–Kuznetsov–Burgers equation with a variable dissipation

In this paper the Lie symmetry analysis is performed for the study of a generalized 2D Zakharov–Kuznetsov–Burgers equation with a variable dissipation. This equation originates from Zakharov–Kuznetsov equation supplemented with parabolic terms as in Burgers equation, which describes the nonlinear development of ion-acoustic waves in a magnetized plasma comprised of cold ions and hot isothermal electrons in the presence of a uniform magnetic field. We use Lie symmetry analysis to study this equation, which leads us reductions equations and solutions for this equation. We also get the low-order conservation laws of the equation that have been obtained using the corresponding symmetries of the family.

Three-Dimensional Isothermal Ion Acoustic Shock Waves in Ultra-Relativistic Degenerate Electron–Positron–Ion Magnetoplasmas

The modifications, which arise from ultra-relativistic and degenerate effects of electrons and positrons as well as the chemical potentials in the basic features of three-dimensional isothermal ion acoustic shock waves (IIASWs) propagating in magnetized electron–positron–ion (e–p–i) plasmas are studied. The cold ions are considered to be magnetized and inertial, while the small particles (i.e., electrons and positrons) are taken to obey the Fermi–Dirac statistics. The well-known reductive perturbation analysis is applied to obtain the nonlinear Zakharov–Kuznetsov–Burgers equation (NZKBE). The analytical shock wave solution is obtained by employing the tanh technique. Furthermore, the asymptotic behavior and the stability of the shock structures are discussed. In the current model, the disturbances of nonlinear isothermal ion acoustic waves are found to exhibit only monotonic IIASWs. The consequences of the chemical potential, the presence of ultra-relativistic degenerate electrons and positrons, and magnetic field on the essential properties of three-dimensional IIASWs are numerically examined. The numerical investigations give rise to significant high lights on the propagation and the dynamic behavior of IIASWs. It is found that the amplitudes of the monotonic IIASWs decrease with chemical potentials of ultra-relativistic degenerate electrons and positrons increase.

Zakharov–Kuznetsov–Burgers equation in a magnetised non-extensive electron–positron–ion plasma

In this paper, we have studied the three-dimensional (3D) electron-acoustic waves (EAWs) in a three-component complex plasma containing q-non-extensive distributed hot electrons and positrons. The propagation characteristics of the 3D electron-acoustic (EA) shock waves under the influence of magnetic field have been studied. Our present plasma model supports the negative potential shocks. Combined action of dissipation $$(\eta )$$ , non-extensivity (q), concentration of positrons ( $$\beta $$ ), temperature ratio of cold electrons to positrons ( $$\sigma $$ ) and magnetic field ( $$\omega _c$$ ) on the EA shock waves has been studied in detail and the findings obtained here will be beneficial in future astrophysical investigations.

Nonlinear excitations in a degenerate relativistic magneto-rotating quantum plasma

An investigation of heavy nucleus-acoustic (HNA) excitations in a degenerate relativistic magnetorotating quantum plasma system comprising relativistically degenerate light nuclei/electrons and inertial nondegenerate heavy nuclei has been presented. The Zakharov-Kuznetsov-Burgers (ZKB) equation has been derived by employing the reductive perturbation method. The solution of the ZKB equation supports only positive potential monotonic and oscillatory HNA shock waves in congruence with the space observations. It is observed that the heavy nucleus viscosity is a source of dissipation and is responsible for the formation of HNA monotonic and oscillatory shock structures. Bifurcation analysis is also examined in the absence of dissipation. It is shown that the combined effects of external magnetic field strength, rotational frequency, and obliqueness significantly modify the basic properties of different HNA nonlinear structures. The results should be utilitarian to understand the characteristics of nonlinear excitations in degenerate relativistic magnetorotating quantum plasma which is present in astrophysical compact objects especially in white dwarfs and neutron stars.

Excitation and Formation Conditions of Monotonic Shock Waves in Magnetized Plasmas with Superthermality Distributed Electrons

Nonlinear excitation and the properties of ion-acoustic shock waves (IASWs) in a magnetoplasma model composed of viscous ions with two-temperature superthermality distributed electrons are studied by employing the well-known reductive perturbation analysis to obtain a nonlinear Zakharov-Kuznetsov-Burgers equation (ZKBE), which admits the excitation of nonlinear IASWs in superthermal plasmas. Applying the tanh method, we discuss the solutions of the ZKBE. The asymptotic behavior and the stability of the analytical shock wave solution are studied. In general, nonlinear ion-acoustic disturbances are found analytically to exhibit only monotonic shock structures in the proposed model. For different situations, the effects of the dispersion and the dissipation coefficients on the profiles of the shock structures are discussed. The findings here demonstrate that the effective features of nonlinear IASWs depend strongly on the dispersion and the dissipation coefficients, which include physical parameters such as the superthermality of cold electrons, the cold superthermal electron-to-ion number density ratio, the ion kinematic viscosity and the ion cyclotron frequency. The current work may be helpful for an advanced comprehension of the physical nature of shock waves in astrophysical plasma situations.

Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term −∂x2u. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in Hs(R2) for s>1/2, and the 2D nonlinear parabolic equation with quadratic derivative nonlinearity is well-posed in Hs(R2) for s≥0. By using the Fourier restriction norm with dissipative effect, we prove the well-posedness for ZKB equation in Hs(R2) for s>−1/2.

([formula omitted])-dimensional ZK–Burgers equation with the generalized beta effect and its exact solitary solution

The (1 ＋1)-dimensional mathematical model had been extensively derived to describe Rossby solitary waves in a line in the past few decades. But as is well known, the (1 ＋1)-dimensional model cannot reflect the generation and evolution of Rossby solitary waves in a plane. In this paper, a (2 ＋1)-dimensional nonlinear Zakharov–Kuznetsov–Burgers equation is derived to describe the evolution of Rossby wave amplitude by using methods of multiple scales and perturbation expansions from the quasi-geostrophic potential vorticity equations with the generalized beta effect. The effects of the generalized beta and dissipation are presented by the Zakharov–Kuznetsov–Burgers equation. We also obtain the new solitary solution of the Zakharov–Kuznetsov equation when the dissipation is absent with the help of the Bernoulli equation, which is different from the common classical solitary solution. Based on the solution, the features of the variable coefficient are discussed by geometric figures Meanwhile, the approximate solitary solution of Zakharov–Kuznetsov–Burgers equation is given by using the homotopy perturbation method. And the amplitude of solitary waves changing with time is depicted by figures. Undoubtedly, these solitary solutions will extend previous results and better help to explain the feature of Rossby solitary waves.

Shock Waves, Variational Principle and Conservation Laws of a Schamel–Zakharov–Kuznetsov–Burgers Equation in a Magnetised Dust Plasma

Abstract In this article, we investigate a (3+1)-dimensional Schamel–Zakharov–Kuznetsov–Burgers (SZKB) equation, which describes the nonlinear plasma-dust ion acoustic waves (DIAWs) in a magnetised dusty plasma. With the aid of the Kudryashov method and symbolic computation, a set of new exact solutions for the SZKB equation are derived. By introducing two special functions, a variational principle of the SZKB equation is obtained. Conservation laws of the SZKB equation are obtained by two different approaches: Lie point symmetry and the multiplier method. Thus, the conservation laws here can be useful in enhancing the understanding of nonlinear propagation of small amplitude electrostatic structures in the dense, dissipative DIAWs’ magnetoplasmas. The properties of the shock wave solutions structures are analysed numerically with the system parameters. In addition, the electric field of this solution is investigated. Finally, we will study the physical meanings of solutions.

Dynamical survey of the dual power Zakharov–Kuznetsov–Burgers equation with external periodic perturbation

We investigate the dynamical behavior of the newly introduced dual power Zakharov–Kuznetsov–Burgers equation. Using bifurcation theory of planar dynamical systems, we study bifurcations of traveling wave solutions of the dual power Zakharov–Kuznetsov–Burgers equation in presence and absence of viscosity (μ) effect. In presence of an external periodic perturbation, we discuss the periodic and chaotic motions of the perturbed dual power Zakharov–Kuznetsov–Burgers equation by analyzing phase projection analysis, time series analysis, Poincaré section and sensitivity analysis. The effect of viscosity (μ) plays an important role in the transition from chaotic motion to periodic motion of the perturbed dual power Zakharov–Kuznetsov–Burgers equation.

Bifurcation and exact traveling wave solutions for dual power Zakharov–Kuznetsov–Burgers equation with fractional temporal evolution

In this article, we introduce the dual power Zakharov–Kuznetsov–Burgers equation with fractional temporal evolution in the sense of modified Riemann–Liouville derivative. We investigate the dynamical behavior, bifurcations and phase portrait analysis of the exact traveling wave solutions of the system with and without damping effect. We apply the (G′∕G)-expansion method in context of fractional complex transformation and seek a variety of exact traveling wave solutions including solitary wave, kink-type wave, breaking wave and periodic wave solutions of the equation. Furthermore, the remarkable features of the traveling wave solutions and phase portraits of dynamical system are demonstrated through interesting figures.

Bifurcations of solitary wave solutions for the three dimensional Zakharov–Kuznetsov–Burgers equation and Boussinesq equation with dual dispersion

In this research, we apply a new technique for solving and obtaining exact and solitary wave solutions of the three dimensional Zakharov–Kuznetsov–Burgers equation for the dust-ion-acoustic waves in dusty plasmas and Boussinesq equation with dual dispersion. We use the improved G′G-expansion method with the aid of Maple 16 which support us with three different kinds of solutions (the hyperbolic functions, the trigonometric functions and the rational functions). This method depends on auxiliary equation and also it is considered as one of general method for solving partial differential equations where this method include the extended G′G-expansion method when σ=0 and also the G′G-expansion method when N takes only positive value and zero. All of these solutions helps us to investigate the physical meaning of each models and the stability of above mentioned model.

(2[formula omitted]1) dimensional Rossby waves with complete Coriolis force and its solution by homotopy perturbation method

In this paper, the effects of both complete Coriolis force and dissipation on equatorial nonlinear Rossby wave are investigated analytically. From the quasi-geostrophic potential vorticity equation, by using methods of multiple scales and perturbation expansions, a (2+1) dimensional nonlinear Zakharov–Kuznetsov–Burgers equation is derived in describing the evolution of Rossby wave amplitude. The effects of generalized beta, the horizontal component of Coriolis parameter and the dissipation are presented from the Zakharov–Kuznetsov–Burgers equation. We also obtain the classical solitary solution of the Zakharov–Kuznetsov equation when the dissipation is absent by elliptic function expansion method, and the complete Coriolis force effect can be seen by the solution. But the method is failed to Zakharov–Kuznetsov-Burgers equation, therefore, we use the efficient homotopy perturbation method to solve the Zakharov–Kuznetsov–Burgers equation.

Mathematical methods and solitary wave solutions of three-dimensional Zakharov-Kuznetsov-Burgers equation in dusty plasma and its applications

Abstract In this research article, a new technique for solving nonlinear complex physical phenomena, arising in different fields of science, is investigated, called Modified extended mapping method. The method is applied to three dimensional Zakharov-Kuznetsov-Burgers (ZKB) equation for the dust-ionacoustic waves in dusty plasmas. As a result, the exact and solitary wave solutions (which represent electric field potential), electric and magnetic fields and quantum statistical pressure for ZKB equation are obtained with the aid of Mathematica. These new exact solitary wave solutions are expressed in the forms of hyperbolic, trigonometric and rational functions. The graphical representations of the electric field potential and electric and magnetic fields are shown. These results demonstrate the efficiency and precision of the method that can be applied to many other mathematical physical problems.

Nonlinear wave solutions of the three-dimensional Zakharov–Kuznetsov–Burgers equation in dusty plasma

The propagation of dust–ion-acoustic waves with high-energy electrons and positrons in three-dimensional is considered. The Zakharov–Kuznetsov–Burgers (ZKB) equations for the dust–ion-acoustic waves in dusty plasmas is obtained. The conservations laws and integrals of motion for the ZKB equation are deduced. In the present study, by applying the modified direct algebraic method, we found the electric field potential, electric field and quantum statistical pressure in form water wave solutions for three-dimensional ZKB equation. The solutions for the ZKB equation are obtained precisely and efficiency of the method can be demonstrated. The stability of the obtained solutions and the movement role of the waves by making the graphs of the exact solutions are discussed and analyzed.

Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers equation

This paper addresses an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers (ZKB) equation. The Lie symmetry analysis leads to many plethora of solutions to the equation. The nonlinear self-adjointness condition for the ZKB equation is established and subsequently used to construct simplified but infinitely many nontrivial and independent conserved vectors. In particular, we also get conservation laws of the equation with the corresponding Lie symmetry.