Dimensional Zakharov-Kuznetsov Equation Research Articles

Temporal wave dynamics, phase portrait and qualitative analysis of the time-dependent (2+1)-dimensional Zakharov-Kuznetsov equation

In this article, we analyze the effect of time-dependent coefficients and the complex wave dynamics of the (2+1)-dimensional Zakharov-Kuznetsov (ZK) equation. This equation provides a detailed, insightful, and realistic description of space physics, plasma physics, controlled fusion, and nonlinear sciences. The wave solutions are established using the generalized Kudryashov, modified simple equation, and modified sine-Gordon expansion techniques and are illustrated by graphical depictions, which provide valuable insight into understanding the complex dynamics of waves across different physical systems. Exact solitary wave solutions offer a dependable approach to investigating the behavior of a system under particular conditions and facilitating a comprehensive understanding of its dynamics. We also conduct a stability analysis and present the phase portrait of the solutions, which are useful in various fields, including physics, plasma physics, chemistry, biology, economics, and sociology. We ascertain that the profiles of 3D and 2D soliton-shaped waves are significantly affected by dynamic changes in coefficients, wave velocity, and associated model parameters. This research could help clarify the dynamics of intricate systems, paving to a better understanding and analysis of the temporal aspects of various phenomena.

Qualitative analysis and new exact solutions for the extended space-fractional stochastic (3 + 1)-dimensional Zakharov-Kuznetsov equation

In this paper, the extended (3 + 1)-dimensional Zakharov-Kuznetsov equation, which describes the propagation of ion-acoustic waves in a magnetic environment, is investigated. Due to the exposure of the propagation to unpredictable factors, the stochastic model is assessed including the Brownian process, in addition to including the recent concept of truncated M-fractional derivative. A fractional stochastic transformation is applied to transform the model into an integer-order ordinary differential equation which in turn is equivalent to a conservative Hamiltonian model. Novel solutions, such as hyperbolic, trigonometric, and Jacobian elliptic functions, are established by employing both of the qualitative analysis of dynamical systems and the first integral of the Hamiltonian model. We explore and graphically display the effects of the fractional derivative order and noise intensity on the solutions structures. In the deterministic instance, i.e. in the absence of noise, solitary and cnoidal solutions among other traveling wave solutions of the Zakharov-Kuznetsov equation, are derived. Further, it is found that the curvature of the wave disturbs and the surface turns substantially flat by increasing the value of noise. While the curve in all cases loses its characteristic shape and degenerates into another deterministic shape by changing the fractional derivative order.

Bright, dark, periodic soliton solutions and other analytical solutions of a time-dependent coefficient (2 + 1)-dimensional Zakharov–Kuznetsov equation

Bright, dark, periodic soliton solutions and other analytical solutions of a time-dependent coefficient (2 + 1)-dimensional Zakharov–Kuznetsov equation

New diverse exact optical solutions of the three dimensional Zakharov–Kuznetsov equation

New diverse exact optical solutions of the three dimensional Zakharov–Kuznetsov equation

Dynamical properties and novel wave solutions of the time-fractional extended (2+1)-dimensional Zakharov–Kuznetsov equation in plasma physics

Dynamical properties and novel wave solutions of the time-fractional extended (2+1)-dimensional Zakharov–Kuznetsov equation in plasma physics

Abundance of Exact Solutions of a Nonlinear Forced (2 + 1)-Dimensional Zakharov–Kuznetsov Equation for Rossby Waves

In this paper, an improved tan (φ/2) expansion method is used to solve the exact solution of the nonlinear forced (2 + 1)-dimensional Zakharov–Kuznetsov equation. Firstly, we analyse the research status of the improved tan (φ/2) expansion method. Then, exact solutions of the nonlinear forced (2 + 1)-dimensional Zakharov–Kuznetsov equation are obtained by the perturbation expansion method and the multi-spatiotemporal scale method. It is shown that the improved tan (φ/2) expansion method can obtain more exact solutions, including exact periodic travelling wave solutions, exact solitary wave solutions, and singular kink travelling wave solutions. Finally, the three-dimensional figure and the corresponding plane figure of the corresponding solution are given by using MATLAB to illustrate the influence of external source, dimension variable y, and dispersion coefficient on the propagation of the Rossby wave.

Impacts of a general power law on soliton for a (2+1)-dimensional Zakharov–Kuznetsov equation in magnetized quantum plasmas

In this article, under investigation is a (2+1)-dimensional Zakharov–Kuznetsov equation with a general power law nonlinearity arising from magnetized quantum plasmas model. The analytical periodic wave, bright soliton and bright–dark soliton solutions are computed for the equation via the F-expansion method. Furthermore, the impacts of the general power law parameter on the soliton is examined in details. It is interesting that the general power law can be used to control the amplitude and propagation speed, which can determine partly the kinetic energy. Particularly, as the general power law parameter increases, the soliton’ energy will decrease.

INVESTIGATION OF THE TIME FRACTIONAL GENERALIZED (2 + 1)-DIMENSIONAL ZAKHAROV–KUZNETSOV EQUATION WITH SINGLE-POWER LAW NONLINEARITY

In this paper, we studied the time fractional generalized [Formula: see text]-dimensional Zakharov–Kuznetsov (ZK) equation with single-power law nonlinearity, which can be used to describe weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions. Through the fractional Lie transformation group idea, the symmetries, similarity reduction, one-parameter Lie transformation group and conservation laws of these considered equations were obtained. In addition, some new exact solutions were also constructed for them. These novel results provide a new perspective for studying the evolutionary mechanism of these considered models.

On group of Lie symmetry analysis, explicit series solutions and conservation laws for the time-fractional (2 + 1)-dimensional Zakharov-Kuznetsov (q,p,r) equation

This research article intends to utilize results on Lie symmetry analysis, explicit series solutions and conservation laws for the time-fractional Zakharov-Kuznetsov (q,p,r) equation in three-dimensional space. Such a fractional equation yields the mathematical model which describes an occurrence of stationary spatial stripe modalities in a three-dimensional system in the framework of the theory of conservation laws. The governing equation is solved analytically by the power series method, where the total derivative in the sense of Riemann-Liouville type. Simulation results are systematically validated through a series of test cases. Strong evidence shows that the model and the method are conservative and robust.

Generalized Variational Principle for the Fractal (2 + 1)-Dimensional Zakharov–Kuznetsov Equation in Quantum Magneto-Plasmas

In this paper, we propose the fractal (2 + 1)-dimensional Zakharov–Kuznetsov equation based on He’s fractal derivative for the first time. The fractal generalized variational formulation is established by using the semi-inverse method and two-scale fractal theory. The obtained fractal variational principle is important since it not only reveals the structure of the traveling wave solutions but also helps us study the symmetric theory. The finding of this paper will contribute to the study of symmetry in the fractal space.

Modified homotopy methods for generalized fractional perturbed Zakharov\u2013Kuznetsov equation in dusty plasma

We propose a new modification of homotopy perturbation method (HPM) called the δ-homotopy perturbation transform method (δ-HPTM). This modification consists of the Laplace transform method, HPM, and a control parameter δ. This control convergence parameter δ in this new modification helps in adjusting and controlling the convergence region of the series solution and overcome some limitations of HPM and HPTM. The δ-HPTM and q-homotopy analysis transform method (q-HATM) are considered to study the generalized time-fractional perturbed (3+1)-dimensional Zakharov–Kuznetsov equation with Caputo fractional time derivative. This equation describes nonlinear dust-ion-acoustic waves in the magnetized two-ion-temperature dusty plasmas. The selection of an appropriate value of δ in δ-HPTM and the auxiliary parameters n and ħ in q-HATM gives a guaranteed convergence of series solution, but the difference between the two techniques is that the embedding parameter p in δ-HPTM varies from zero to nonzero δ, whereas the embedding parameter q in q-HATM varies from zero to frac{1}{n}, ngeq{1}. We examine the effect of fractional order on the considered problem and present the error estimate when compared with exact solution. The outcomes reveal complete reliability and efficiency of the proposed algorithm for solving various types of physical models arising in sciences and engineering. Furthermore, we present the convergence and error analysis of the two methods.

Symmetry analysis and conservation laws of a further modified 3D Zakharov-Kuznetsov equation

We study a further modified (3 + 1)-dimensional(3D) Zakharov-Kuznetsov equation. The classical Lie symmetry method will be employed to obtain solutions of a further modified 3D Zakharov-Kuznetsov equation. We will also establish the multiplier method to derive conservation laws of the underlying equation. Moreover, a brief physical interpretation of the derived conserved quantities is conferred.

Construction of traveling waves patterns of [formula omitted]-dimensional modified Zakharov-Kuznetsov equation in plasma physics

In this research, we examine the modified model of (1+n)-dimensional Zakharov-Kuznetsov (ZK) equation, which will be used to analyze the nature of weakly nonlinear traveling waves in the existence of a constant magnetic area in a plasma comprising in cold ions and hot isothermal electrons. The modified Zakharov-Kuznetsov (mZK) equation will have solutions describing the traveling solitary waves, using the extended (G′G2)-expansion method and extended direct algebraic method gives way to the mZK equation regulating the transmission of ion dynamics for nonlinear traveling waves in a plasma. The sufficient conditions for the stability and existence of the traveling wave solutions are reported. Semi-dark, rational, and singular solitary wave solutions are computed. Graphical interpretations of certain practical solutions for specific values of parameters have also been available. The research findings reported throughout this evaluation are fresh and from which this model is employed to analyze waves in numerous plasmas, could be valuable and important. Subsequently, there are concluding remarks mentioned.

On group analysis of the time fractional extended (2+1)-dimensional Zakharov–Kuznetsov equation in quantum magneto-plasmas

In this article, the time fractional extended (2+1)-dimensional Zakharov–Kuznetsov (Z–K) equation in quantum magneto-plasmas, is executed. First of all, the symmetry of this considered equation by using group analysis approach with the sense of Riemann–Liouville (R–L) fractional derivative, is obtained. Then, the symmetry of the above yielded, the optimal system of one-dimensional subalgebras for this equation is also found. Subsequently, the original equation can be reduced into (1+1)-dimensional fractional differential equation with the adding of extended Erdélyi–Kober fractional differential operator. Further, the one parameter group, invariant solutions and non-invariant solutions are constructed. Finally, the conservation laws are also shown with a new conservation theorem. We believe that these beautiful results can help us to discover more evolutionary mechanisms of the considered equation.

Localized solitary waves and their dynamical stabilities in magnetized dusty plasma

We consider a magnetized dusty plasma, which composed of low-temperature and high-temperature ions, electrons, and dust particles. The dynamical behaviors can be described by a (3+1)-dimensional Zakharov–Kuznetsov equation (ZKE). Interestingly, a type of completely localized solitary waves, which are different from the line solitons, of ZKE are obtained analytically and approximately for the first time. This kind of solitary wave is also confirmed numerically by the Petviashvili method. Both the analytical and numerical results indicate that the amplitude of the localized wave is proportional to its velocity and inverse proportional to the nonlinear interaction strength. A finite difference scheme with second-order accuracy is presented to make the long-time nonlinear evolution of ZKE. The numerical results indicate that the localized solitons are always dynamically stable. Moreover, the collision between two solitary waves is investigated numerically. The results show that both elastic and inelastic collision exist when two localized solitary waves colliding.

General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics

This paper is devoted to improving the general Kudryashov method by a new and general auxiliary equation. So, a new method is introduced, which we call ‘the general improved Kudryashov method’, to produce exact solutions for nonlinear evolution equations arising in mathematical physics. As application examples, exact traveling wave solutions for the combined Korteweg–de Vries and modified Korteweg–de Vries (KdV–mKdV) equation and the (2+1)-dimensional Zakharov–Kuznetsov equation are obtained. These solutions can be classified as solitary and periodic wave solutions. Some of the obtained solutions are graphically sketched.

The new exact solitary solutions for the (3+1)-dimensional Zakharov-Kuznetsov equation using the Riccati equation

In this paper, a non-linear (3+1)-dimensional Zakharov-Kuznetsov equation is investigated by employing the subsidiary equation method, which arises in quantum magneto plasma. The periodic solutions, rational wave solutions, soliton solutions for the quantum Zakharov-Kuznetsov equation which play an important role in mathematical physics are obtained with the help of the Riccati equation expan?sion method. Meanwhile, the electrostatic potential can be accordingly obtained. Compared to the other methods, the exact solutions obtained will extend on earlier reports by using the Riccati equation.

Investigation of solitary wave solutions for the (3 + 1)-dimensional Zakharov–Kuznetsov equation

In this paper, the new traveling wave solutions containing the trigonometric functions, hyperbolic functions and rational functions of [Formula: see text]-dimensional Zakharov–Kuznetsov equation are obtained. The graphs of the solution functions are presented by giving specific values to the constants. Numerical solutions are obtained by using finite difference method with new initial condition. Von Neumann’s Stability, Consistency and Linear Stability analysis of the equation are performed and [Formula: see text], [Formula: see text] norm errors are also examined with the truncation error. The exact solution obtained is presented via numerical solutions and absolute error graphs, and the analysis of exact solution and the numerical solutions are performed. Complex operations and graphical drawings were made using the computer package program.

Lie point symmetries, conservation laws and exact solutions of ( $$1+ n$$ 1 + n )-dimensional modified Zakharov–Kuznetsov equation describing the waves in plasma physics

In this study, we explore the modified form of (\(1+n\))-dimensional Zakharov–Kuznetsov equation, which is used to investigate the waves in dusty and magnetised plasma. It is proved that the equation follows the property of nonlinear self-adjointness. Lie point symmetries are calculated and conservation laws in the framework of the new general conservation theorem of Ibragimov are obtained. The \((1/G^{\prime })\), \((G^{\prime }/G)\)-expansion and modified Kudryshov methods are applied to extract exact analytical solutions. The so-called bright, dark and singular solutions are also found using the solitary wave ansatz method. The results obtained in this study are new and may be of significant importance where this model is used to study the waves in different plasmas.

Solitary wave solution and conservation laws of higher dimensional Zakharov‐Kuznetsov equation with nonlinear self‐adjointness

In this paper, we consider (1 + n)‐dimensional Zakharov‐Kuznetsov (ZK) equation for nonlinear self‐adjointness and symmetries. It is proved that the equation is nonlinearly self‐adjoint and its symmetries are computed. The conservation laws are obtained by using new general conservation theorem of Ibragimov for (1 + n)‐dimension and particularly for (1 + 3)‐dimensional ZK equation. Multiplier method is also used to find more conservation laws for (1 + 3)‐dimensional ZK equation and symmetry reduction technique is used to calculate the solitary wave solution.