Modified Zakharov Kuznetsov Equation Research Articles

A NOVEL COMPUTATIONAL APPROACH TO THE LOCAL FRACTIONAL (3+1)-DIMENSIONAL MODIFIED ZAKHAROV–KUZNETSOV EQUATION

The fractional derivatives have been widely applied in many fields and has attracted widespread attention. This paper extracts a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKe) with the local fractional derivative (LFD) for the first time. Two special functions, namely, the [Formula: see text] and [Formula: see text] functions that are derived on the basis of the Mittag-Leffler function (MLF) defined on the Cantor set (CS), are employed to construct the auxiliary trial function to look into the exact solutions (ESs). Aided by Yang’s non-differentiable (ND) transformation, six groups of the ND ESs are found. The ND ESs on the CS for [Formula: see text] are depicted graphically. Additionally, as a comparison, the ESs of the classic (3+1)-dimensional MZKe for [Formula: see text] are also illustrated. The outcomes reveal that the derived method is powerful and effective, and can be used to deal with the other local fractional PDEs.

New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion

The aim of the present paper is to obtain and analyze new exact travelling wave solutions and bifurcation behavior of modified Zakharov-Kuznetsov (mZK) equation with higher-order dispersion term. For this purpose, the first and second simplest methods are used to build soliton solutions of travelling wave solutions. Furthermore, the bifurcation behavior of traveling waves including new types of quasiperiodic and multi-periodic traveling wave motions have been examined depending on the physical parameters. Multistability for the nonlinear mZK equation has been investigated depending on fixed values of physical parameters with various initial conditions. The suggested methods for the analytical solutions are powerful and beneficial tools to obtain the exact travelling wave solutions of nonlinear evolution equations (NLEEs). Two and three-dimensional plots are also provided to illustrate the new solutions. Bifurcation and multistability behaviors of traveling wave solution of the nonlinear mZK equation with higher-order dispersion will add some value to the literature of mathematical and plasma physics.

NONLINEAR DYNAMIC BEHAVIORS OF THE FRACTIONAL (3+1)-DIMENSIONAL MODIFIED ZAKHAROV–KUZNETSOV EQUATION

This paper derives a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation based on the conformable fractional derivative for the first time. Some new types of the fractal traveling wave solutions are successfully constructed by applying a novel approach which is called the fractal semi-inverse variational method. To our knowledge, the obtained results are all new and have not reported in the other literature. In addition, the dynamic characteristics of the different solutions on the fractal space are discussed and presented via the 3D plots, 2D contour and 2D curves. It can be found that: (1) The fractal order can not only affect the peak value of the fractal traveling waves, but also affect the wave structures, that is, the smaller the fractional order value is, the more curved the waveform is, and the slower waveform changes. (2) In the fractal space, the fractal wave keeps its shape unchanged in the process of the propagation and still meets the energy conservation. The methods in this paper can be used to study the other fractal PDEs in the physics, and the findings are expected to bring some new thinking and inspiration toward the fractal theory in physics.

W-shaped soliton solutions to the modified Zakharov-Kuznetsov equation of ion-acoustic waves in (3+1)-dimensions arise in a magnetized plasma

<abstract><p>This paper is presented to investigate the exact solutions to the modified Zakharov-Kuznetsov equation that have a critical role to play in mathematical physics. The $ \tan \left(\phi \left(\zeta \right)/2 \right) $-expansion, $ (m+G'(\zeta)/G(\zeta)) $-expansion and He exponential function methods are used to reveal various analytical solutions of the model. The equation regulates the treatment of weakly nonlinear ion-acoustic waves in a plasma consisting of cold ions and hot isothermal electrons throughout the existence of a uniform magnetic field. Solutions in forms of W-shaped, singular, periodic-bright and bright are constructed.</p></abstract>

Modified Zakharov–Kuznetsov Equation for Describing Low-Frequency Nonlinear Perturbations in Plasma of the Dusty Moon Exosphere

The nonlinear equation is obtained describing the dynamics of nonlinear wave structures in the dusty plasma above the illuminated surface of the Moon in the case of low frequencies and pancake-like shape of wave packet in the direction along the external magnetic field. This equation is the modified Zakharov–Kuznetsov equation. The analytical formula for the one-dimensional soliton solution is derived. The analysis of the stability of one-dimensional soliton solution was performed.

Well-posedness for the Cauchy Problem of the Modified Zakharov-Kuznetsov Equation

This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(\mathbb{R}^2)$ for $s \geq 1/4$. If $d \geq 3$, by employing $U^p$ and $V^p$ spaces, we establish the small data global well-posedness in the scaling critical Sobolev space $H^{s_c}(\mathbb{R}^d)$ where $s_c = d/2-1$.

Transverse instability of dust acoustic solitary waves in magnetized dusty plasma composed of vortex-like distribution ions

The dynamical behaviors of dusty plasma can be described by a (3+1)-dimensional modified Zakharov-Kuznetsov equation (mZKE) when the distribution of ions is vortex-like. The critical stable conditions for the line solitons are obtained by the linear stability analysis, which are also confirmed by the nonlinear dynamic evolution. An interesting phenomenon is found from the numerical results, maybe the first time, that the unstable line solitons of the mZKE will evolve into one or more completely localized soliton(s) after a long time evolution. Subsequently, we numerically studied the collision process of two line solitons. The results show that two stable line solitons can restore to their original states. However, if one of the two solitons or both of them are unstable, one or more completely localized solitons will appear during the collision. The results indicate that there are both elastic and inelastic collisions between line solitons.

Mass-concentration of low-regularity blow-up solutions to the focusing 2D modified Zakharov–Kuznetsov equation

We consider the focusing modified Zakharov–Kuznetsov (mZK) equation in two space dimensions. We prove that solutions which blow up in finite time in the \(H^1(\mathbb {R}^{2})\) norm have the property that they concentrate a non-trivial portion of their mass (more precisely, at least the amount equal to the mass of the ground state) at the blow-up time. For finite-time blow-up solutions in the \(H^s(\mathbb {R}^2)\) norm for \(\frac{17}{18}< s < 1\), we prove a slightly weaker result. Moreover, we prove that the stronger concentration result can be extended to the range \( \frac{17}{18} < s \le 1\) under an additional assumption on the upper bound of the blow-up rate of the solution. The main tools used here are the I-method and a profile decomposition theorem for a bounded family of \(H^1(\mathbb {R}^{2})\) functions.

A variety of exact solutions of the (2+1)-dimensional modified Zakharov–Kuznetsov equation

From the point of view of two distinct methods, we will construct new multiple types of private exact solutions of the (2+1)-dimensional modified Zakharov–Kuznetsov equation (MZKE) which is a famous model in plasma physics. The suggested methods for this purpose are the extended simple equation method (ESEM) and the Paul–Painleve approach method (PPAM). Moreover, the numerical solutions corresponding to the achieved solutions are demonstrated in the framework of the variational iteration method (VIM). Furthermore, we will demonstrate a good comparison not only between our achieved solutions but also with that realized previously by other authors who studied this model before.

Investigation of supernonlinear and nonlinear ion-acoustic waves in a magnetized electron-ion plasma with generalized (r, q) distributed electrons

Bifurcations of nonlinear and supernonlinear ion-acoustic waves (IAWs) are studied in an electron-ion plasmas with generalized (r, q)-distributed electrons. The IAWs are examined under the Zakharov–Kuznetsov (ZK) and modified ZK equations using the reductive perturbation technique. Transforming both the ZK and mZK equations into their corresponding dynamical systems, all possible phase spaces and potential energy functions are analyzed. The nonlinear periodic and solitary wave solutions are obtained under the ZK and mZK equations. A newly discovered supernonlinear wave, in particular, supernonlinear periodic wave under the modified ZK equation in electron-ion magnetized plasma with (r, q)-distributed electrons is reported for the first time in the literature. Nonlinear and supernonlinear wave solutions are shown under the influence of physical parameters. The proposed study contributes to new wave motions in slow solar wind streams and astrophysical plasmas.

Dynamics of dust-ion acoustic cnoidal and solitary pulses in a magnetized collisional complex plasma

ABSTRACT The diagnostic of magnetized radio frequency complex plasma via probing techniques and spectroscopy is challenging. However, a self-excited dust ion acoustic waves (DIAWs) have been observed employing laser scattering. Inspired by the current high enthusiasm for the theoretical description of complex plasma, we study the propagation of DIAWs in a three-dimensional collisional and magnetized dusty plasma medium. We consider the fluid model for dust and ion particles, while we neglect the inertia of electrons considering the Boltzmann distribution. A linear dispersion relation is obtained and a nonlinear modified Zakharov–Kuznetsov equation is derived utilizing the reductive perturbation technique. The effects of ion-neutral and dust-neutral collisions on the dynamics of the nonlinear DIAWs are considered. The phase portrait analysis and numerical illustration of a rarefactive DIA cnoidal waves and solitons are also presented. Our findings agree with experimental observations by Choudhary M, Bergert R, Mitic S, and Thomas MH. [Influence of external magnetic field on dust acoustic waves in a capacitive rf discharge. Contrib Plasma Phys. 2020;60:e201900115] as the growth of nonlinear periodic waves is recorded when the magnetic field is roughly 0.05 T and vanishes when the magnetic field is close to 0.1 T. Moreover, the model reveals that the electrostatic cnoidal and soliton waves propagate with low frequencies up to 140 Hz.

Approximate Analytical Solutions of Generalized Zakharov–Kuznetsov and Generalized Modified Zakharov–Kuznetsov Equations

Approximate Analytical Solutions of Generalized Zakharov–Kuznetsov and Generalized Modified Zakharov–Kuznetsov Equations

Supernonlinear wave and multistability in magneto-rotating plasma with (r, q) distributed electrons

Supernonlinear ion-acoustic waves (IAWs) and their multistability are studied under the Zakharov-Kuznetsov (ZK) and modified ZK (mZK) equations in a rotating magnetized electron-ion plasma that consists of generalized (r, q)-distributed electrons. Bifurcation of IAW is presented through phase plane analysis and existence of IAW solutions is also shown through graphs of potential energy function. Supernonlinear wave exists for system corresponding to mZK equation with nonlinear periodic and solitary wave solutions. Effects of strength (q) and flatness (r) of the (r, q)-distribution on IAW solutions are shown along with other parameters, such as speed of the wave (V) and direction cosine (n). Furthermore, introducing an extraneous periodic perturbation, multistability property of the perturbed dynamical system is examined, and it is displayed using phase space and time series plots. Features of coexisting trajectories are examined for different values of spectral indices (r, q). Existence of two different types of chaos are supported through Lyapunov exponent graphs. Here, acceptable parametric values of spectral indices (r, q) are considered from data values of slow solar wind streams. Hence, supernonlinear periodic IAW and multistability property under the modified ZK equation are reported for the first time in rotating electron-ion magnetized plasma with (r, q)-distributed electrons. This study is applicable to understand supernonlinear wave features and multistability property of ion-acoustic wave motions featured in slow solar wind streams.

An analytical Technique for Solving New Computational Solutions of the Modified Zakharov-Kuznetsov Equation Arising in Electrical Engineering

The modified (G'/G)-expansion method is an efficient method that has appeared in recent times for solving new computational solutions of nonlinear partial differential equations (NPDEs) arising in electrical engineering. This research has applied this process to seek novel computational results of the developed Zakharov-Kuznetsov (ZK) equation in electrical engineering. With 3D and contour graphical illustration, mathematical results explicitly exhibit the proposed algorithm's complete honesty and high performance.

Asymptotic 𝐾-soliton-like solutions of the Zakharov-Kuznetsov type equations

We study here the Zakharov-Kuznetsov equation in dimension 2 2 , 3 3 and 4 4 and the modified Zakharov-Kuznetsov equation in dimension 2 2 . Those equations admit solitons, characterized by their velocity and their shift. Given the parameters of K K solitons R k R^k (with distinct velocities), we prove the existence and uniqueness of a multi-soliton u u such that \[ ‖ u ( t ) − ∑ k = 1 K R k ( t ) ‖ H 1 → 0 as t → + ∞ . \| u(t) - \sum _{k=1}^K R^k(t) \|_{H^1} \to 0 \quad \text {as} \quad t \to +\infty . \] The convergence takes place in H s H^s with an exponential rate for all s ≥ 0 s \ge 0 . The construction is made by successive approximations of the multi-soliton. We use classical arguments to control of H 1 H^1 -norms of the errors (inspired by Martel [Amer. J. Math. 127 (2005), pp. 1103–1140]), and introduce a new ingredient for the control of the H s H^s -norm in dimension d ≥ 2 d\geq 2 , by a technique close to monotonicity.

Instability of Dust–Acoustic Waves in Plasmas with Two-Temperature Nonthermal Ions

Theoretical and numerical investigations are carried out for the instability of the dust–acoustic waves (DAWs) under the transverse perturbation in a magnetized dusty plasma with two-temperature nonthermal ions. By the reductive perturbation technique, the Zakharov–Kuznetsov (ZK) equation and modified ZK equation of the DAWs are derived. Under the higher-order transverse perturbations, the instability growth rate of the soliton solutions of the ZK equations is studied. It is found that the most unstable wave mode exists in system. It is also noted that solitary waves are unstable when the ratio of the perturbation wave length to solitary wave width in the experimental coordinate is larger than a certain critical value.

Comment on “Construction of traveling waves patterns of ([formula omitted])-dimensional modified Zakharov–Kuznetsov equation in plasma physics” by Adil Jhangeer and et al. [RINP, 19 2020, 103330

We analyze the paper “Construction of traveling waves patterns of (1+n)-dimensional modified Zakharov-Kuznetsov equation in plasma physics” by Adil Jhangeer and et al. [RINP, 2020, 103330] and demonstrate that results of this paper are wrong. The correct results are given.

Initial-boundary value problems on a half-strip for the modified Zakharov–Kuznetsov equation

Initial-boundary value problems on a half-strip with different types of boundary conditions for the modified Zakharov–Kuznetsov equation are considered. Results on local and global well-posedness in classes of mild and regular solutions, internal regularity of mild solutions and long-time decay of both mild and regular solutions are established. The solutions are considered in weighted at infinity Sobolev spaces.

Propagation of dispersive wave solutions for (3 + 1)-dimensional nonlinear modified Zakharov–Kuznetsov equation in plasma physics

In the current study, we instigate the four-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation. The NLmZK equation guides the attitude of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Two different methods are used, namely the sine-Gordon expansion method (SGEM) and the [Formula: see text]-expansion method to the proposed model. We have successfully constructed some topological, non-topological, and wave solutions. In addition, the 2D, 3D, and contour graphs of the solutions are also plotted under the choice of appropriate values of the parameters.

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.