Zakharov-Kuznetsov Equation Research Articles

On acoustic solitary waves in a multispecies degenerate relativistic magnetized plasma using physics informed neural networks

In this paper, we investigate the nonlinear electrostatic wave propagation in a two-dimensional magnetized plasma. The plasma consists of electron and positron components with relativistic degeneracy and stationary ions for neutralizing its background. Using the basic equations for this type of plasma in combination with the reductive perturbation method, we derived the Zakharov–Kuznetsov equation using the Lorentz transformation stretching method (LT). For the first time, we compared the results of the Galilean transformation stretching method (GT) and the LT method to investigate the effect of plasma parameters, such as the relativistic degeneracy parameter of electron particles (re0), the density ratio of ion to electrons (δ), and the normalized electron cyclotron (Ωe), on the amplitude and width of the wave solutions. The plasma parameters used in this research are representative of compact astrophysical objects. Numerical results showed that the amplitude of wave solutions obtained by the LT method is smaller than the GT method, but the width is greater. We provide a physical explanation for these differences. Furthermore, we present a physics-informed neural network (PINN) approach to directly recover the intrinsic nonlinear dynamics from spatiotemporal data. The PINN model uses a deep neural network constrained by the governing equations to learn the optimal parameters, with the aim of enhancing the predictive capabilities of the system. The results of this study provide valuable insight into the propagation of nonlinear waves in white dwarfs, where relativistic effects are significant. These findings could substantially advance the development of emerging machine learning applications in astrophysics.

On the analytical soliton approximations to damping forced Zakharov-Kuznetsov equation arising in dissipative nonthermal magnetized plasma

Abstract This paper aims to analyse the nonlinear characteristics of ion-acoustic solitary waves (IASWs) in a plasma system that is collisional and homogeneous in nature. The system under consideration comprises positive ions, negative ions, and nonthermal electrons and is influenced by an ambient magnetic field. To analyse the system, we utilize the reductive perturbation technique (RPT) and obtain the damped forced Zakharov-Kuznetsov (DFZK) equation. The DFZK equation provides a solitary wave solution when an external periodic force is present. Through numerical analysis of DFZK, we have observed that different parameters, e.g., the nonthermal parameters (r,q), the direction cosine, the temperature ratio of a positive ion to an electron, the temperature ratio of a negative ion to an electron, the density ratio of a negative ion to a positive ion source strength, and the source frequency, affect the phase velocity and the structures of IASWs significantly. This finding could help clarify the plasmas observed in the D- and F-regions of Earth's ionosphere, as well as in laboratory experiments where pair ions and nonthermal electrons are key characteristics.

Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

New Processing Technique of Jacobian Elliptic Equation and Its Application to the (3+1)-Dimensional Modified Korteweg de Vries–Zakharov–Kuznetsov Equation

This article introduces two kinds of processing techniques to solve Jacobian elliptic equations and obtain rich periodic wave solutions. Then, the equation was used as an auxiliary equation to solve the (3+1)-dimensional modified Korteweg de Vries–Zakharov–Kuznetsov (mKDV-ZK) equation. Combined with the mapping method, a large number of new types of exact periodic wave solutions were obtained, many of which were rarely found in previous research. Numerical simulations have demonstrated the evolution of various periodic waves in (3+1)-dimensional mKDV-ZK. The solutions and wave phenomena obtained in this article will help expand our understanding of the equation.

Exploring the dynamics of the heavy magnetized dusty plasma in laboratory experiments and solar wind interaction with lunar terminator: Theoretical predictions into the role of polarization force

This study is motivated by the lack of research that connects the dynamics of space dusty plasma (DP) with the existing experimental findings and the role of the magnetic field in the case of massive DP. For that purpose, we presented a theoretical interpretation of the dynamics of strongly (experimental) and weakly (space) magnetized DP, including the influence of the polarization force (PF). We used the multifluid model to describe the dynamics of dust particles while the hot plasma of electrons and ions is described by the Boltzmann distribution. We derived the dispersion relation to investigate the behavior of the compressive dust-acoustic waves (DAWs). To get a full picture of the mechanism of dust particles, we used the reductive perturbation technique to obtain the periodic solution for the Zakharov–Kuznetsov equation (ZK), which is coincidental with the observed wave profiles. In the case of the experimental application, we observed that the non-propagating instability starts at a small value of the polarization parameter (R), which is 0.02. Therefore, for values of R less than 0.02, the linear analysis revealed that increasing the value of the polarization parameter is not significant in distinguishing the growth rates. Moreover, we noted that both R and the applied magnetic field enhance the strength of the wave electric field. Interestingly, calculating the value of the electric field required to levitate the dust particles that are subject to a high magnetic field agrees with our theoretical findings. The current model has also been applied to DP observed in the lunar terminator. Our results reveal no role for the PF in the lunar terminator. The analysis conducted using the New Hampshire Dispersion Relation Solver (NHDS) confirms the dominance of electrostatic modes in response to the solar wind interaction with the lunar terminator. Moreover, at certain values of plasma parameters such as the temperature of solar wind electrons and ions and the density of dust particles and solar wind electrons, there is a cutoff in the fluid dispersion curve corresponding to certain values of the normalized wavenumber. Such electrostatic profiles could contribute to the observed glow in the horizontal direction of the lunar terminator.

An intriguing numerical strategy for Zakharov–Kuznetsov equation through graph-theoretic polynomials

This paper explores graph-theoretic polynomials to find the approximate solution of the (2+1)D Time-fractional Zakharov-Kuznetsov(TF-Z-K) equation. The Zakharov-Kuznetsov equations govern the behavior of nonlinear acoustic waves in the plasma of hot isothermal electrons and cold ions in the presence of a homogeneous magnetic field. Independence polynomials of the Ladder-Rung graph serve as the polynomial approximation for the suggested Independence Polynomial Collocation Method (IPCM). The Caputo fractional derivatives are adopted to determine the fractional derivatives in the TF-Z-K equation. The TF-Z-K equation is converted into a system of nonlinear algebraic equations using the collocation points in IPCM. The Newton-Raphson approach yields the solution of the suggested method by solving the resulting system. We’ve compared a few scenarios with the tangible outcomes to validate the procedure. Quantitative outcomes match the current findings and validate the exactness of IPCM compared t o the recent numerical and semi-analytical approaches in the literature.

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.

Electromagnetic effects on solitons propagation of the (3+1)-dimensional extended Zakharov-Kuznetsov dynamical model with applications

This paper investigates wave solutions and electromagnetic wave phenomena governed by the (3+1)-dimensional extended Zakharov-Kuznetsov equation (EZKE) utilizing the Sardar sub-equation method. With a focus on electromagnetic wave generation and propagation, we rigorously analyze fundamental properties, soliton solutions, and dynamic behaviors of the EZKE. Through this analytical technique, we unravel the complex interplay among various wave types, including solitary waves and electromagnetic structures, elucidating their formation mechanisms and interaction dynamics. Furthermore, we delve into the stability characteristics of the EZKE, enhancing our understanding of its mathematical and physical implications. Our findings not only contribute to theoretical insights into nonlinear wave phenomena in (3+1)-dimensional space but also hold practical significance in plasma physics, nonlinear optics, and electromagnetic wave propagation. This study advances the development of innovative wave manipulation and control techniques, with applications ranging from plasma confinement in fusion devices to the design of advanced photonic devices for telecommunications and sensing purposes.

Asymptotic stability of a finite sum of solitary waves for the Zakharov–Kuznetsov equation

We prove the asymptotic stability of a finite sum of well-ordered solitary waves for the Zakharov–Kuznetsov equation in dimensions two and three. We also derive a qualitative version of the orbital stability result, which will be useful for studying the collision of two solitary waves in a forthcoming paper.The proof extends the ideas of Martel, Merle and Tsai for the sub-critical gKdV equation in dimension one to the higher-dimensional case. It relies on monotonicity properties on oblique half-spaces and rigidity properties around one solitary wave introduced by Côte, Muñoz, Pilod and Simpson in dimension two, and by Farah, Holmer, Roudenko and Yang in dimension three.

A complete dynamical analysis of discrete electric lattice coupled with modified Zakharov–Kuznetsov equation

The behavior of nonlinear waves within a modified Zakharov–Kuznetsov equation and their interactions with discrete electric lattice structures are examined in this study. The ϕ6−model expansion method is utilized to acquire substantial knowledge into the complex dynamics of the system under consideration, particularly with regard to the discrete electric lattice and analytical electrical solitons. By incorporating higher-order effects and improving accuracy in representing specific physical conditions, the study achieves a more realistic portrayal of nonlinear wave dynamics. The investigation also sheds light on the relationship between non-linearity, discreteness, and equation dynamics by exploring the conditions that lead to the formation of solitons and other nonlinear structures. In addition, a unique set of electrical solitons is defined to explore dynamic behaviors such as chaotic, quasi-periodic, and periodic motions under various parameterized conditions, including an external damping force. Phase plane analysis is visualized by using dynamic structure 3D and 2D phase plots, is used for bifurcation and sensitivity inspections. Finally, time series graphs are offered as mathematical depictions of solitary waves, and Lyapunov exponents with real and complex eigenvalues are used to study the stability and chaotic behaviors of the system.

Zakharov–Kuznetsov Equation for Describing Low-Frequency Nonlinear Dust Acoustic Perturbations in Saturn’s Dusty Magnetosphere

A description is given of low-frequency nonlinear dust acoustic waves in Saturn’s dusty magnetosphere, which contains electrons of two types (hot and cold) obeying the kappa distribution, magnetospheric ions, and charged dust particles. For the corresponding conditions, the derivation of the Zakharov–Kuznetsov equation is given, which describes the nonlinear dynamics of dust acoustic waves in the case of low frequencies and a pancake-shaped wave packet along an external magnetic field. It is shown that under the conditions of Saturn’s magnetosphere there exist solutions of the Zakharov–Kuznetsov equation in the form of one-dimensional and three-dimensional solitons. Possible observations of the considered solitons in future space missions are discussed.

Analytical solutions to (modified) Korteweg–de Vries–Zakharov–Kuznetsov equation and modeling ion-acoustic solitary, periodic, and breather waves in auroral magnetoplasmas

Analytical solutions to (modified) Korteweg–de Vries–Zakharov–Kuznetsov equation and modeling ion-acoustic solitary, periodic, and breather waves in auroral magnetoplasmas

On decay properties for solutions of the Zakharov–Kuznetsov equation

This work mainly focuses on spatial decay properties of solutions to the Zakharov–Kuznetsov equation. For the two- and three-dimensional cases, it was established that if the initial condition u0 verifies 〈σ⋅x〉ru0∈L2(σ⋅x≥κ), for some r∈N, κ∈R, being σ be a suitable non-null vector in the Euclidean space, then the corresponding solution u(t) generated from this initial condition verifies 〈σ⋅x〉ru(t)∈L2σ⋅x>κ−νt, for any ν>0. Additionally, depending on the magnitude of the weight r, it was also deduced some localized gain of regularity. In this regard, we first extend such results to arbitrary dimensions, decay power r>0 not necessarily an integer, and we give a detailed description of the gain of regularity propagated by solutions. The deduction of our results depends on a new class of pseudo-differential operators, which is useful for quantifying decay and smoothness properties on a fractional scale. Secondly, we show that if the initial data u0 has a decay of exponential type on a particular half space, that is, ebσ⋅xu0∈L2(σ⋅x≥κ), then the corresponding solution satisfies ebσ⋅xu(t)∈Hpσ⋅x>κ−t, for all p∈N, and time t≥δ, where δ>0. To our knowledge, this is the first study of such property. As a further consequence, we also obtain well-posedness results in anisotropic weighted Sobolev spaces in arbitrary dimensions.Finally, as a by-product of the techniques considered here, we show that our results are also valid for solutions of the Korteweg–de Vries equation.

Diving into plasma physics: dynamical behaviour of nonlinear waves in (3 + 1)-D extended quantum Zakharov–Kuznetsov equation

Diving into plasma physics: dynamical behaviour of nonlinear waves in (3 + 1)-D extended quantum Zakharov–Kuznetsov equation

Qualitative analysis and soliton solutions of nonlinear extended quantum Zakharov-Kuznetsov equation

Qualitative analysis and soliton solutions of nonlinear extended quantum Zakharov-Kuznetsov equation

Symmetry analysis, conservation laws and exact soliton solutions for the $$(n+1)$$-dimensional modified Zakharov–Kuznetsov equation in plasmas with magnetic fields

Symmetry analysis, conservation laws and exact soliton solutions for the $$(n+1)$$-dimensional modified Zakharov–Kuznetsov equation in plasmas with magnetic fields

Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation

Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation

Electron acoustic solitary waves in relativistically quantum magnetoplasmas with electron exchange correlation effects

The propagating of electron acoustic solitary waves (EASWs) in relativistically magnetized quantum plasmas which consists inertial cold electrons, inertialess hot electrons and stationary background ions have been theoretically investigated. The modified Zakharov-Kuznetsov (ZK) equation is derived by adopting the standard reductive perturbation method to describe the nonlinear evolution of the EASWs. After discussing the effects of various parameter values on the solitary structures, it is shown that the solitary structures are very sensitive to the relativistic degeneracy effect, the cold to hot electrons number density ratio and the electron exchange correlation effects. The numerical analysis can be well described by the existing analytical theory. Our results may be useful in explaining the physics behind the formation of the EASWs in both astrophysical and laboratory plasmas.

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.

Propagation of nonlinear ion-acoustic fluctuations in the mantle of Venus

Motivated by the observations of ion-acoustic fluctuations with the Parker Solar Probe (PSP) and earlier by the Pioneer Venus Orbiter (PVO) in the Venusian magnetosheath, we investigate the nature of ion-acoustic solitary and double-layer (DL) structures in the mantle. We employed a hydrodynamic description along with reductive perturbation theory to derive the nonlinear Zakharov—Kuznetsov equation that elucidates the dynamics of three-dimensional ion-acoustic wave packets. Using the spacecraft measurements of the plasma configuration at Venus, we carried out a parametric analysis of these structures, including the influence of the magnetic field strength and the relative densities and temperatures, considering two cases: quasi-parallel and oblique propagation. Moreover, we determined the structural characteristics of these waves, where oblique (quasi-parallel) solitary waves have a potential of 0.4 V (0.4 V) and a maximum electric field amplitude Em ~ 0.024 mV m−1 (8 m V m−1) across spatial and temporal widths of ~40–80 km (~140–200 m) and 0.4 s (1.6 ms). These waves produce low-frequency electrostatic activity in the frequency range of 1.6–10 Hz (630–3160 Hz). Quasi-parallel DLs have potential drops of (6.5–13) V and Em ~ (0.16–0.35) mV m−1 with a width and duration of (100–120) m and ~1 ms, and a frequency range of ~630–3980 Hz. These outcomes can explain the detected electrostatic fluctuations above the ionosphere via PVO in the frequency channels of 730 Hz and 5.4 kHz. Furthermore, the DL features estimated in this work are in line with the recent PSP measurements of the DLs propagating in the magnetosheath of Venus.