Korteweg De Vries Zakharov Kuznetsov Equation Research Articles

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

Exact solutions and bifurcations for the (3+1)-dimensional generalized KdV-ZK equation

In this paper, a class of (3+1)-dimensional generalized Korteweg–de Vries-Zakharov-Kuznetsov (KdV-ZK) equation is studied by utilizing the bifurcation theory of the planar dynamical systems and the Fan sub-function method. This model can be used to explain the effects of magnetic fields on weakly nonlinear ion-acoustic waves investigated in plasma fields composed of cold and hot electrons. Under the different parameter conditions, the phase portraits and bifurcations are derived, and new exact solutions including soliton, periodic, kink and breaking wave solutions for the model are constructed. Moreover, some exact solutions, which contain soliton, kink, trigonometric function, hyperbolic function, Jacobi elliptic function solutions, are derived via the improved Fan sub-function method. The types of solutions obtained completely correspond to the types of the orbits acquired above, which verifies the validity of the method. Finally, the physical structures of some exact solutions are analyzed in graphical forms.

Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis

In this paper, we explore analytical solutions for the (3+1)-dimensional time-fractional modified Korteweg–de Vries Zakharov-Kuznetsov equation, which incorporates a conformable derivative. Our interest in this model is driven by its significant role in simulating ion-acoustic waves in magnetized plasma. We adopt the unified Riccati equation expansion method and the new Kudrashov method to discover soliton solutions. Our approach uncovers various soliton types, such as kink, singular, periodic-singular, and bright solitons. We conduct a thorough analysis of how different parameters affect wave propagation, enhancing our study with descriptive figures and insightful observations. Furthermore, we delve into the modulation instability characteristic of this model. The influence of specific parameters, like wave number and the order of the conformable derivative, on wave dynamics is demonstrated through detailed visualizations. We also present 2D and 3D graphical representations of these solutions.

Analytical soliton solutions and wave profiles of the (3+1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation

In this research paper, the improved generalized Riccati equation mapping (IGREM) method is employed to get analytical soliton solutions of (3+1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV-ZK) equation. By giving appropriate parametric values, we obtain soliton solutions of various types, such as kink soliton, singular soliton, and periodic-singular soliton. The description provided in physical variables allows for the study of genuine multispecies plasmas, plasma models, and frequency regimes. These solutions are essential in illuminating several physical phenomena in engineering and other applied sciences. In addition, 3D and 2D graphs of some selected solutions are sketched for the best physical characterization of the obtained results. This method can also be implemented in many non-linear equations in contemporary research areas. Comparison with previous studies shows that the discovery of singular soliton is novel. Therefore, our research represents a significant advancement in the field and contributes to expanding the knowledge regarding soliton behavior. The applications of this study include electron and ion-acoustic modes in regular plasmas with hot Boltzmann electron species, as well as ion and dust-acoustic modes, depending on the specific modeling of the heavier components.

Some traveling wave solutions to the generalized (3 + 1)‐dimensional Korteweg–de Vries–Zakharov–Kuznetsov equation in plasma physics

The generalized Korteweg–de Vries–Zakharov–Kuznetsov model is one of the dominant nonlinear complex structures to exhibit the influence of magnetic fields on weak ion‐acoustic waves in plasma made up of cool and hot electrons. In this study, the nonlinear higher dimensional model is subjected to the extended ‐expansion approach and the ‐expansion strategy in order to discover some novel traveling waves solutions and other exact solutions. There have been several solutions discovered, such as bright, dark, periodic, bell‐shaped, and single bell‐shaped wave structures. Furthermore, the stability analysis of the model is also reported along with the dynamical visualization of the charismatic behavior of various solitons with the aid of 3D, 2D, and contour plots. The proposed techniques can also be implemented to analyze a wide range of nonlinear evolution models emerging in diverse disciplines of science and technology including mathematical physics and plasma physics.

Dust–ion acoustic solitary waves in a collisionless magnetized five components plasma

Abstract We have derived a Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation to study the nonlinear behavior of dust–ion acoustic waves in a collisionless magnetized five components dusty plasma consisting of warm adiabatic ions, nonthermal hot electrons, isothermal cold electrons, nonthermal positrons and static negatively charged dust particulates. It is found that the coefficient of the nonlinear term of the KdV-ZK equation vanishes along different family of curves in different compositional parameter planes. In this situation, to describe the nonlinear behavior of dust–ion acoustic waves, we have derived a modified KdV-ZK (MKdV-ZK) equation. When the coefficients of the nonlinear terms of both KdV-ZK and MKdV-ZK equations are simultaneously equal to zero, then we have derived a further modified KdV-ZK (FMKdV-ZK) equation which effectively describes the nonlinear behavior of dust–ion acoustic waves. Analytically and numerically, we have investigated the solitary wave solutions of different evolution equations propagating obliquely to the direction of the external static uniform magnetic field. We have seen that the amplitude of the KdV soliton strictly increases with increasing β e, whereas the amplitude of the MKdV soliton strictly decreases with increasing β e, where β e is the nonthermal parameter associated with the hot electron species. Also, there exists a critical value β r ( c ) ${\beta }_{\text{r}}^{(\text{c})}$ of β e such that the FMKdV soliton exists within the interval β r ( c ) < β e ≤ 4 7 ${\beta }_{\text{r}}^{(\text{c})}< {\beta }_{\text{e}}\le \frac{4}{7}$ , whereas the FMKdV soliton does not exist within the interval 0 < β e < β r ( c ) $0< {\beta }_{\text{e}}< {\beta }_{\text{r}}^{(\text{c})}$ . We have also discussed the effect of different parameters of the system on solitary waves obtained from the different evolution equations.

Weakly nonlinear electron-acoustic waves in the fluid ions propagated via a (3+1)-dimensional generalized Korteweg–de-Vries–Zakharov–Kuznetsov equation in plasma physics

In present work, we discussed the (3+1)-dimensional generalized Korteweg–de-Vries–Zakharov–Kuznetsov equation (gKdV-ZKe) which describes the effect of magnetic field on weak nonlinear ion-acoustic waves studied in the field of plasma comprised of cool and hot electrons. The solutions of gKdV-ZKe are acquired by new modified extended direct algebraic method (MEDAM) and are observed in the form of solitary, shock, singular, shock-singular, solitary-shock and double singular soliton solutions. Moreover, the families of rational solutions are also emerged during the derivation. The results we derive, in this article, are useful to study and verify the analytical solutions with numerical and experimental solutions in plasma physics. The obtained solutions are also illustrated graphically.

Exact solutions of time fractional Korteweg–de Vries–Zakharov–Kuznetsov equation

In this study, an analytic method based on the Jacobi elliptic functions has been presented to obtain the exact solutions of time fractional Korteweg–de Vries–Zakharov–Kuznetsov (KdV–ZK) equation. This equation is reduced to a nonlinear ordinary differential equation. The elementary and elliptic function solutions of this equation are investigated. Many exact solutions containing the rational, complex, trigonometric, and hyperbolic functions are also found. Besides, some of the solutions are demonstrated by the graphics.

Modified KdV–Zakharov–Kuznetsov dynamical equation in a homogeneous magnetised electron–positron–ion plasma and its dispersive solitary wave solutions

Propagation of three-dimensional nonlinear ion-acoustic solitary waves and shocks in a homogeneous magnetised electron–positron–ion plasma is analysed. Modified extended mapping method is introduced to find ion-acoustic solitary wave solutions of the three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, solitary wave solutions (which represent electrostatic field potential), electric fields, magnetic fields and quantum statistical pressures are obtained with the aid of Mathematica. These new exact solitary wave solutions are obtained in different forms such as periodic, kink and antikink, dark soliton, bright soliton, bright and dark solitary wave etc. The results are expressed in the forms of hyperbolic, trigonometric, exponential and rational functions. The electrostatic field potential and electric and magnetic fields are shown graphically. These results demonstrate the efficiency and precision of the method that can be applied to many other mathematical and physical problems.

New exact solutions to the space–time fractional nonlinear wave equation obtained by the ansatz and functional variable methods

In this paper, the ansatz method and the functional variable method are employed to find new analytic solutions for the space–time nonlinear fractional wave equation, the space–time fractional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation and the space–time fractional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, some exact solutions are obtained in terms of hyperbolic and periodic functions. It is shown that the proposed methods provide a more powerful mathematical tool for constructing exact solutions for many other nonlinear fractional differential equations occurring in nonlinear physical phenomena. We have also presented the numerical simulations for these equations by means of three dimensional plots.

Electron acoustic solitary waves in a magnetized plasma with nonthermal electrons and an electron beam

A theoretical investigation is carried out to study the obliquely propagating electron acoustic solitary waves having nonthermal hot electrons, cold and beam electrons, and ions in a magnetized plasma. We have employed reductive perturbation theory to derive the Korteweg-de-Vries-Zakharov-Kuznetsov (KdV-ZK) equation describing the nonlinear evolution of these waves. The two-dimensional plane wave solution of KdV-ZK equation is analyzed to study the effects of nonthermal and beam electrons on the characteristics of the solitons. Theoretical results predict negative potential solitary structures. We emphasize that the inclusion of finite temperature effects reduces the soliton amplitudes and the width of the solitons increases by an increase in the obliquity of the wave propagation. The numerical analysis is presented for the parameters corresponding to the observations of “burst a” event by Viking satellite on the auroral field lines.

Soliton solutions for a (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a plasma

Under investigation in this paper is a (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation, which describes the nonlinear behaviors of ion-acoustic waves in a magnetized plasma where the cooler ions are treated as a fluid with adiabatic pressure and the hot isothermal electrons are described by a Boltzmann distribution. With the Hirota method and symbolic computation, we obtain the one-, two- and three-soliton solutions for such an equation. We graphically study the solitons related with the coefficient of the cubic nonlinearity [Formula: see text]. Amplitude of the one soliton increases with increasing [Formula: see text], but the width of one soliton keeps unchanged as [Formula: see text] increases. The two solitons and three solitons are parallel, and the amplitudes of the solitons increase with increasing [Formula: see text], but the widths of the solitons are unchanged. It is shown that the interactions between the two solitons and among the three solitons are elastic.

Stability analysis solutions for nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a magnetized electron–positron plasma

The nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov ​(mKdV–ZK) equation governs the behavior of weakly nonlinear ion-acoustic waves in magnetized electron–positron plasma which consists of equal hot and cool components of each species. By using the reductive perturbation procedure leads to a mKdV–ZK equation governing the oblique propagation of nonlinear electrostatic modes. The stability of solitary traveling wave solutions of the mKdV–ZK equation to three-dimensional long-wavelength perturbations is investigated. We found the electrostatic field potential and electric field in form traveling wave solutions for three-dimensional mKdV–ZK equation. The solutions for the mKdV–ZK equation are obtained precisely and efficiency of the method can be demonstrated.

Deriving the New Traveling Wave Solutions for the Nonlinear Dispersive Equation, KdV-ZK Equation and Complex Coupled KdV System Using Extended Simplest Equation Method

In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov–Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.

Stability of ion–acoustic solitary waves in a multi-species magnetized plasma consisting of non-thermal and isothermal electrons

AbstractA theoretical study of the first-order stability analysis of an ion–acoustic solitary wave, propagating obliquely to an external uniform static magnetic field, has been made in a plasma consisting of warm adiabatic ions and a superposition of two distinct populations of electrons, one due to Cairns et al. and the other being the well-known Maxwell–Boltzmann distributed electrons. The weakly nonlinear and the weakly dispersive ion–acoustic wave in this plasma system can be described by the Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation and different modified KdV-ZK equations depending on the values of different parameters of the system. The nonlinear term of the KdV-ZK equation and the different modified KdV-ZK equations is of the form [φ(1)]ν(∂φ(1)/∂ζ), where ν = 1, 2, 3, 4; φ(1) is the first-order perturbed quantity of the electrostatic potential φ. For ν = 1, we have the usual KdV-ZK equation. Three-dimensional stability analysis of the solitary wave solutions of the KdV-ZK and different modified KdV-ZK equations has been investigated by the small-k perturbation expansion method of Rowlands and Infeld. For ν = 1, 2, 3, the instability conditions and the growth rate of instabilities have been obtained correct to order k, where k is the wave number of a long-wavelength plane-wave perturbation. It is found that ion–acoustic solitary waves are stable at least at the lowest order of the wave number for ν = 4.

Alternative ion-acoustic solitary waves in magnetized low beta plasma consisting of warm adiabatic ions, nonthermal electrons, and electrons having a vortexlike distribution

The solitary wave structures of the ion-acoustic waves propagating obliquely to an external uniform magnetic field have been considered in a low beta plasma consisting of warm adiabatic ions, nonthermal electrons, due to Cairns et al. [Geophys. Res. Lett. 22, 2709 (1995)], which generates the fast energetic electrons and electrons having a vortexlike distribution, due to Schamel [Plasma Phys. 13, 491 (1971); 14, 905 (1972)], taking care of both free and trapped electrons, immersed in a uniform static magnetic field. The nonlinear dynamics of ion-acoustic waves in such a plasma is shown to be governed by Schamel’s modified Korteweg–de Vries–Zakharov–Kuznetsov equation. When the coefficient of the nonlinear term of this equation vanishes, the vortexlike velocity distribution function of electrons simply becomes the isothermal velocity distribution function of electrons and the nonlinear behavior of the same ion-acoustic wave is described by a Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation. A combined Schamel’s modified Korteweg–de Vries–Zakharov–Kuznetsov (S-KdV-ZK) equation is shown to describe the nonlinear behavior of ion-acoustic wave when the vortexlike velocity distribution function of electrons approaches the isothermal velocity distribution function of electrons, i.e., when the contribution of trapped electrons tends to zero. This combined S-KdV-ZK equation admits an alternative solitary wave solution having profile different from sech4 or sech2. The condition for the existence of this alternative solitary wave solution has been obtained. It is found that this alternative solitary wave solution approaches the solitary wave solution (sech2-profile) of the KdV-ZK equation when the contribution of trapped electrons tends to zero.

Ion-acoustic solitary waves in a multi-species magnetized plasma consisting of non-thermal and isothermal electrons

AbstractThe Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation describes the nonlinear behaviour of long-wavelength weakly nonlinear ion-acoustic waves propagating obliquely to an external uniform (space independent) static (time independent) magnetic field in a plasma consisting of warm adiabatic ions and a superposition of two distinct population of electrons, one due to Cairns et al. (1995Geophys. Res. Lett.22, 2709), which generates the fast energetic electrons, and the other the well-known Maxwell–Boltzman distributed electrons. It is found that the compressive or rarefactive nature of the ion-acoustic solitary wave solution of the KdV-ZK equation does not depend on the ion temperature if σc<0 or σc>1, where σcis a function of β1,nscand σsc. This β1is the non-thermal parameter associated with the non-thermal velocity distribution function of electrons (Cairns et al.),nscis the ratio of the unperturbed number density of the isothermal electrons to that of the non-thermal electrons and σscis the ratio of the average temperature of the non-thermal electrons to that of the isothermal electrons. The KdV-ZK equation describes compressive or rarefactive ion-acoustic solitary wave according to whether σc<0 or σc>1. When 0 ≤ σc≤ 1, the KdV-ZK equation describes compressive or rarefactive ion-acoustic solitary wave according to whether σ>σcor σ<σc, where σ is the ratio of the average temperature of ions to the effective temperature of electrons. If σ takes the value σcwith 0 ≤ σc≤ 1, the coefficient of the nonlinear term of the KdV-ZK equation vanishes and for this case the nonlinear evolution equation of the ion-acoustic wave is a modified KdV-ZK (MKdV-ZK) equation. It is found that the four-dimensional parameter space, originated from the physically admissible values of the four-parameters β1, σ, σscandnscof the present extended plasma system, can be decomposed into five mutually disjoint subsets with respect to the critical values of the different parameters, and the nonlinear behaviour of the same ion acoustic wave in those subsets can be described by different modified KdV-ZK equations. A general method of perturbation of the dependent variables has been developed to obtain the different evolution equations. The applicability of the different evolution equations and their solitary wave solutions (along with the conditions for their existence) have been investigated analytically and graphically.

Modified Korteweg–de Vries–Zakharov–Kuznetsov solitons in symmetric two-temperature electron–positron plasmas

AbstractSolitary waves are investigated in a magnetized electron–positron plasma consisting of equal hot and cool components of each species. The hot components have a Boltzmann distribution and the cool components are described by the fluid equations. A modified Korteweg–de Vries–Zakharov–Kuznetsov equation governing the oblique propagation of nonlinear electrostatic modes is derived using the reductive-perturbation technique. Soliton amplitudes are studied as a function of plasma parameters such as the particle number densities and the temperatures. Such results may be of relevance to the magnetosphere of pulsars.

Alternative ion-acoustic solitary waves in magnetized plasma consisting of warm adiabatic ions and non-thermal electrons having vortex-like velocity distribution: existence and stability

AbstractThe solitary structures of the ion-acoustic waves have been considered in a plasma consisting of warm adiabatic ions and non-thermal electrons (due to the presence of fast energetic electrons) having a vortex-like velocity distribution function (due to the presence of trapped electrons), immersed in a uniform (space-independent) and static (time-independent) magnetic field. The nonlinear dynamics of ion-acoustic waves in such a plasma is governed by the Schamel's modified Korteweg–de Vries–Zakharov–Kuznetsov (S-ZK) equation. This equation admits solitary wave solutions having a profile sech4. When the coefficient of the nonlinear term of this equation vanishes, the vortex-like velocity distribution function of electrons simply becomes the non-thermal velocity distribution function of electrons and the nonlinear behaviour of the same ion-acoustic wave is described by a Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation. This equation admits solitary wave solutions having a profile sech2. A combined S–KdV–ZK equation more efficiently describes the nonlinear behaviour of an ion-acoustic wave when the vortex-like velocity distribution function of electrons approaches the non-thermal velocity distribution function of electrons, i.e. when the contribution of trapped electrons tends to zero. This combined S-KdV-ZK equation admits an alternative solitary wave solution having a profile different from either sech4or sech2. The condition for the existence of this alternative solitary wave solution has been derived. It is found that this alternative solitary wave solution approaches the solitary wave solution (the sech2profile) of the KdV-ZK equation when the contribution of trapped electrons tends to zero. The three-dimensional stability of these solitary waves propagating obliquely to the external uniform and static magnetic field has been investigated by the multiple-scale perturbation expansion method of Allen and Rowlands. The instability condition and the growth rate of the instability have been derived at the lowest order. It is also found that the instability condition and growth rate of instability of the alternative solitary waves are exactly the same as those of the solitary waves as determined from the KdV-ZK equation (the sech2profile) when the contribution of trapped electrons tends to zero.

Existence and stability of alternative ion-acoustic solitary wave solution of the combined MKdV-KdV-ZK equation in a magnetized nonthermal plasma consisting of warm adiabatic ions

The purpose of this paper is to present the recent work of Das et al. [J. Plasma Phys. 72, 587 (2006)] on the existence and stability of the alternative solitary wave solution of fixed width of the combined MKdV-KdV-ZK (Modified Korteweg-de Vries-Korteweg-de Vries-Zakharov-Kuznetsov) equation for the ion-acoustic wave in a magnetized nonthermal plasma consisting of warm adiabatic ions in a more generalized form. Here we derive the alternative solitary wave solution of variable width instead of fixed width of the combined MKdV-KdV-ZK equation along with the condition for its existence and find that this solution assumes the sech profile of the MKdV-ZK (Modified Korteweg-de Vries-Zakharov-Kuznetsov) equation, when the coefficient of the nonlinear term of the KdV-ZK (Korteweg-de Vries-Zakharov-Kuznetsov) equation tends to zero. The three-dimensional stability analysis of the alternative solitary wave solution of variable width of the combined MKdV-KdV-ZK equation shows that the instability condition and the first order growth rate of instability are exactly the same as those of the solitary wave solution (the sech profile) of the MKdV-ZK equation, when the coefficient of the nonlinear term of the KdV-ZK equation tends to zero.