Modified KdV Zakharov Kuznetsov Research Articles

Investigation of the wave solutions of two space–time fractional equations in physics

In this study, two fractional differential equations describing the characteristics of three-dimensional nonlinear ion-acoustic waves (IAWs) in plasmas are considered with the conformable fractional derivative (CFD). 3+1-dimensional space–time fractional Jimbo–Miwa (JM) equation and 3+1-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV-ZK) equation are reduced into ordinary differential equations and an investigation into the exact solutions is carried out through the modified exponential function method (MEFM). These exact wave solutions are beneficial in interpreting the interior structure and characteristics of physical phenomena modeled by the aforementioned nonlinear evolution equations (NLEEs) arising in magnetized plasma. Various analytical methods were applied to such equations. Therefore, it is vital to determine the proper technique for the present problem. The proposed scheme has demonstrated a different spectrum of wave structures. The portrait of the exact solutions permits the exploration of plasmas, plasma models, and optics. The software system Mathematica is employed and the rational, exponential, trigonometric, and hyperbolic function solutions are acquired. The graphical results are presented for the physical nature of the solutions. The results are compared with the integer-order models.

Fractal solitary waves of the (3+1)-dimensional fractal modified KdV-Zakharov-Kuznetsov

In this work, the fractal (3+1)-D modified KdV-Zakharov-Kuznetsov (MKdV-ZK) model is studied, which can represent weakly non-linear waves under the unsmooth boundary. With the help of the fractal traveling wave transformation and the semi-inverse method, a fractal variational principle is obtained, which is a strong minimum one according to the He-Weierstrass function. From the variational principle, a fractal solitary wave solution is obtained, and the influence of un-smooth boundary on solitary waves is studied and the behaviors of the solutions are presented via 3-D plots. This paper shows that the fractal dimensions can affect the wave pattern, but cannot influence its crest value.

GENERALIZED VARIATIONAL STRUCTURE OF THE FRACTAL MODIFIED KDV–ZAKHAROV–KUZNETSOV EQUATION

A fractal modification of the modified KdV–Zakharov–Kuznetsov equation is suggested and its fractal generalized variational structure is established by means of the semi-inverse method. Furthermore, the obtained fractal generalized variational structure is discussed and verified through the two-scale transform from another dimension field in detail. The obtained fractal generalized variational structure reveals the conservation laws via the energy form in the fractal space and can be employed to study the fractal solitary wave properties.

A variety of closed-form solutions, Painlevé analysis, and solitary wave profiles for modified KdV–Zakharov–Kuznetsov equation in (3+1)-dimensions

The major goal of the article is to use the generalized exponential rational function method to seek abundant closed-form solutions and dynamics of solitary wave profiles to the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov (MKdV–ZK) equation. We also apply the Painlevé analysis with Kruskal simplification to investigate the integrability of the said equation. The (3+1)-dimensional MKdV–ZK model is a significant model with great advantages in various disciplines of physics, such as nonlinear optics, fluid dynamics, plasma physics, mathematical physics, and quantum mechanics. By taking advantage of the generalized exponential rational function technique, we extracted various soliton solutions, including exponential form, trigonometric form, and hyperbolic form function solutions. From both mathematical and physical points of view, it is important to obtain bright–dark soliton forms and other solitonic wave profiles. Furthermore, the results have been graphically revealed using 3D, 2D, and contour plots to demonstrate the underlying dynamics of the interactive waves. The obtained findings indicate the method’s efficacy and dependability, allowing it to be widely applied in a variety of advanced nonlinear evolution equations.

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

On the Conservation Laws and Exact Solutions to the (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation

In this paper, we consider conservation laws and exact solutions of the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov equation. Firstly, we construct conservation laws of the given equation with the help of the conservation theorem; the developed conservation laws are modified conservation laws. Then, we obtain exact solutions of the given equation via the (G′/G,1/G)-expansion method. The obtained solutions are classified as trigonometric solutions, hyperbolic solutions and rational solutions. Furthermore, graphical representations of the obtained solutions are given.

On the complex solutions to the (3+1)-dimensional conformable fractional modified KdV–Zakharov–Kuznetsov equation

This paper presents some exponential function solutions of the (3[Formula: see text]+[Formula: see text]1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK). The improved Bernoulli sub-equation function method (IBSEFM) is used in carrying out this computation. Choosing the suitable parameters, the 2D and 3D surfaces of solutions are plotted. The constructed results may be useful in explaining the physical meaning of the studied model.

Construction of solitary wave solutions of some nonlinear dynamical system arising in nonlinear water wave models

The higher order of nonlinear partial differential equations in mathematical physics is studied. We used the analytical mathematical methods of the nonlinear (3+1)-dimensional extended Zakharov–Kuznetsov dynamical, modified KdV–Zakharov–Kuznetsov and generalized shallow water wave equations to demonstrate the efficiency and validity of the proposed powerful technique. The shallow water wave models have been applied in tidal waves and weather simulation. Exact wave solutions of these models in various forms such as Kink and anti-Kink solitons, bright–dark soliton, solitary wave and periodic solutions are constructed that have plenty of applications in diverse areas of physics. Graphically, we presented the movement of some obtained solitary wave solutions that aids in understanding the physical phenomena of these models.

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Abstract In the current paper, we carry out an investigation into the exact solutions of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (fractional mKdV–ZK) equation. Based on the conformable fractional derivative and its properties, the fractional mKdV–ZK equation is reduced into an ordinary differential equation which has been solved analytically by the variable separated ODE method. Various types of analytic solutions in terms of hyperbolic functions, trigonometric functions and Jacobi elliptic functions are derived. All conditions for the validity of all obtained solutions are given.

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.

Analysis of Lie symmetries with conservation laws for the (3+1) dimensional time-fractional mKdV–ZK equation in ion-acoustic waves

In this paper, symmetry properties and conservation laws of the (3+1) dimensional time-fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation have been studied with Riemann–Liouville fractional derivative. Fractional Lie symmetry method has been used here for getting symmetry properties of (3+1) dimensional time-fractional mKdV–ZK equation. Here, the reduction of (3+1) dimensional time-fractional mKdV–ZK equation into fractional ordinary differential equation has been done by using Erdelyi–Kober fractional differential operator. Also, by using the new conservation theorem, the new conserved vectors for (3+1) dimensional time-fractional mKdV–ZK equation have been constructed with the help of formal Lagrangian with a detail derivation.

Exact solutions to the space–time fractional Schrödinger–Hirota equation and the space–time modified KDV–Zakharov–Kuznetsov equation

In this paper, the first integral method and the functional variable method are used to establish exact traveling wave solutions of the space–time fractional Schrodinger–Hirota equation and the space–time fractional modified KDV–Zakharov–Kuznetsov equation in the sense of conformable fractional derivative. The results obtained confirm that proposed methods are efficient techniques for analytic treatment of a wide variety of the space–time fractional partial differential equations.

The Exact Solution of the Space-Time Fractional Modified Kdv-Zakharov-Kuznetsov Equation

In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based on a fractional complex transform. These solutions have physics meanings in natural sciences. This method can be used to other nonlinear fractional differential equations.

Lie symmetry analysis for similarity reduction and exact solutions of modified KdV–Zakharov–Kuznetsov equation

In this paper, the Lie group analysis is used to carry out the similarity reduction and exact solutions of the \((3+1)\)-dimensional modified KdV–Zakharov–Kuznetsov equation. This research deals with the similarity solutions of mKdV–ZK equation. The mKdV–ZK equation has been reduced into a new partial differential equation with less number of independent variables, and again using Lie group symmetry method, the new partial differential equation is reduced into an ordinary differential equation. We have obtained the infinitesimal generators, commutator table of Lie algebra, symmetry group, and similarity reduction for the mKdV–ZK equation. In addition to that, solitary wave solutions of the mKdV–ZK equation are derived from the reduction equation. Thus, we obtain some new exact explicit solutions of the \((3+1)\)-dimensional mKdV–ZK equation which describes the dynamics of nonlinear waves in plasmas.

Different methods for [formula omitted]-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

In this paper, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the exp-function method, the (G′G)-expansion method and the generalized Kudryashov method are used to construct exact solutions for (3+1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation. This fractional equation can be turned into another nonlinear ordinary differential equation by fractional complex transformation and then these three methods are applied to solve it. As a result, some new exact solutions are obtained. The three methods demonstrate power, reliability and efficiency.

Improved fractional sub-equation method for (3+1) -dimensional generalized fractional KdV–Zakharov–Kuznetsov equations

In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics, namely the time fractional KdV–Zakharov–Kuznetsov (KdV–ZK) and space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equations by using improved fractional sub equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie’s modified Riemann–Liouville sense.

Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous Burgers equation

Mathematical modeling of many physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear partial differential equations (NPDEs) plays a significant role in the study of nonlinear physical phenomena. In this article, we construct the traveling wave solutions of modified KDV-ZK equation and viscous Burgers equation by using an enhanced (G '/G) -expansion method. A number of traveling wave solutions in terms of unknown parameters are obtained. Derived traveling wave solutions exhibit solitary waves when special values are given to its unknown parameters.Mathematics subject classification35C07; 35C08; 35P99

Exact and solitary wave solutions for the Tzitzeica–Dodd–Bullough and the modified KdV–Zakharov–Kuznetsov equations using the modified simple equation method

The modified simple equation (MSE) method is thriving in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in engineering and mathematical physics. In this study, we bring to bear the MSE method to look for the exact solutions via the Tzitzeica–Dodd–Bullough and the modified KdV–Zakharov–Kuznetsov equations involving parameters. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. By means of this scheme, we found some fresh traveling wave solutions of the above mentioned equations.