Abundant Traveling Wave Solutions Research Articles

Nonlinear Dynamic Behaviors of the (3+1)-Dimensional B-Type Kadomtsev—Petviashvili Equation in Fluid Mechanics

This paper provides an investigation on nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation, which is used to model the propagation of weakly dispersive waves in a fluid. With the help of the Cole—Hopf transform, the Hirota bilinear equation is established, then the symbolic computation with the ansatz function schemes is employed to search for the diverse exact solutions. Some new results such as the multi-wave complexiton, multi-wave, and periodic lump solutions are found. Furthermore, the abundant traveling wave solutions such as the dark wave, bright-dark wave, and singular periodic wave solutions are also constructed by applying the sub-equation method. Finally, the nonlinear dynamic behaviors of the solutions are presented through the 3-D plots, 2-D contours, and 2-D curves and their corresponding physical characteristics are also elaborated. To our knowledge, the obtained solutions in this work are all new, which are not reported elsewhere. The methods applied in this study can be used to investigate the other PDEs arising in physics.

N-SOLITON, BREATHER, LUMP SOLUTIONS AND DIVERSE TRAVELING WAVE SOLUTIONS OF THE FRACTIONAL (2 + 1)-DIMENSIONAL BOUSSINESQ EQUATION

The [Formula: see text]-dimensional Boussinesq equation plays a key role in modeling the shallow water. In this work, we derive a new fractional [Formula: see text]-dimensional Boussinesq equation based on the conformable fractional derivative for the first time. By means of the Hirota bilinear method, we obtain the [Formula: see text]-soliton, breather and lump solutions. In addition, the abundant traveling wave solutions like bright solitary, dark solitary wave solutions are investigated by applying the variational method. The solutions are presented through the 3D plots and 2D contours by assigning the proper parameters. The corresponding physical interpretations are also elaborated. The findings in this work are expected to open some new horizons on the study of fractional PDEs in physics.

INVESTIGATION TO THE LOCAL FRACTIONAL FOKAS SYSTEM ON CANTOR SET BY A NOVEL TECHNOLOGY

This paper derives a new fractional Fokas system with the aid of the local fractional derivative. A novel technology named the extended rational fractal sine–cosine method is presented for the first time ever to develop the abundant traveling wave solutions. Two families (six sets) of the traveling wave solutions on Cantor set are successfully constructed. The dynamic behaviors of the solutions on Cantor sets are provided via numerical simulations. The obtained results in this work strongly prove that the proposed approach is simple but effective, which is expected to shed a new light on the study of the traveling wave theory of the local fractional equations.

On abundant wave structures of the unsteady korteweg-de vries equation arising in shallow water

The aim of this paper is to study the unsteady korteweg-de vries equation that plays an important role in describing the shallow water. Two analytical techniques namely the Sardar-subequation method and the energy balance method are employed to seek the abundant traveling wave solutions for the first time. By these two methods, plenty of traveling wave solutions such as the bright solitary wave solutions, dark solitary wave solutions, singular periodic wave solutions and perfect periodic wave solution that expressed in terms of the generalized hyperbolic functions, generalized trigonometric functions and the cosine function are obtained. Finally, the dynamic behaviors of the solutions are described through the 3D plot and 2D curve. The results in this paper demonstrate that the proposed methods are powerful and effective to construct the traveling wave solutions of the nonlinear evolution equations in ocean engineering and science.

Abundant travelling wave solutions of 3+1 dimensional Boussinesq equation with dual dispersion

This study presents utilization of Jacobi elliptic function expansion method to obtain the di¤erent types solutions of 3+1dimensional Boussinesq equation with dual dispersion. By using this method hyperbolic solutions and trigonometric functionsolutions are also obtained. The resulting outcomes verify that the preferred method is valid and reliable for the analytical technique of an extensive application of nonlinear phenomena.

Abundant stable and accurate solutions of the three-dimensional magnetized electron-positron plasma equations

This paper tries constructing analytical and numerical solutions of three-dimensional magnetized electron-positron plasma equations. In magnetized electron-positron plasma with equally hot and cool components of each species, these waves demonstrate weakly nonlinear ion-acoustic characteristics that make studying their dynamical behavior an essential target for its applications. A reduction perturbation method was previously employed to convert this model into the three-dimensional modified Korteweg–de Vries-Zakharov–Kuznetsov (3D KdV–ZK) equation. Operating two recent analytical techniques to the transformed equation gets abundant traveling wave solutions that are used to construct the requested conditions for applying a semi-analytical scheme. Furthermore, the stability property of obtained analytical solutions has been studied. The purposed analytical and semi-analytical method’s performance is investigated to show their power, effectiveness for applying to some nonlinear evolution equations.

Abundant traveling wave and numerical solutions for Novikov-Veselov system with their stability and accuracy

Solutions such as symmetric, bright soliton and periodic solutions play a prominent role in the field of differential equations, and they can be used to investigate several phenomena in nonlinear sciences. Some waves such as ion and magneto-sound waves in plasma are investigated by using some partial differential equations (PDEs) such as Novikov-Veselov (NV) equations. In this work, the improved -expansion approach is utilized to extract numerous soliton solutions for NV system. Hamiltonian system is invoked to analyze the stability of some solutions. The finite difference approach is successfully applied to achieve the numerical simulations of the proposed equations. We also introduce the stability and the accuracy of the numerical scheme. In order to validate the correctness of the accomplished results, we compare the exact solutions with the numerical solutions analytically and graphically. The presented techniques are very convenient and adequate and can be employed to other types of nonlinear evolution equations.

Traveling wave solutions of conformable time-fractional Klien-Fock-Gordon equation by the improved [formula omitted]-expansion method

This paper considers the improved tan(Ψ(ζ)/2)-expansion method to retrieve the traveling wave solutions of the time-fractional Klien-Fock-Gordon equation. The quadratic case has been discussed for the proposed model. The conformable derivative has been applied on the proposed model for investigating the fractional effects. As a result, abundant traveling wave solutions are found. The method contributes four kinds of solutions which are exponential, hyperbolic, trigonometric and rational function solutions for the aforementioned model. The obtained results are also explained graphically for some particular choices of fractional order α,0<α<1. The fractional effects on solutions have been illustrated through 3D plots and line graphs.

Traveling wave solutions of the Gardner equation in dusty plasmas

In this paper, we mainly study the Gardner equation that models the propagation of dust ion acoustic waves. By using the Exp-function method, abundant traveling wave solutions like bright solitary, bright-dark solitary, dark solitary (type-1, type-2, type-3, type-4) and periodic wave solutions or compact-like solutions (type-1, type-2, type-3, type-4) are constructed. The behaviors of the solutions are illustrated through the 3-D contours and 2-D curves. The obtained results show that the proposed method is straightforward, powerful and can provide more forms of the traveling wave solutions, which are expected to be helpful for the study of the traveling wave theory in dusty plasmas.

On new abundant exact traveling wave solutions to the local fractional Gardner equation defined on Cantor sets

In this article, we derive a new local fractional Gardner equation based on the local fractional derivative. A novel traveling wave transform of the non‐differentiable type is utilized to convert the local fractional Gardner equation into a nonlinear local fractional ordinary differential equation. Then a new method called the Mittag–Leffler function based method is proposed for the first time ever to construct the abundant traveling wave solutions. By this method, three families (six sets) of the traveling wave solutions on Cantor set are obtained. Finally, the performances of the various solutions on Cantor set are presented through the three‐dimensional contours. It shows that the proposed method is straightforward, powerful and can provide more forms of the traveling wave solutions for the local fractional equations, which is expected to be helpful for the study of the traveling wave theory for local fractional equations.

Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation

This paper deals with cross-kink waves in the (2+1)-dimensional KP–BBM equation in the incompressible fluid. Based on Hirota’s bilinear technique, cross-kink solutions related to KP–BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He’s variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational [Formula: see text] method on the generalized Hirota–Satsuma–Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

Abundant traveling wave solutions to an intrinsic fractional discrete nonlinear electrical transmission line

The main idea of this paper is to search for all traveling wave solutions to an intrinsic fractional discrete nonlinear electrical transmission line, which plays a essential role in obtaining the new insights in nonlinear voltage dynamics. We first give a brief introduction how to transform the discrete system into a continuous one, which is described by a fractional-order partial differential equation. After that, this equation is handled by conformable fractional transformation and the complete discrimination system for polynomial method (CDSPM). By applying the advanced method, the whole of the exact traveling wave solutions emerged in existing articles are obtained, especially we obtain the solitary wave solutions and elliptic functions solutions which are hardly founded by other methods. Notably, the elliptic functions solutions in rational form are discovered for the first time. Finally, the electrical characteristics and the fractional nature are revealed via graphical represents. By the depicted graphs, we intuitively observe the existence of the phenomena for periodic wave and solitary wave, and the time-fractional derivative is proved do has important influence on the behaviors of the solutions. Considering the significance of the nonlinear electrical transmission line, the acquired results would have wide application in electrical engineering and nonlinear voltage dynamics, liking analyzing and predicting the complex voltage wave propagation phenomenon in realistic electrical transmission system.

Abundant Traveling Wave and Numerical Solutions of Weakly Dispersive Long Waves Model

In this article, plenty of wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony ((2 + 1)-D KP-BBM) model are constructed by employing two recent analytical schemes (a modified direct algebraic (MDA) method and modified Kudryashov (MK) method). From the point of view of group theory, the proposed analytical methods in our article are based on symmetry, and effectively solve those problems which actually possess explicit or implicit symmetry. This model is a vital model in shallow water phenomena where it demonstrates the wave surface propagating in both directions. The obtained analytical solutions are explained by plotting them through 3D, 2D, and contour sketches. These solutions’ accuracy is also tested by calculating the absolute error between them and evaluated numerical results by the Adomian decomposition (AD) method and variational iteration (VI) method. The considered numerical schemes were applied based on constructed initial and boundary conditions through the obtained analytical solutions via the MDA, and MK methods which show the synchronization between computational and numerical obtained solutions. This coincidence between the obtained solutions is explained through two-dimensional and distribution plots. The applied methods’ symmetry is shown through comparing their obtained results and showing the matching between both obtained solutions (analytical and numerical).

Constructions of new abundant traveling wave solutions for system of the ion sound and Langmuir waves by the variational direct method

In this paper, a novel and simple approach called the variational direct method (VDM) is applied to construct the exact traveling wave solutions of the system of ion sound and Langmuir waves (SEISLWs), which is under the influence of the ponderomotive force caused by a nonlinear force experienced by a charged particle in an inhomogeneous oscillating electromagnetic field due to high-frequency field. These exact solutions include the bright soliton, bright-like soliton, kinky-bright soliton, bright-dark soliton and periodic wave solutions, which are all presented through the numerical results in the form of 3-D plots. Obtained results show that the VDM is simple, straightforward and powerful, which is expected to open new perspectives for the traveling wave theory.

Optical envelope patterns perturbation with full nonlinearity for Gerdjikov–Ivanov equation by trial equation method

In this paper, the classification of all envelope traveling wave solutions of Gerdjikov–Ivanov model with full nonlinearity is obtained by using complete discriminant system for polynomial method and trial equation method. All the results show that the Gerdjikov–Ivanov model has abundant traveling wave solutions, including isolated solutions, periodic solutions, rational singular solutions, double periodic continuous solutions and discontinuous solutions. Among them, some new solutions are obtained.

Bidirectional solitons and interaction solutions for a new integrable fifth-order nonlinear equation with temporal and spatial dispersion

A new nonlinear integrable fifth-order equation with temporal and spatial dispersion is investigated, which can be used to describe shallow water waves moving in both directions. By performing the singularity manifold analysis, we demonstrate that this generalized model is integrable in the sense of Painleve for one set of parametric choices. The simplified Hirota method is employed to construct the one-, two-, three-soliton solutions with non-typical phase shifts. Subsequently, an extended projective Riccati expansion method is presented and abundant travelling wave solutions are constructed uniformly. Furthermore, several new interaction solutions between periodic waves and kinky waves are also derived via a direct method. The rich interactions including overtaking collision, head-on collision and periodic-soliton collision are analyzed by some graphs.

Abundant traveling wave solutions to the resonant nonlinear Schrödinger’s equation with variable coefficients

In this paper, some new traveling wave solutions to the resonant nonlinear Schrödinger’s equation (R-NLSE) with time-dependent coefficients are constructed. The well-known auxiliary equation method is applied to develop numerous interesting classes of nonlinearities, namely the Kerr law and parabolic law. Such approach provides an extensive mathematical tool to develop a family of traveling wave solutions such as bright, dark, singular and optical solutions to the nonlinear evolution model. Moreover, with the aid of symbolic computation the three-dimensional plot and contour plot have been carried out to demonstrate the dynamical behavior of the nonlinear complex model.

New Analytical Solutions for Time Fractional Benjamin-Ono Equation Arising Internal Waves in Deep Water

In this article, the author sets up the abundant traveling wave solutions for time fractional Benjamin-Ono equation which was introduced to describe internal waves in stratified fluids by using Jacobi elliptic function expansion method. The traveling wave solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. It can be seen that the obtained results are found to be important for the statement of some physical demonstrations of problems in mathematical physics and ocean engineering. In ocean engineering Benjamin-Ono equations are generally used in computer simulation for the water waves in deep water and open seas.

Abundant travelling wave solutions for nonlinear Kawahara partial differential equation using extended trial equation method

ABSTRACTIn this paper, we use the extended trial equation method (ETEM) to construct the exact solutions in many different functions such as the trigonometric function, elliptic integral function, rational function, hyperbolic function and Jacobi elliptic functions for nonlinear evolution equation in mathematical physics via the fifth-order modified nonlinear Kawahara equation. In this method, the balance number is not constant as we have shown in other methods, but it is changed by changing the trial equation derivative definition. This method is powerful, reliable, effective and simple for solving more complicated nonlinear partial differential equations in mathematical physics.

A variety of exact analytical solutions of extended shallow water wave equations via improved (G’/G) -expansion method

In this present work, we have established exact solutions for (2+1) and (3+1) dimensional extended shallow-water wave equations in-volving parameters by applying the improved (G’/G) -expansion method. Abundant traveling wave solutions with arbitrary parameter are successfully obtained by this method, and these wave solutions are expressed in terms of hyperbolic, trigonometric, and rational functions. The improved (G’/G) -expansion method is simple and powerful mathematical technique for constructing traveling wave, solitary wave, and periodic wave solutions of the nonlinear evaluation equations which arise from application in engineering and any other applied sciences. We also present the 3D graphical description of the obtained solutions for different cases with the aid of MAPLE 17.