Exact Traveling Wave Solutions Research Articles

Classification and Traveling Wave Solutions for the Gilson–Pickering Equation

The main purpose of this paper is to form a complete group classification for the Gilson–Pickering equation. Exact traveling wave solutions and bifurcations of the Gilson–Pickering equation are studied by the method of dynamical systems. The study on the wave solutions of the model derives a planar Hamiltonian system. Based on phase portraits, many exact explicit parametric representations of wave solutions are obtained under different parametric conditions.

Bifurcations and Traveling Wave Solutions of Lakshmanan–Porsezian–Daniel Equation with Parabolic Law Nonlinearity

In this paper, we investigate the bifurcations and exact traveling wave solutions of the Lakshmanan–Porsezian–Daniel model with parabolic law nonlinearity. Employing the bifurcation theory of planar dynamical system, we obtain the phase portraits of the corresponding traveling wave system. The existence of the two singular straight lines leads to the dynamical behavior of solutions with two scales. Corresponding to some special level curves, we give the explicit exact solutions under different parameter conditions, containing peakon solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic wave solutions and periodic peakon solutions. Moreover, we collect all these traveling solutions into a theorem. Finally, we discuss a small perturbation of the LPD equation and persistence of the traveling waves.

Analysis of a population model with advection and an autocatalytic-type growth

This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth. As opposed to a logistic growth where the rate of growth of the population decreases from the onset, an autocatalytic growth has a point of inflection where the rate of growth switches from an increasing trend to a decreasing trend. Employing the idea of Painlevé property, we show that a variety of exact traveling wave solutions can be obtained for this model depending on the choice of the advection term. In particular, due to situations in resource distribution or environmental variations, if the advection is represented as a decaying function in time or an oscillating function in time, we are able to find exact solutions with interesting behavior. We also carry out a computational study of the model using an exponentially upwinded numerical scheme and illustrate the movement of the solutions and their characteristics pictorially.

Analytical and Numerical Solutions for a Kind of High-Dimensional Fractional Order Equation

In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. It is rarely studied because the order of the highest-order derivative term of this equation is higher than the common generalized (4+1)-dimensional Fokas equation. Firstly, the (4+1)-dimensional time-fractional Fokas equation with the Riemann–Liouville fractional derivative is derived by the semi-inverse method and variational method. Further, the symmetry of the time-fractional equation is obtained by the fractional Lie symmetry analysis method. Based on the symmetry, the conservation laws of the time fractional equation are constructed by the new conservation theorem. Then, the G′G-expansion method is used here to solve the equation and obtain the exact traveling wave solutions. Finally, the spectral method in the spatial direction and the Gru¨nwald–Letnikov method in the time direction are considered to obtain the numerical solutions of the time-fractional equation. The numerical solutions are compared with the exact solutions, and the error results confirm the effectiveness of the proposed numerical method.

The Application of the Functional Variable Method for Solving the Loaded Non-linear Evaluation Equations

In this article, we construct exact traveling wave solutions of the loaded Korteweg-de Vries, the loaded modified Korteweg-de Vries, and the loaded Gardner equation by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary and periodic wave solutions. All solutions to these equations have been examined and 3D graphics of the obtained solutions have been drawn by using the Matlab program. We get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, and solitary wave solutions of kink type. Our results reveal that the method is a very effective and straightforward way of formulating the exact traveling wave solutions of non-linear wave equations arising in mathematical physics and engineering.

Classification of All Single Traveling Wave Solutions of ( 3 + 1 )-Dimensional Jimbo-Miwa Equation with Space-Time Fractional Derivative

In this paper, the complete discrimination system method is used to construct the single traveling wave solutions for the ( 3 + 1 )-dimensional Jimbo-Miwa equations with space-time fractional derivative. As a result, we get the exact traveling wave solutions of the ( 3 + 1 )-dimensional Jimbo-Miwa equation with space-time fractional derivative, which include rational function solutions, Jacobian elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. Some graphical representations of the solutions are also provided. Finally, the obtained solution is compared with the existing literature.

EXACT TRAVELING WAVE SOLUTIONS FOR THE LOCAL FRACTIONAL KADOMTSOV–PETVIASHVILI–BENJAMIN–BONA–MAHONY MODEL BY VARIATIONAL PERSPECTIVE

In this work, the Kadomtsov–Petviashvili–Benjamin–Bona–Mahony model is described by the local fractional derivative (LFD) on Cantor sets. A novel algorithm is presented to seek the exact traveling wave solution of the nondifferentiable type for the local fractional Kadomtsov–Petviashvili–Benjamin–Bona–Mahony model based on the variational theory, which is called variational wave transform method (VWTM). This new algorithm provides a new idea for seeking the exact traveling wave solutions in fractal space with simplicity and efficiency. The physical properties of traveling wave solutions are described by some 3D simulation figures.

A NOVEL PERSPECTIVE TO THE LOCAL FRACTIONAL BIDIRECTIONAL WAVE MODEL ON CANTOR SETS

In this paper, the bidirectional wave model is described by using the local fractional derivative (LFD) on Cantor sets for the first time. A novel algorithm is established to obtain the exact traveling-wave solution of the non-differential type for the local fractional bidirectional wave model (LFBWM), which is called the local fractional wave method (LFWM). The advantages of the LFWM are simple, efficient and accurate. The LFWM sheds a new light on solving the local fractional wave equations (LFWE) in physics and engineering. Finally, the physical properties of the obtained exact traveling-wave solution of the non-differential type are elaborated by some simulation figures.

New traveling wave exact solutions to the coupled Klein–Gordon system of equations

A system of coupled Klein–Gordon nonlinear partial differential equations are considered in this paper and several new families of exact solutions for this system are derived. Since this system is highly nonlinear, exact solutions are very difficult to obtain. A rational ansatz form is assumed to derive the required exact solutions in terms of the doubly periodic functions. We derive all the possible real solutions having this ansatz form and which are functions of Jacobi elliptic functions. Particular solutions corresponding to each family of solutions are presented graphically.

Spatio-temporal dynamic solitary wave solutions and diffusion effects to the nonlinear diffusive predator-prey system and the diffusion-reaction equations

The coupled parabolic nonlinear evolution equations that describe the temporal-spatio dynamics of predator-prey systems and the nonlinear diffusion-reaction equations with diffusion effects have broad applications in biology, population dynamics, ecology, and many other fields where nonlinearity and diffusivity are key features of wave evolution. As a result, exact traveling wave solutions of such models are highly effective in numerical and analytical theories. In this article, we find the soliton solutions of the above stated biological models by contriving the enhanced modified simple equation method which is related to the exponential, hyperbolic, and trigonometric functions by balancing the highest order nonlinearity and diffusivity associated with biological groups. Graphical representations of some of the obtained solutions in three- and two-dimensional layouts are provided to estimate their behavior. The results revealed that the stated method is categorical, robust, and efficient in finding exact solutions to a variety of nonlinear evolution equations.

Approximate Analytical and Numeric Solutions to a Forced Damped Gardner Equation.

In this paper, some exact traveling wave solutions to the integrable Gardner equation are reported. The ansatz method is devoted for deriving some exact solutions in terms of Jacobi and Weierstrass elliptic functions. The obtained analytic solutions recover the solitary waves, shock waves, and cnoidal waves. Also, the relation between the Jacobi and Weierstrass elliptic functions is obtained. In the second part of this work, we derive some approximate analytic and numeric solutions to the nonintegrable forced damped Gardner equation. For the approximate analytic solutions, the ansatz method is considered. With respect to the numerical solutions, the evolution equation is solved using both the finite different method (FDM) and cubic B-splines method. A comparison between different approximations is reported.

Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics.

Bifurcations and Exact Traveling Wave Solutions in the Generalized Sasa–Satsuma Equation

This paper studies traveling wave solutions of a generalized Sasa–Satsuma equation introduced in [Adem et al., 2020]. For the cases of [Formula: see text], under given parameter conditions, the bifurcations of traveling wave solutions in the parameter space are investigated for the corresponding traveling systems. All possible explicit exact parametric representations of various solutions are obtained.

Explicit, periodic and dispersive soliton solutions to the Schamel-KdV equation with constant coefficients

In plasma physics, the Schamel–Korteweg-de Vries (S-KdV) model determines the electrostatic potential for a given electron distribution in velocity space and is used to explore ion acoustic solitons during the trapping of electrons. The key model reduces to the generalised KdV equation for α=0, and the Schamel equation when β=0. In this article, some new traveling wave solutions to the Schamel-KdV equation are constructed with the help of the classical Khater and the modified Khater methods. The observed results are very encouraging, which provide more powerful mathematical tool for analyzing exact traveling wave solution to the nonlinear equations arising in the recent era of applied sciences and engineering.

Novel exact and approximate solutions to the family of the forced damped Kawahara equation and modeling strong nonlinear waves in a plasma

In this investigation, a set of novel exact and approximate analytic solutions to the family of the forced damped Kawahara equation (KE) are derived in detail. This work is divided into four main parts: in the first part, the ansatz method is employed to obtain some novel exact solutions to the integrable Extended KE (EKE). In this part, we use some different hypotheses to derive many exact traveling wave solutions (including solitary waves (SWs) and cnoidal waves (CWs)). In the second part, the ansatz method is devoted for deriving some exact SWs and CWs solutions to the forced KE. In the same part, two different general formulas to the analytical approximations to the forced damped KE are obtained. In the third part, the family of forced damped modified KE (mKE) is analyzed comprehensively via several different techniques in order to obtain several approximations. At the beginning, two different general formulas with different accuracy to the forced damped mKE are derived. Also, a particular soliton solution to the forced damped mKE is obtained using a direct ansatz. After that, a general formula for the approximate solution to the forced undamped mKE is obtained. In the fourth part, we focus our efforts on obtaining some approximations to forced EKE. The validity of all exact solutions is checked. Moreover, the maximum global residual error to the analytical approximations is estimated. This investigation can help all researchers interested in studying the characteristics of many nonlinear phenomena in various fields of science such as physics of plasmas, physics of fluids, nonlinear optics, Veins and capillaries, Oceans and seas, etc.

Diverse and novel soliton structures of coupled nonlinear Schrödinger type equations through two competent techniques

Nonlinear evolution equations play enormous significant roles to work with complicated physical phenomena located across the nature world. The Schrödinger type equations bearing nonlinearity are important models that flourished with the wide-ranging arena concerning plasma physics, nonlinear optics, fluid flow and the theory of deep-water waves. In this exploration, we retrieve the soliton and other solutions in an appropriate form to the coupled nonlinear Schrödinger equations by means of the improved tanh method and the rational [Formula: see text]-expansion method. The suggested system of nonlinear Schrödinger equations is turned into a differential equation of a single variable through executing some operations. Thereupon, successful implementation of the advised techniques regains the abundant exact traveling wave solutions. The obtained solutions are figured out in the profiles of three-dimensional (3D), two-dimensional (2D) and contour by assigning suitable values of the involved unknown constants. These diverse graphical appearances enable the researchers to understand the underlying mechanisms of intricate phenomena of the leading equations. The individual performances of the employed methods are praiseworthy which deserve further application to unravel any other nonlinear partial differential equations arising in various branches of science.

An efficient technique of [formula omitted]–expansion method for modified KdV and Burgers equations with variable coefficients

The extended generalized G′G–expansion is well defined and an efficient technique, which is used to obtain the exact traveling wave solutions to the governing nonlinear equations with constant coefficients as well as variable coefficients. In this paper, the modified Korteweg–de Vries (mKdV) equation and Burgers equation with variable coefficients are investigated through the extended generalized G′G–expansion method, which are exceptional cases of the nonlinear evolution equations widely used in a two-layer fluid system, in fluid-filled elastic tubes, in an atmospheric and oceanic dynamical system, traffic flow, turbulence in fluid dynamics, dusty plasma, ion-acoustic waves in a plasma system. New families of exact closed-form solutions are obtained in hyperbolic, trigonometric, and rational function solutions with the available free constants. With the help of computerized symbolic computation work, the newly formed closed-form solutions are validated by back substituting them into the equations using the computational mathematical software. Furthermore, the graphical representations of all these obtained solutions are discussed and demonstrated by giving the suitable best values of arbitrary functions and constants via three-dimensional surface and density plots. The dynamics of the solution profiles demonstrate the annihilation of 3D kink-type soliton waves, shock waves, double solitons, and multi-soliton wave structures.

New Exact Solutions to the Lakshmanan–Porsezian–Daniel Equation with Kerr Law of Nonlinearity

In this study, some new exact travelling wave solutions to the Lakshmanan–Porsezian–Daniel (LPD) equation with Kerr law of nonlinearity are retrieved by the complete discrimination system for the polynomial method. Under the travelling wave transformation, the LPD equation is reduced to an ordinary differential equation. The new exact travelling wave solutions including rational solutions, triangle function solutions, solitary wave solutions, and Jacobian elliptic function solutions are obtained and graphically illustrated using three-dimensional and two-dimensional graphs. Comparing with the previous results for LPD equation, some of new solutions in this work such as elliptical solution are not studied, which shows the complete discrimination system method is efficient.

Some novel analytical solutions of a new extented (2 + 1)-dimensional Boussinesq equation using a novel method

There are many analytical methods for solving partial differential equations (PDEs), such as the (G′G)-expansion, the Ricatti transformation method, and many more. In this manuscript, applying another analytical method, i.e., the (G′G′+G+A)-expansion approach, we find novel exact travelling wave solutions for the (2 + 1)-dimensional Boussinesq equation. Some new exact solutions for different cases are derived, which demonstrate the efficiency of the suggested approach and are used to describe the physical interpretation of nonlinear processes. The obtained solutions represent periodically solitary waves, which are simulated by Maple-18.

Extended exp (-φ (ξ))-expansion method for some exact solutions of (2+1) and (3+1)-dimensional constant coefficients KdV equations

We gained some exact solutions of the (2+1)- and (3+1)-dimensional constant coefficients KdV equations by using extended exp(−φ(ξ))-expansion method. These solutions we gained are hyperbolic, trigonometric, exponential and rational solutions. In addition, we provided both two-dimensional and three-dimensional graphics of the solutions we found in −10≤x≤10 and −1≤x≤1 intervals. Then, we observed, via Mathematica 11.2, that the solutions prove the equations. This method has been used for obtaining exact travelling wave solutions of nonlinear partial differential equations (NLPDEs).