Rational Function Solutions Research Articles

Closed-form solutions to the new coupled Konno–Oono equation and the Kaup–Newell model equation in magnetic field with novel statistic application

This paper employs the new robust solver to retrieve exact solutions to a Kaup–Newell model equation and the nonlinear coupled Konno–Oono equation. This solver yields the closed formula for the solutions. Different types of travelling wave solutions, i.e., rational function, hyperbolic function and trigonometric function solutions with many capricious parameters are revealed. Subsequently, by utilizing Matlab 18 we plot 2D and 3D surfaces of obtained analytical solutions for suitable values of the free parameters. The depiction of the solver is direct, vital, sturdy and can be applied to other nonlinear partial differential equations. We also show that some proposed rational solutions can be rapprochement with some known probability distributions.

Dynamics of combined soliton solutions of unstable nonlinear fractional-order Schrodinger equation by beta-fractional derivative

In this article, a new version of the trial equation method is suggested. This method allows new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear fractional-order Schr¨odinger equation in fractional time derivative form of order. Some exact solutions of the fractional-order fractional PDE are attained by employing the new powerful expansion approach using by beta-fractional derivatives which are used to get many solitary wave solutions by changing various parameters. New exact solutions are expressed with rational hyperbolic function solutions, rational trigonometric function solutions, 1-soliton solutions, dark soliton solitons, and rational function solutions. We can say that the unstable nonlinear Schr¨odinger equation exists I different dynamical behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.

Dynamics of lump-periodic and breather waves solutions with variable coefficients in liquid with gas bubbles

Lump solutions are empirical rational function solutions found in all directions in space. One of the essential solutions to both linear and nonlinear partial differential equations is lump solutions. The current work studies a class of lump interaction phenomena to the generalized -dimensional nonlinear-wave equation with time-dependent-coefficient. Variable-coefficient nonlinear partial differential equations offer us with more real aspects in the inhomogeneities of media and nonuniformities of boundaries than their counterparts constant-coefficient in many physical cases. The Hirota bilinear form is the fundamental concept that has been used to derive the novel lump-periodic and breather wave solutions. The acquired solutions are constructed using symbolic computations called Maple. The physical characteristics of the acquired solutions are shown with three-dimensional and contour plots in order to shed more light on the acquired novel solutions.

Soliton solutions, Painleve analysis and conservation laws for a nonlinear evolution equation

In this paper, we investigate a reputed nonlinear partial differential equation (NLPDE) known as geophysical Korteweg-de Vries (GPKdV) equation. We implement a renowned Unified method (UM) of nonlinear (NL) sciences for the extraction of polynomial and rational function solutions of GPKdV equation, which degenarate to various wave solutions like solitary, soliton (dromions) and elliptic wave solutions. Further more, for the analysis of the integrability of our governing model, we apply Painlevé (P) algorithm to check the singularities structure of the model. The fulfillment of all the requirements of the P test indicates the solvability of the governing equation with the help of inverse scattering transformation (IST) or some linear techniques. Moreover, we calculate conservation laws (CLs) in polynomial form as conserved fluxes and densities by implementing dilation symmetry. We utilize Euler and Homotopy operators for the evaluation of the intended conserved quantities.

New Exact Solutions of the Fractional Complex Ginzburg–Landau Equation

In this paper, we apply the complete discrimination system method to establish the exact solutions of the fractional complex Ginzburg–Landau equation in the sense of the conformable fractional derivative. Firstly, by the fractional traveling wave transformation, time-space fractional complex Ginzburg–Landau equation is reduced to an ordinary differential equation. Secondly, some new exact solutions are obtained by the complete discrimination system method of the three-order polynomial; these solutions include solitary wave solutions, rational function solutions, triangle function solutions, and Jacobian elliptic function solutions. Finally, two numerical simulations are imitated to explain the propagation of optical pulses in optic fibers. At the same time, the comparison between the previous results and our results are also given.

New Exact Traveling Wave Solutions of the Time Fractional Complex Ginzburg-Landau Equation via the Conformable Fractional Derivative

In this study, the exact traveling wave solutions of the time fractional complex Ginzburg-Landau equation with the Kerr law and dual-power law nonlinearity are studied. The nonlinear fractional partial differential equations are converted to a nonlinear ordinary differential equation via a traveling wave transformation in the sense of conformable fractional derivatives. A range of solutions, which include hyperbolic function solutions, trigonometric function solutions, and rational function solutions, is derived by utilizing the new extendedG′/G-expansion method. By selecting appropriate parameters of the solutions, numerical simulations are presented to explain further the propagation of optical pulses in optic fibers.

On the exact and numerical solutions to a new (2 + 1)-dimensional Korteweg-de Vries equation with conformable derivative

Abstract The aim of this paper is to introduce a novel study of obtaining exact solutions to the (2+1) - dimensional conformable KdV equation modeling the amplitude of the shallow-water waves in fluids or electrostatic wave potential in plasmas. The reduction of the governing equation to a simpler ordinary differential equation by wave transformation is the first step of the procedure. By using the improved tan(φ/2)-expansion method (ITEM) and Jacobi elliptic function expansion method, exact solutions including the hyperbolic function solution, rational function solution, soliton solution, traveling wave solution, and periodic wave solution of the considered equation have been obtained. We achieve also a numerical solution corresponding to the initial value problem by conformable variational iteration method (C-VIM) and give comparative results in tables. Moreover, by using Maple, some graphical simulations are done to see the behavior of these solutions with choosing the suitable parameters.

Lie symmetry analysis for obtaining exact soliton solutions of generalized Camassa–Holm–Kadomtsev–Petviashvili equation

The prime objective of this paper is to obtain the exact soliton solutions by applying the two mathematical techniques, namely, Lie symmetry analysis and generalized exponential rational function (GERF) method to the (2+1)-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili (g-CHKP) equation. First, we obtain Lie infinitesimals, possible vector fields, and commutative product of vectors for the g-CHKP equation. By the means of symmetry reductions, the g-CHKP equation reduced to various nonlinear ODEs. Subsequently, we implement the GERF method to the reduced ODEs with the help of computerized symbolic computation in Mathematica. Some abundant exact soliton solutions are obtained in the shapes of different dynamical structures of multiple-solitons like one-soliton, two-soliton, three-soliton, four-soliton, bell-shaped solitons, lump-type soliton, kink-type soliton, periodic solitary wave solutions, trigonometric function, hyperbolic trigonometric function, exponential function, and rational function solutions. Consequently, the dynamical structures of attained exact analytical solutions are discussed through 3D-plots via numerical simulation. A comparison with other results is also presented.

Resonant optical solitons with conformable time-fractional nonlinear Schrödinger equation

We successfully apply extended rational sine-cosine/sinh-cosh and advance expansion function techniques in this paper. Various kinds of soliton solutions to the conformable time-fractional resonant nonlinear Schrödinger equation (RNLSE) in optical fiber are extracted. We discuss the system with the effects of coefficients like group velocity dispersion, non-Kerr nonlinearity, resonant nonlinearity, and construct optical soliton solution in terms of rational, trigonometric and hyperbolic function solutions with arbitrary parameters. The recovered solutions reveal that the applied approaches are straightforward and efficient to work out the excellent contribution for analyzing several classes of nonlinear partial differential equations (NLPDEs) in engineering and sciences. Moreover, 3D, 2D and contour graphs are sketched with suitable selection of the parameters under the criteria of constraints conditions.

Solitary Wave Solutions of the Coupled Konno-Oono Equation by using the Functional Variable Method and the Two Variables (G'/G,1/G) -Expansion Method

In this work, we investigate solitary wave solutions of the coupled Konno-Oono equation with the aid of the functional variable method (FVM) and the two variables (G'/G,1/G)-expansion method. We obtain solitary wave solutions in form of trigonometric function and hyperbolic function and rational function solutions. We also draw two and three-dimensional graphics for some solutions with help of Mathematica 7.

Bifurcation and exact solutions for the (2+1)-dimensional conformable time-fractional Zoomeron equation

In this paper, the bifurcation and new exact solutions for the (2+1)-dimensional conformable time-fractional Zoomeron equation are investigated by utilizing two reliable methods, which are generalized (G'/G)-expansion method and the integral bifurcation method. The exact solutions of the (2+1)-dimensional conformable time-fractional Zoomeron equation are obtained by utilizing the generalized (G'/G)-expansion method, these solutions are classified as hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Giving different parameter conditions, many integral bifurcations, phase portraits, and traveling wave solutions for the equation are obtained via the integral bifurcation method. Graphical representations of different kinds of the exact solutions reveal that the two methods are of significance for constructing the exact solutions of fractional partial differential equation.

Exact Analytical Solutions of Nonlinear Fractional Liouville Equation by Extended Complex Method

The extended complex method is investigated for exact analytical solutions of nonlinear fractional Liouville equation. Based on the work of Yuan et al., the new rational, periodic, and elliptic function solutions have been obtained. By adjusting the arbitrary values to the constants in the constructed solutions, it can describe the physical phenomena to the traveling wave solutions, since traveling wave has significant value in applied sciences and engineering. Our results indicate that the extended complex technique is direct and easily applicable to solve the nonlinear fractional partial differential equations (NLFPDEs).

Dynamics of interaction between solitary and rogue wave of the space-time fractional Broer–Kaup models arising in shallow water of harbor and coastal zone

Under inquisition, we consider the space-time fractional Broer–Kaup model that replicate the real coastal profile to describe bidirectional wave transmission of long waves in shallow water. We integrate of the model via the unified scheme to derive exact solitary wave solutions. The achieved results present the hyperbolic, trigonometric, and rational functions solutions. All of these solutions convey the periodic and solitonic natures. In particular, we present combo waves that imply a curvy periodic waves in which rogue waves occur in both sides of each waves. The dynamics of obtained nonlinear wave solutions are demonstrated in 3-D and 2-D shapes with definite parametric values. It is shown that the obtained solutions of the model pact a very rich dynamical behavior and can be used to describe seismic type nonlinear wave motion in shallow water waves of the coastal and harbor region.

Peakon, rational function and periodic solutions for Tzitzeica–Dodd–Bullough type equations

Abstract The fourth-order complete discrimination system method is employed to the seek exact solutions of the Tzitzeica–Dodd–Bullough equation as well as the double sinh-Gordon equation. We obtain a range of new solutions, including a periodic solution, a triangular function solution, rational function solutions, peakon solutions and Jacobian elliptic function solutions.

Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order α . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

Dynamics of combined soliton solutions of unstable nonlinear Schrodinger equation with new version of the trial equation method

In this article, a new version of the trial equation method is suggested. With this method, it is possible to find the new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear Schrödinger equation. New exact solutions are expressed with Jacobi elliptic function solutions, 1-soliton solutions and rational function solutions. When the obtained results are examined, we can say the unstable nonlinear Schrödinger equation shows different dynamic behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.

Analytical Solutions to the Caudrey-Dodd-Gibbon-Sawada-Kotera Equation via Symbol Calculation Approach

In this paper, we derive analytical solutions of the Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation via symbol calculation approach. Applying the exp−φz-expansion method, we achieve the trigonometric, exponential, hyperbolic, and rational function solutions for the CDGSK equation. By choosing the appropriate parameters, we give some computer simulation to the analytical solutions of the CDGSK equation.

New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

The major purpose of this article is to seek for exact traveling wave solutions of the nonlinear space-time Sharma–Tasso–Olver equation in the sense of conformable derivatives. The novel ( G ′ G ) -expansion method and the generalized Kudryashov method, which are analytical, powerful, and reliable methods, are used to solve the equation via a fractional complex transformation. The exact solutions of the equation, obtained using the novel ( G ′ G ) -expansion method, can be classified in terms of hyperbolic, trigonometric, and rational function solutions. Applying the generalized Kudryashov method to the equation, we obtain explicit exact solutions expressed as fractional solutions of the exponential functions. The exact solutions obtained using the two methods represent some physical behaviors such as a singularly periodic traveling wave solution and a singular multiple-soliton solution. Some selected solutions of the equation are graphically portrayed including 3-D, 2-D, and contour plots. As a result, some innovative exact solutions of the equation are produced via the methods, and they are not the same as the ones obtained using other techniques utilized previously.

On the travelling wave solutions of Ostrovsky equation

In this paper, extended trial equation method (ETEM) is applied to find exact solutions of (1+1) dimensional nonlinear Ostrovsky equation. We constitute some exact solutions such as soliton solutions, rational, Jacobi elliptic and hyperbolic function solutions of this equation via ETEM. Then, we submit the results obtained by using this method.

Soliton solutions of higher-order nonlinear schrödinger equation (NLSE) and nonlinear kudryashov's equation

This article secures new soliton solutions of the higher-order nonlinear Schrödinger equation (NLSE) and the nonlinear Kudryashov's equation by means of two analytical techniques, namely the extended (G′G2)-expansion method and the first integral method. Many exact traveling wave solutions such as hyperbolic function solutions, trigonometric function solutions and rational function solutions with free parameters are characterized. Periodic solitons, dark solitons, singular solitons, combo solitons and plane waves are obtained. The existence criteria for such solutions are also provided. Moreover, the modulation instability analysis is used to examine the stabilities of both equations, by which the modulation instability gain spectrums of both equations are obtained. The obtained solutions are also presented graphically.