Rational Function Solutions Research Articles

Searching for Analytical Solutions of the (2+1)‐Dimensional KP Equation by Two Different Systematic Methods

In this paper, we derive analytical solutions of the (2+1)‐dimensional Kadomtsev‐Petviashvili (KP) equation by two different systematic methods. Using the exp⁡(−ψ(z))‐expansion method, exact solutions of the mentioned equation including hyperbolic, exponential, trigonometric, and rational function solutions have been obtained. Based on the work of Yuan et al., we proposed the extended complex method to seek exact solutions of the (2+1)‐dimensional KP equation. The results demonstrate that the applied methods are efficient and direct methods to solve the complex nonlinear systems.

Solitary Wave and other Solutions of Nonlinear Space-Time Fractional Differential Equation Systems

In this study, we have successfully found some travelling wave solutions of the variant Boussinesq system and fractional system of two-dimensional Burgers' equations of fractional order by using the -expansion method. These exact solutions contain hyperbolic, trigonometric and rational function solutions. The fractional complex transform is generally used to convert a partial fractional differential equation (FDEs) with modified Riemann-Liouville derivative into ordinary differential equation. We showed that the considered transform and method are very reliable, efficient and powerful in solving wide classes of other nonlinear fractional order equations and systems.

Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model

In this article, we employ Lie group analysis to obtain symmetry reduction of a class of (3+1)-dimensional nonlinear model. This nonlinear model plays a critical role in the study of nonlinear sciences. By the exp$(-\varphi(z))$-expansion method, we construct explicit solutions for the proposed equation. Four types of explicit solutions are obtained, which are hyperbolic, exponential, trigonometric and rational function solutions.

New sub-equation method to construct solitons and other solutions for perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber materials

In a previous work, Zayed and Al-Nowehy have applied the Riccati equation mapping method combined with the generalized extended (G′/G)-expansion method and found new exact solutions of the nonlinear KPP equation. In the present article, we propose a different method, namely, a new sub-equation method consists of the Riccati equation mapping method and the (G′/G, 1/G)-expansion method to find new exact solutions of the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber materials. This proposed method is not found elsewhere. Hyperbolic, trigonometric and rational function solutions are given. New solutions of the generalized Riccati equation are presented for the first time which are not reported previously. The solutions of the given nonlinear equation can be applied in ocean engineering for calculating the height of tides in the ocean.

Fractional effects on solitons in a 1D array of rectangular ferroelectric nanoparticles

ABSTRACT In this paper, we present non-local effects on solitons in a one-dimensional (1D) array of ferroelectric nanoparticles. Using the discrete tangent hyperbolic method, several fractional function solutions are obtained such as fractional hyperbolic, trigonometric and rational function solutions. As results, we derive many solitons such as dark, bright and rogue waves. We show that the fractional parameter α has an effect on the width and on the amplitude of the above-obtained solitons. Their width and their amplitude are modified by the value of the fractional parameter α. Fractional derivative has a memory effect on ferroelectric nanoparticles. Applications of the obtained results are also presented.

Optical solitons and Peregrine solitons for nonlinear Schrödinger equation by variational iteration method

The current study is dedicated for operating the variational iteration method (VIM) to the self-focusing nonlinear Schrödinger (NLS) equation. Consequently, optical solitons and singular soliton solutions are formally derived. The VIM, representing the resultant solution in a series structure, does not require linearization of the equation and accelerates the convergence of resultant solution. The presented analysis shows the pertinent features of this method. Furthermore, the powerful G′G2-expansion method has been adopted for the aforementioned equation for culminating hyperbolic, trigonometric and rational function solutions with rich mathematical structures, leading to diverse types of solitons.

The (G′/G)-expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines

ABSTRACTIn this paper, we apply the (G′/G)-expansion method based on three auxiliary equations, namely, the generalized Riccati equation , the Jacobi elliptic equation and the second order linear ordinary differential equation (ODE) to find many new exact solutions of a nonlinear partial differential equation (PDE) describing the nonlinear low-pass electrical lines. The given nonlinear PDE has been derived and can be reduced to a nonlinear ODE using a simple transformation. Soliton wave solutions, periodic function solutions, rational function solutions and Jacobi elliptic function solutions are obtained. Comparing our new solutions obtained in this paper with the well-known solutions is given. Furthermore, plotting 2D and 3D graphics of the exact solutions is shown.

On symmetries of roots of rational functions and the classification of rational function solutions of functional equations arising from multiplication of quantum integers with prime semigroup supports

The study of quantum integers and their operations is closely related to the studies of symmetries of roots of polynomials and of fundamental questions of decompositions in Additive Number Theory. In his papers on quantum arithmetics, Melvyn Nathanson raises the question of classifying solutions of functional equations arising from the multiplication of quantum integers, starting with polynomial solutions and then generalizing to rational function solutions. The classification of polynomial solutions with fields of coefficients of characteristic zero and support base P has been completed. In a paper concerning the Grothendieck group associated to the collection of polynomial solutions, Nathanson poses a problem which asks whether the set of rational function solutions strictly contains the set of ratios of polynomial solutions. It is now known that there are infinitely many rational function solutions $$\Gamma $$ with fields of coefficients of characteristic zero not constructible as ratios of polynomial solutions, even in the purely cyclotomic case, which is the case most similar to the polynomial solution case. The classification of polynomial solutions is thus not sufficient, in essential ways, to resolve the classification problem of all rational function solutions with fields of coefficients of characteristic zero. In this paper we study symmetries of roots of rational functions and the classification of the important class-the last and main obstruction to the classification problem-of rational function solutions, the purely cyclotomic, purely nonrational primitive solutions with fields of coefficients of characteristic zero and support base P, which allows us to complete the classification problem raised by Nathanson.

Application of $$\tan (\Phi (\xi )/2)$$tan(Φ(ξ)/2)-expansion method to solve some nonlinear fractional physical model

Based on the $$\tan (\Phi (\xi )/2)$$-expansion method, five nonlinear fractional physical models for obtaining the solutions containing three types hyperbolic function, trigonometric function and rational function solutions are investigated. These equations are the time fractional biological population model, time fractional Burgers, time fractional Cahn–Hilliard, space–time fractional Whitham–Broer–Kaup, space–time fractional Fokas equations. The fractional derivative is described in the Caputo sense. We obtained the exact solutions for the aforementioned nonlinear fractional equations. A generalized fractional complex transform is appropriately used to convert these equations to ordinary differential equations which subsequently resulted into number of exact solutions.

Optical solitons for perturbed Gerdjikov–Ivanov equation

The current article investigates the optical solitons for the perturbed Gerdjikov–Ivanov equation. The algorithms namely: exp(−ϕ(ξ))-expansion method and G′G2-expansion method are adopted for the aforementioned equation. As a result, we formally derive hyperbolic, trigonometric or rational function solutions. The acquired solutions leads to diverse types of optical solitons by stipulating the free parameters along with requisite limitations that ensures the sustainability of such optical solitons.

New exact solutions to the high dispersive cubic–quintic nonlinear Schrödinger equation

The complete discrimination system method is employed to find exact solutions for a dispersive cubic–quintic nonlinear Schrödinger equation with third order and fourth order time derivatives. As a result, we derive a range of solutions which include triangular function solutions, kink solitary wave solutions, dark solitary wave solutions, Jacobian elliptic function solutions, rational function solutions and implicit analytical solutions. Numerical simulations are presented to visualize the mechanism of Eq. (1) by selecting appropriate parameters of the solutions. The comparison between our results and other's works are also given.

Traveling wave and exact solutions for the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

The nonlinear Schrodinger equation (NLSE) with the aid of three order dispersion terms is investigated to find the exact solutions via the extended $ (\frac{G'}{G^{2}})$ -expansion method and the first integral method. Many exact traveling wave solutions, such as trigonometric, hyperbolic, rational, soliton and complex function solutions, are characterized with some free parameters of the problem studied. It is corroborated that the proposed techniques are manageable, straightforward and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Some figures are plotted to describe the propagation of traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions.

Analytical study of solitons for Lakshmanan–Porsezian–Daniel model with parabolic law nonlinearity

This paper retrieves new solitary wave solutions to the Lakshmanan–Porsezian–Daniel model with parabolic law nonlinearity. The modified trail equation method is used to get dark soliton, rational function and periodic solutions. We also use improved (G′/G)-expansion method to obtain exact and traveling wave solutions.

Optical dark and dark-singular soliton solutions of (1+2)-dimensional chiral nonlinear Schrodinger’s equation

ABSTRACTIn this study, the dynamical analysis of optical dark and singular solitons is carried out for chiral (1+2)-dimensional nonlinear Schrödinger’s equation with the implementation of extended direct algebraic and extended trial equation method independently. The constraint conditions guarantee the perseverance of these soliton solutions. Along with optical dark and singular solitons, these integration techniques yield other wave solutions such as Jacobi elliptic function, rational function, and hyperbolic function solutions as outgrowth.

Abundant closed form solutions of the conformable time fractional Sawada-Kotera-Ito equation using (G′/G)-expansion method

In current study, we use the (G′/G)-expansion method to construct the closed form solutions of the seventh order time fractional Sawada-Kotera-Ito (TFSKI) equation based on conformable fractional derivative. As a result, trigonometric, hyperbolic and rational functions solutions with arbitrary constants are obtained. When the arbitrary constants are taken some special values, the periodic and soliton solutions are obtained from the travelling wave solutions. The obtained solutions are new and not found elsewhere. The effect of the fractional order on some of these solutions are represented graphically to illustrate the behavior of the exact solutions when the parameter take some special choose.

New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod

Abstract This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the longitudinal wave equation (LWE) that arises in mathematical physics with dispersion caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method.

Simplest equation method for some time-fractional partial differential equations with conformable derivative

The conformable fractional derivative was proposed by R. Khalil et al. in 2014, which is natural and obeys the Leibniz rule and chain rule. Based on the properties, a class of time-fractional partial differential equations can be reduced into ODEs using traveling wave transformation. Then the simplest equation method is applied to find exact solutions of some time-fractional partial differential equations. The exact solutions (solitary wave solutions, periodic function solutions, rational function solutions) of time-fractional generalized Burgers equation, time-fractional generalized KdV equation, time-fractional generalized Sharma–Tasso–Olver (FSTO) equation and time-fractional fifth-order KdV equation, (3+1)-dimensional time-fractional KdV–Zakharov–Kuznetsov (KdV–ZK) equation are constructed. This method presents a wide applicability to solve some nonlinear time-fractional differential equations with conformable derivative.

Soliton solutions of fractional complex Ginzburg–Landau equation with Kerr law and non-Kerr law media

In this study, new soliton solutions of the fractional complex Ginzburg–Landau equation, that models soliton propagation in the presence of detuning factor, have been constructed. The Kerr law, power law, dual-power law and log law nonlinearity have been considered. The exp(−ϕ(ξ))-expansion method has been utilized for finding new exact solutions of fractional complex Ginzburg–Landau equation. Different forms of solutions, including the hyperbolic, trigonometric and rational function solutions are formally extracted. The method suggests a useful and efficient technique to look for the exact solutions of a wide range of nonlinear fractional partial differential equations.

The $$\left( \frac{\boldsymbol{G}^{\prime }}{\boldsymbol{G}},\frac{\boldsymbol{1}}{\boldsymbol{G}}\right)$$ G ′ G , 1 G -expansion method and its applications for constructing many new exact solutions of the higher-order nonlinear Schrödinger equation and the quantum Zakharov–Kuznetsov equation

In this article, we apply the two variable $$\left( \frac{G^{\prime }}{G}, \frac{1}{G}\right)$$ -expansion method with the aid of symbolic computation to construct many new exact solutions for two higher-order nonlinear partial differential equatuions (PDEs) namely, the higher-order nonlinear Schro dinger (NLS) equation with derivative non-kerr nonlinear terms describing pulse of the propagation beyond ultrashort range in optical communication systems and the higher-order nonlinear quantum Zakharov–Kuznetsov (QZK) equation which arises in quantum magneto plasma . Also, based on Lienard equation, we find many other diffrent new soliton solutions of the above NLS equation. Soliton solutions, periodic solutions, rational functions solutions and Jacobi elliptic functions solutions are obtained. Comparing our new solutions obtained in this article with the well-known solutions are given.

Some new analytical solutions of the variant Boussinesq equations

In this manuscript, the exact analytical solutions for the variant Boussinesq equations are constructed by utilizing the improved $$\tanh (\phi (\xi )/2)$$ -expansion method (ITEM) and the improved $$(G'{/}G)$$ -expansion method (IGEM). Using wave transformation, we converted the problem under consideration to an ordinary differential equation and then employed the ITEM and IGEM for hyperbolic function, trigonometric function and rational function solutions. The Mathematica software is used throughout for the solution of the system of algebraic equations obtained along the way and also for the graphical illustrations, respectively.