Hyperbolic Function Solutions Research Articles

Time-fractional generalized fifth-order KdV equation: Lie symmetry analysis and conservation laws

The purpose of this study is to apply the Lie group analysis method to the time-fractional order generalized fifth-order KdV (TFF-KdV) equation. We examine applying symmetry analysis to the TFF-KdV equation with the Riemann–Liouville (R–L) derivative, employing the G′/G-expansion approach to yield trigonometric, hyperbolic, and rational function solutions with arbitrary constants. The discovered solutions are unique and have never been studied previously. For solving non-linear fractional partial differential equations, we find that the G′/G-expansion approach is highly effective. Finally, conservation laws for the equation are well-built with a full derivation based on the Noether theorem.

Bifurcations and traveling wave solitons in optical fibers with the nonlinear Kaup–Newell system

The main objective of this work is to study the dynamical behavior and traveling wave solution for the nonlinear Kaup–Newell system in optical fibers. Here, we have constructed periodic wave solutions, kink-shaped solitary solutions and anti-kink solitary solutions through the theory of planar dynamical system. In addition, some other traveling wave solutions of the nonlinear Kaup–Newell system are derived by means of the polynomial complete discriminant system method and symbolic computation, respectively. The trigonometric function solutions, solitary wave solutions, rational function solutions, the hyperbolic function solutions and the Jacobian function solutions are obtained. Finally, 3D-diagram and 2D-diagram of the nonlinear Kaup–Newell system are also plotted.

Bifurcation analysis, stationary optical solitons and exact solutions for generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and quintuple power-law of refractive index in optical fibers

In this paper, the bifurcation, stationary optical solitons and exact solutions in optical fiber propagation are studied. The propagation model is described by a generalized nonlinear Schrödinger equation with Kudryashov’s quintuple power law of refractive index. First, the equation is transformed into a second-order nonlinear ordinary differential equation by an appropriate transformation. Then, the bifurcation of the phase diagram of the obtained ordinary differential equation is established using planar dynamic system theory, and the types of optical solitons and exact solutions are determined. What is more, the stationary optical solitons and exact solutions are obtained by the polynomial complete discriminant system, and are demonstrated in 3D and 2D graphics by using the symbolic computation software Maple. Finally, the obtained stationary optical solitons and exact solutions include the Jacobi elliptic function solution, hyperbolic function solution, rational function solution, trigonometric function solution and exponential function solution. Moreover, the results provide a method to further study stationary optical solitons and exact solutions of propagation pulses in optical fibers.

On exact solutions of the (2+1)-dimensional time conformable Maccari system

For the ([Formula: see text])-dimensional time conformable Maccari system with beta-derivative, exact solutions are obtained in this paper using the improved sub-equation method. This technique has been used to find rational solutions, generalized hyperbolic function solutions and generalized trigonometric function solutions. The new structures of the solutions are shown after the proper parameter values have been given. It is easier to understand the intricate physical processes of these dynamical models by viewing the physical surfaces of some obtained solutions in various forms. The outcomes demonstrate the efficiency of the applied strategy, which may be used to solve a number of nonlinear physical problems. ([Formula: see text])-dimensional time conformable Maccari system is considered and the graphs of solutions are sketched using MAPLE software after its exact solutions which are acquired using the improved sub-equation method.

Solitary wave dynamics of thin-film ferroelectric material equation

Ferroelectrics are dielectric materials which exhibit nonlinear behaviors. Thin films made from the ferroelectric materials are used various modern electronics devices. The propagation of solitary polarization in thin-film ferroelectric materials can be described using the nonlinear evolution equations. In this paper, the solitary wave dynamics of the thin-film ferroelectric material equation is investigated. The traveling wave transformation is employed along with the extended sinh-Gordon equation expansion method and the Riccati equation method to construct the soliton and other wave solutions of the thin-film ferroelectric material equation. The hyperbolic, trigonometric and rational function solutions are constructed along with the constraint conditions. A variety of soliton and other traveling wave solutions have been retrieved including kink solitons, bright solitons and periodic wave solutions. To understand the physical structure of the obtained solutions, graphical analysis is presented using 3D-plots, density-plots, and 2D-contours.

New single traveling wave solution of the Fokas system via complete discrimination system for polynomial method

<abstract> <p>In this paper, the traveling wave solution of the Fokas system which represents the irregular pulse propagation in monomode optical fibers is studied by using the complete discriminant system method of polynomials. Firstly, the Fokas system is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the Jacobian function solutions, the trigonometric function solutions, the hyperbolic function solutions and the rational function solution of Fokas system are obtained by using the complete discriminant system method of polynomials. Finally, in order to show the propagation of Fokas system in monomode optical fibers, three-dimensional diagram, two-dimensional diagram, contour plot and density plot of some solutions are drawn by using Maple software.</p> </abstract>

EXPLICIT SOLUTIONS FOR THE CONFORMABLE REGULARIZED LONG WAVE BURGER'S EQUATION

In this paper, a dynamical analysis of the conformable regularized long-wave burgers equation is carried out with help of improved $ \tan\big(\frac{\phi(\eta)}{2}\big) $-expansion method. Fractional complex transform converts a nonlinear fractional differential equation in an ordinary differential form which resulted into a number of exact solutions like exponential function solutions, hyperbolic function solutions, trigonometric function solutions and rational function solutions. The constarint conditions are also given for each solution. The physical profiles of proposed solutions are portrayed by 3D and 2D graphs as well as the influence of fractional parameter is also studied for some solutions. Our proposed results showed that improved $ \tan\big(\frac{\phi(\eta)}{2}\big) $-expansion method is reliable method to solve the nonlinear equation in mathematical physics.

ANALYTICAL SOLUTIONS FOR TIME-FRACTIONAL RADHAKRISHNAN–KUNDU–LAKSHMANAN EQUATION

In this paper, two algebraic methods are applied for solving a class of conformable fractional partial differential equations (FPDEs). We use these methods for the time-fractional Radhakrishnan–Kundu–Lakshmanan equation. With these methods, further solutions can be obtained compared with other approaches and techniques. The exact particular solutions include the exponential solution, trigonometric function solution, rational solution and hyperbolic function solution. These methods are very effective to obtain exact solutions of many fractional differential equations.

Traveling-wave and numerical solutions to a Novikov-Veselov system via the modified mathematical methods

<abstract><p>In this article, we have achieved new solutions for the Novikov-Veselov system using several methods. The present solutions contain soliton solutions in the shape of hyperbolic, rational, and trigonometric function solutions. Magneto-sound and ion waves in plasma are examined by employing partial differential equations, such as, the Novikov-Veselov system. The Generalized Algebraic and the Modified F-expansion methods are employed to achieve various soliton solutions for the system. The finite difference method is well applied to convert the proposed system into numerical schemes. They are used to obtain the numerical simulations for NV. I also present a study of the stability and Error analysis of the numerical schemes. To verify the validity and accuracy of the exact solutions obtained using exact methods, we compare them with the numerical solutions analytically and graphically. The presented methods in this paper are suitable and acceptable and can be utilized for solving other types of non-linear evolution systems.</p></abstract>

Bifurcations of traveling wave solutions for the mixed Korteweg-de Vries equation

<abstract><p>In this paper, the bifurcation theory of planar dynamical systems is employed to investigate the mixed Korteweg-de Vries (KdV) equation. Under different parameter conditions, the bifurcation curves and phase portraits of corresponding Hamiltonian system are given. Furthermore, many different types of exact traveling waves are obtained, which include hyperbolic function solution, triangular function solution, rational solution and doubly periodic solutions in terms of the Jacobian elliptic functions. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors, and could be applicable to explain diversity in qualitative features of wave phenomena.</p></abstract>

Superposition formulas of multi-solution to a reduced (3+1)-dimensional nonlinear evolution equation

We gave the localized solutions, the interaction solutions and the mixed solutions to a reduced (3+1)-dimensional nonlinear evolution equation. These solutions were characterized by superposition formulas of positive quadratic functions, the exponential and hyperbolic functions. According to the known lump solution in the outset, we obtained the superposition formulas of positive quadratic functions by plausible reasoning. Next, we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory. These two kinds of solutions contained superposition formulas of positive quadratic functions, which were turned into general ternary quadratic functions, the coefficients of which were all rational operation of vector inner product. Then we obtained linear superposition formulas of exponential and hyperbolic function solutions. Finally, for aforementioned various solutions, their dynamic properties were showed by choosing specific values for parameters. From concrete plots, we observed wave characteristics of three kinds of solutions. Especially, we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.

Dynamics Investigation and Solitons Formation for (2+1) -Dimensional Zoomeron Equation and Foam Drainage Equation

The prime goal of this study is to investigate novel solutions of two well-known nonlinear models namely the (2+1)-dimensional Zoomeron equation and the foam drainage equation by utilizing a powerful technique; the extended left(frac{G'}{G^{2}}right)-expansion method. Using this methodology, the hyperbolic function solutions, the trigonometric function solutions and the rational function solutions are constructed. Abundant soliton solutions are retrieved from the obtained results. The dynamical structures of the solutions are illustrated graphically through 3-dimensional graphs and the corresponding contour plots. The reported results depict the effectiveness and capability of the extended left(frac{G'}{G^{2}}right)-expansion method for handling different nonlinear partial differential equations.

Employing the exp $(-\varphi(z))$ - Expansion Method to Find Analytical Solutions for a (2+1)-dimensional Combined KdV-mKdV Equation

In this paper, we obtain exact solutions of the (2+1)-dimensional combined KdV-mKdV equation by using a symbol calculation approach. First, we give some background on the equation. Second, the exp$(-\varphi(z))$-expansion method will be introduced to solve the equation. After, using the exp$(-\varphi(z))$-expansion method to solve the equation, we can get four types of exact solutions, which are hyperbolic, trigonometric, exponential, and rational function solutions. Finally, we can observe the characteristics of the exact solutions via computer simulation more easily.

On the multiple explicit exact solutions to the double‐chain DNA dynamical system

The nonlinear dynamics of deoxyribonucleic acid (DNA) is the subject of this research. For an organism to flourish and reproduce, it needs the genetic information that is contained in its DNA. In the DNA molecule, the hydrogen bonds between the base pairs of the two polynucleotide chains are represented by an elastic membrane (or a series of linear springs) that connects the model's two long elastic homogeneous strands (or rods). Using two integration norms, namely, the new auxiliary equation method (NAEM) and the extended sinh‐Gordon equation expansion method (ShGEEM), we obtain a variety of nonlinear dynamical solitary wave structures in various shapes, including hyperbolic, trigonometric, and exponential function solutions, as well as some unique solitary wave solutions, including bell‐shaped, kink, singular, combined, and multiple solitons. Using the logarithmic transformation assistant, we examine a variety of wave structures and recover singular periodic wave solutions with unknown parameters, along with the validity constraints. According to the results, the governing model contains an unusually large number of theoretically exact solitary wave solutions. The physical characterization of some of the presented results can be visualized in 3D and 2D profiles by selecting the appropriate parameter values.

Lie symmetry analysis, exact solutions, optimal system and conservation laws for variable-coefficients Kundu–Mukherjee–Nasker equation

This paper investigates the (2+1)-dimensional variable coefficients Kundu–Mukherjee–Nasker equation which obtained by some transformations. The infinitesimal generators of the equation are obtained by the Lie group method. Next, the representative elements of the one-dimensional subalgebras optimal system are determined with adjoint representation. Based on the optimal system, (1+1)-dimensional partial differential equations can be obtained by similarity reductions. With the help of [Formula: see text]-expansion method, several exact solutions including hyperbolic function solutions, trigonometric function solutions and rational function solutions of variable coefficients Kundu–Mukherjee–Nasker equation are obtained. Finally, the conservation laws of the variable coefficients Kundu–Mukherjee–Nasker equation can be obtained by using the conservation theorem proposed by Ibragimov.

Application of the unified method to solve the ion sound and Langmuir waves model

We present the unified method and use it to integrate the ion sound and Langmuir waves (ISLW) model to retrieve optical soliton solutions. Some new dynamical optical solitons involving the combo of rational, trigonometric, and hyperbolic function solutions are added in this study. The derived optical soliton solutions display various properties such as beat pattern and oscillation with increasing, decreasing, and simultaneously increasing and decreasing amplitudes. Moreover, kink, dark bell, singular kink, single breather, multiple breathers, dark-, bright-, and dark bright periodic waves are founded. Finally, some dynamical characteristics of the acquired solutions are depicted.

Assorted hyperbolic and trigonometric function solutions of fractional equations with conformable derivative in shallow water

We present the modified simple equation method for solving nonlinear fractional-order partial differential equations (fPDEs). With the presented method, some important fPDEs are solved. We used conformable derivatives to solve these equations. The solutions obtained are both hyperbolic and trigonometric function solutions. Most of these solutions are new solutions not found in the literature.

On Travelling Wave Solutions of Dullin-Gottwald-Holm Dynamical Equation and Strain Wave Equation

In this study, extended trial equation method (ETEM) is implemented to obtain exact solutions of the Dullin-Gottwald-Holm Dynamical equation (DGHDE) and the strain wave equation. We constitute some exact solutions such as soliton solutions, rational, Jacobi elliptic, periodic wave solutions and hyperbolic function solutions of these equations via ETEM. Then, we present results that we obtained by using this method.

Construction of new solutions of Korteweg-de Vries Caudrey-Dodd-Gibbon equation using two efficient integration methods

Korteweg-de Vries Caudrey-Dodd-Gibbon (KdV-CDG) equation describes many physical phenomena in plasma physics, optical fibers, dynamics of the ocean, quantum mechanics, acoustic waves and laser optical applications. In this paper, the KdV-CDG equation is analyzed via two reliable and efficient integrating approaches. The suggested techniques; the extended -expansion method and exponential (ψ(ξ))-expansion method successfully extract hyperbolic function solutions, trigonometric function solutions and rational function solutions. The existence criteria for all the obtained solutions are also discussed in this paper. At the end, various 3D and 2D contour plots have been constructed for better understanding of constructed solutions.

Closed-Form Solutions in a Magneto-Electro-Elastic Circular Rod via Generalized Exp-Function Method

In this study, the dispersal caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod is taken into consideration using the nonlinear longitudinal wave equation (LWE), a mathematical physics problem. Using the generalized exp-function method, we investigate the families of solitary wave solutions of one-dimensional nonlinear LWE. Using the computer program Wolfram Mathematica 10, these new exact and solitary wave solutions of the LWE are derived as trigonometric function, periodic solitary wave, rational function, hyperbolic function, bright and dark solitons solutions, sinh, cosh, and sech2 function solutions of the LWE. These solutions represent the electrostatic potential and pressure for LWE as well as the graphical representation of electrostatic potential and pressure.