Trigonometric Solutions Research Articles

Optical and analytical soliton solutions to higher order non-Kerr nonlinear Schrödinger dynamical model

This article studies the propagation of the short pulse optical model governed by higher order nonlinear Schrödinger equation (HNLSE) with non-Kerr nonlinearity. The model is used to describe the propagation of ultrashort photons in highly nonlinear media. Upon establishing the general framework, we discuss the statics and dynamics of HNLSE by employing an extended modified auxiliary equation mapping (EMAEM) architectonic to obtain some new solitary wave solutions like bright dromion (soliton), domain wall, singular, periodic, doubly periodic, trigonometric, rational and hyperbolic solutions etc. Obtained optical soliton solutions are analysed graphically to represent features like as width, amplitude, and structure of solitons. We will also discuss our governing model for M-shaped, Homoclinic breathers, multiwave, kink cross rational, and interaction of M-shaped with 1 and 2 kink solutions with the help of various ansatz transformations.

Travelling Wave Solutions of Nonlinear Evolution Problem Arising in Mathematical Physics through (G′/G)-Expansion Scheme

This article deals with the application of well-known (G′/G)-expansion to investigate the travelling wave solutions of nonlinear evolution problems including the Boussinesq equation, Klein–Gordon equation, and sine-Gordon equation as these problems appear frequently in mathematical physics. The beauty of the suggested method is to transform the highly nonlinear evolution equation into a system of nonlinear algebraic equations by means of trial solution and auxiliary equation. It is found that the presented approach is simple, efficient, has a less computational cost, and produced rational trigonometric solutions. In order to investigate the novel results, various simulations have been executed. It is renowned that all solutions are in the form of soliton with a single hump and singular which is travelling as time increases gradually. It travels as time travel with the same shape. Moreover, as A2 decreases the amplitude of the solutions decreases. A comparative study illustrates that some of the obtained solution matches with the existing results against particular values of parameters and various new travelling wave solution attained the first time. The method seems more appropriate by means of a computational work. It can also be extended to demonstrate the behavior of other physical models of physical nature.

The higher-order modified Korteweg-de Vries equation: Its soliton, breather and approximate solutions

In this study, the fifth-order modified Korteweg-de Vries (F-MKdV) equation is first addressed using Hirota’s bilinear method. Thereafter, the exact and approximative solutions of the generalized form of the F-MKdV equation are investigated using the modified Kudryashov method, the Riccati equation and its Backlund transformation method, the solitary wave ansatz method, and the homotopy perturbation transform method (HPTM). As a result, solitons, breather, and solitary wave solutions are derived from these methods. In particular, we obtain some new solutions such as the dark soliton, bright soliton, singular soliton, periodic trigonometric, exponential, hyperbolic, and rational solutions. The constraint conditions associated with the resulting solutions are also discussed in detail. The HPTM is employed to construct approximate solutions to the aforementioned generalized model due to its strong nonlinear terms and only a few terms are required to obtain accurate solutions. These findings may help to understand complex nonlinear phenomena.

On dynamical behavior for optical solitons sustained by the perturbed Chen–Lee–Liu model

This study investigates the perturbed Chen–Lee–Liu model that represents the propagation of an optical pulse in plasma and optical fiber. The generalized exponential rational function method is used for this purpose. As a result, we obtain some non-trivial solutions such as the optical singular, periodic, hyperbolic, exponential, trigonometric soliton solutions. We aim to express the pulse propagation of the generated solutions, by taking specific values for the free parameters existed in the obtained solutions. The obtained results show that the generalized exponential rational function technique is applicable, simple and effective to get the solutions of nonlinear engineering and physical problems. Moreover, the acquired solutions display rich dynamical evolutions that are important in practical applications.

A variety of solitons on the oceans exposed by the Kadomtsev Petviashvili-modified equal width equation adopting different techniques

The diverse patterns of waves on the oceans yielded by the Kadomtsev Petviashvili-modified equal width (KP-mEW) equation are highlighted in this paper. Two recent established approaches such as the improved auxiliary equation technique and the enhanced rational (G′/G)-expansion scheme are utilized to construct wave solutions of the proposed governing model. Numerous rational, trigonometric, exponential, and hyperbolic wave solutions bearing many free parameters are successfully acquired in appropriate form. The obtained solutions are plotted in various profiles as three-dimension, two-dimension, and contour to illustrate their physical appearances. The plotting outlines appear in the shapes of singular kink, anti-kink, kink, compacton, anti-compacton, bell, anti-bell, periodic, singular periodic etc. The computational software Maple is used for plotting and checking the validity of the found solutions. This paper claims to be novel for generating new results regarding the earlier results.

Perturbation of dispersive optical solitons with Schrödinger–Hirota equation with Kerr law and spatio-temporal dispersion

Objective:The principal purpose of this paper is to examine the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity which governs the propagation of dispersive pulses in optical fibers by proposing and using a direct algebraic form of the enhanced modified extended tanh expansion method for the first time. Our aim is not only restricted to obtaining different and more soliton solutions by proposed method for the first time in this study, but also includes examining the effect of the coefficients of self-steepening and nonlinear dispersion terms to the soliton propagation in the investigated problem. Methodology:Utilizing a traveling wave transformation, the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity can be transformed into an nonlinear ordinary differential equation (NODE). Then, the NODE is convert into a set of algebraic equations by taking account into the Riccati differential equation. Solving the set of algebraic equations, we acquire the analytical soliton solutions of the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity. In the proposed method, the modified extended tanh function method is enhanced by presenting more solutions of Riccati differential equations with the direct algebraic form, is utilized. Results:The more solutions have been established to the literature with new significant physical properties of the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity. We indicated that the presented method are effective, easily computable, and reliable in solving such nonlinear problems. Moreover, we demonstrate the dynamical behaviors and physical significance of some soliton solutions at appropriate values of parameters. Originality:A variety of soliton solutions to the perturbed Schrödinger–Hirota equation with Kerr law non-linearity by the direct algebraic form of enhanced modified extended tanh expansion method have been acquired. These solutions are dark–bright, trigonometric, hyperbolic, periodic, and singular soliton solutions. 3D, contour and 2D plots of some obtained solutions have been demonstrated to interpret the physical meaning of the equation. For some parameter values in the equation, the behavior of soliton solutions has been examined. The constraint conditions are established to confirm the existence of valid solutions. The obtained results can be effective in interpreting the physical meaning of this nonlinear system. We have seen that the proposed direct algebraic form of the enhanced modified extended tanh expansion method is a powerful mathematical technique which can be utilized to acquire the analytical solutions to different complex nonlinear mathematical models.

Computational Technique to Study Analytical Solutions to the Fractional Modified KDV-Zakharov-Kuznetsov Equation

In this article, we study and investigate the analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov-Kuznetsov (mKDV-ZK) equation. We have got new exact solutions of the fractional mKDV-ZK equation by using first integral method; we found new types of hyperbolic solutions and trigonometric solutions by symbolic computation.

Fractional-Stochastic Solutions for the Generalized ( 2 + 1 )-Dimensional Nonlinear Conformable Fractional Schrödinger System Forced by Multiplicative Brownian Motion

In this paper, the ( 2 + 1 )-dimensional nonlinear conformable fractional stochastic Schrödinger system (NCFSSS) generated by the multiplicative Brownian motion is treated. To get new rational, trigonometric, hyperbolic, and elliptic stochastic solutions, we use two different methods: the sine-cosine and the Jacobi elliptic function methods. Moreover, we use the MATLAB tools to plot our figures to introduce a variety of 2D and 3D graphs to highlight the effect of the multiplicative noise on the exact solutions of the NCFSSS. Finally, we illustrate that the multiplicative Brownian motion stabilizes the solutions of NCFSSS a round zero.

Applications of Standard Methods for Solving the Electric Train Mathematical Model With Proportional Delay

In electric trains, the current is collected via a certain device, called the Pantograph. The governing mathematical model of such physical problem is well-known as the Pantograph delay differential equation (PDDE): y’(t)=ay(t)+by(ct), where c is a proportional delay parameter. In the literature, a special case of the PDDE was analyzed when c=−1. The objective of this paper is to determine the general solution of the PDDE for arbitrary c. In addition, it will be shown that the obtained general solution reduces to exact one at c=−1. Such exact solution is expressed in terms of several types of functions, e.g., hyberbolic, Mittag-Leffler, and trigonometric functions. Moreover, it is declared that the exact trigonometric solution is periodic with periodicity 2π/√b2−a2 which agrees with the corresponding results in the literature. Furthermore, the solution of PDDE is provided at almost all possible cases of the involved parameters a, b, and c. Finally, the solution Ambartsumian delay equation (ADE), which has an application in the theory of surface brightness in the Milky Way, will be recovered as a special case of our results.

Novel exact solutions and bifurcation analysis to dual-mode nonlinear Schrödinger equation

This work discusses the dual-mode form of the nonlinear Schrödinger equation, which depicts the augmentation or absorption of dual waves. This model scrutinizes the contemporaneous generation of two characteristic waves dealing with dual-mode and introduces three physical parameters: nonlinearity, phase velocity, and dispersive factor. The wave phenomena of the obtained solutions are applied to water wave mechanics, fluid dynamics, ocean engineering, and science. We use two different methods for the dual-mode nonlinear Schrödinger equation (DMNLSE): the new extended direct algebraic method (NEDAM) and the dynamical system method (bifurcation analysis). These methods were not applied to the DMNLSE before. Firstly, by using the NEDAM, we observe that the DMNLSE has the shape of mixed-trigonometric solutions, shock solutions, singular solutions, complex dark-bright solutions, mixed-singular solutions, trigonometric solutions, different types of complex-combo solutions, periodic and mixed-periodic solutions, mixed-hyperbolic solutions, and a plane solution. Indeed, thanks to the NEDAM, the families of rational solutions have also appeared during the derivation. Secondly, the bifurcation analysis of the model in question is performed, and the fixed points are systematically generated. Thus, other solutions of various types are disclosed. To show the physical significance of the respected model, some three-dimensional and contour plot graphs of the obtained outcomes are illustrated with the help of Mathematica under the appropriate option concerning parameter values. We have made comparisons between our solution and other solutions in the previous literature. The obtained outcomes are beneficial to studying and corroborating the analytical solutions with numerical and experimental work in nonlinear dynamics modeled by the equation. Consequently, it revealed that the mentioned techniques can be a conceivable tool for creating unique precise soliton solutions for different needs, which play a paramount role in applied science and engineering.

The modified simplest equation procedure for conformable time-fractional Boussinesq equations

In this paper, we used the traveling wave transformation to find out the exact solutions of the conformable time fractional differential equations with the help of the Modified simplest equation procedure. We applied the given procedure to two different equations which are conformable time-fractional Boussinesq and coupled Boussinesq equations. First, we reduced the given equations to ordinary differential equations using wave transformations. Then, we applied the procedure to the given equations. We obtained hyperbolic, trigonometric and rational solutions. We gave the 3D plots, 2D plots and contour plots for some obtained solutions. According to our results, we can say the given method is effective, powerful and gives simple, reliable solutions compared to the other methods. All obtained solutions were checked with the help of Mathematica.

Elliptic solitons and “freak waves”

It is shown that elliptic solutions to the AKNS hierarchy equations can be obtained by exploring spectral curves that correspond to elliptic solutions of the KdV hierarchy. This also allows one to get the quasirational and trigonometric solutions for AKNS hierarchy equations as a limit case of the elliptic solutions mentioned above.

Stable abundant computational solitary wave structures of the perturbed time-fractional NLS equation

The conformable fractional derivatives Khater II (Khat. II) method and the Adomian decomposition (AD) approach are used to investigate the analytical and semi-analytical wave solutions to the perturbed time-fractional nonlinear Schrödinger (NLS) model. This model describes the dynamics of optical solitons propagating via nonlinear optical fibers. Several novel solitary wave solutions are constructed in distinct formats such as hyperbolic, trigonometric, rational, dark, brilliant, combined dark-bright, singular, combination singular, and periodic wave solutions. Additionally, the Adomian decomposition technique determines the absolute error associated with analytical and semi-analytical wave solutions. The Hamiltonian system examines the stability of discovered solutions to determine their acceptability for usage in the model’s application.

The Optical Solutions of the Stochastic Fractional Kundu–Mukherjee–Naskar Model by Two Different Methods

In this paper, we focus on the stochastic fractional Kundu–Mukherjee–Naskar equation perturbed in the Stratonovich sense by the multiplicative Wiener process. To gain new elliptic, rational, hyperbolic and trigonometric stochastic solutions, we use two different methods: the Jacobi elliptic function method and the (G′/G)-expansion method. Because of the significance of the Kundu-Mukherjee equation in a magnetized plasma, the obtained solutions are useful in understanding many remarkable physical phenomena. Furthermore, we show the effect of the multiplicative Wiener process on the obtained solutions of the Kundu–Mukherjee–Naskar equation.

Effects of the Wiener Process on the Solutions of the Stochastic Fractional Zakharov System

We consider in this article the stochastic fractional Zakharov system derived by the multiplicative Wiener process in the Stratonovich sense. We utilize two distinct methods, the Riccati–Bernoulli sub-ODE method and Jacobi elliptic function method, to obtain new rational, trigonometric, hyperbolic, and elliptic stochastic solutions. The acquired solutions are helpful in explaining certain fascinating physical phenomena due to the importance of the Zakharov system in the theory of turbulence for plasma waves. In order to show the influence of the multiplicative Wiener process on the exact solutions of the Zakharov system, we employ the MATLAB tools to plot our figures to introduce a number of 2D and 3D graphs. We establish that the multiplicative Wiener process stabilizes the solutions of the Zakharov system around zero.

Dynamics of invariant solutions of mKdV-ZK arising in a homogeneous magnetised plasma

Classical Lie symmetry analysis is proposed to get a new variety of similarity solutions to a nonlinear (3+1)-modified Korteweg de Vries-Zakharov–Kuznetsov equation. The equation is often used to control the type of weakly nonlinear ion-acoustic waves in a homogeneous magnetised electron-positron plasma. In a magnetised plasma, including some hot and cold ions, such waves exist. By comparing the results reported earlier, new variety of analytical solutions are created and validated. Trigonometric, hyperbolic, rational, and exponential solutions of various types are derived. To prove their physical significance, animation profiles are plotted using MATLAB. Dominated dynamical behaviour of the animation profile is represented in each case. Electrostatic potential dynamics reveal elastic single solitons to multi solitons, elastic multi solitons, kink to stationary, and bell-shaped to asymptotic behaviour. Equations (13) and (21) in this study that were solved trivially can lead to more research in this area.

Dispersive optical solitons along with integrability test and one soliton transformation for saturable cubic-quintic nonlinear media with nonlinear dispersion

Three well known properties of solitons will be discussed in this paper, like Painleve test (P-test), Hirota bilinear method (HBM) and extended modified auxiliary equation mapping (EMAEM) architectonic in saturable cubic quintic nonlinear media (SCQNM) in the presence of nonlinear dispersion. The P-test will be used to examine the integrability of our governing model. For finding multiple soliton interaction HBM scheme will be adopted. EMAEM architectonic will be implemented in order to obtain some new solitary wave solutions like bright dromion (soliton), domain wall, singular, periodic, doubly periodic, trigonometric, rational and hyperbolic solutions etc. Graphical outcomes (3D, 2D, and density plots) will be shown to reflect the actual behavior of results.

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).

On exact solutions of some important nonlinear conformable time-fractional differential equations

The nonlinear fractional Boussinesq equations are known as the fractional differential equation class that has an important place in mathematical physics. In this study, a method called $$\Big (\frac{G'}{G^2}\Big )$$ -extension method which works well and reveals exact solutions is used to examine nonlinear Boussinesq equations with conformable time-fractional derivative. This method is a very useful approach and extremely utility compared to other analytical methods. With the proposed method, there are three unique types of solutions such as hyperbolic, trigonometric and rational solutions. This approach can similarly be applied to other nonlinear fractional models.

Some new type optical and the other soliton solutions of coupled nonlinear Hirota equation

In this study, we obtained some exact traveling wave solutions of coupled nonlinear Hirota equation (CNHE) by using generalized G'G-expansion method. The solutions we obtained are hyperbolic, trigonometric and exponential solutions. Analytical results for the obtained solutions for the CNHE have been corroborated through the Mathematica 11.2 symbolic calculation. Solutions have been illustrated in 3D graphs, and 2D plots. The present method has been used to obtain exact traveling wave solutions for several nonlinear partial differential equations.