sinh-Gordon Equation Expansion Method Research Articles

Soliton solutions of the improved perturbed nonlinear Schrödinger equation having parabolic law with non-local nonlinearity in the presence of chromatic and spatio-temporal dispersion terms

This paper was organized to examine the analytical solutions of the improved perturbed parabolic-law nonlinear Schrodinger equation including non-local nonlinearity in the presence of chromatic and spatio-temporal dispersion. This model mostly makes use of studying the propagation of optical pulses in fiber optic communication systems. We performed the Sinh-Gordon equation expansion method so that we produce the analytical solutions of the model under consideration. It was confirmed that the acquired solutions satisfy the main model. Therefore, bright and dark soliton solutions were retrieved; besides, various 3D and 2D graphical illustrations of the solitons were demonstrated via appropriate values of the parameters. Furthermore, this manuscript focused on the parameters’ effect on the acquired solitons behavior.

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

A nonlinear electrical lattice is an effective experimental tool for dealing with nonlinear dispersive media and creating possibilities for the realistic modeling of electrical soliton. In this work, we discuss the dynamical behavior of the governing (1+1)-dimensional time–space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions are obtained by the two analytical approaches, namely the modified generalized exponential rational function method (MGERFM) and the extended sinh-Gordon equation expansion method (shGEEM). The MGERF approach is applied to construct exact solitary wave solutions containing trigonometry, hyperbolic and exponential functions, while the extended shGEE technique is employed to obtain periodic wave solutions containing hyperbolic and Jacobi elliptic functions via performing the symbolic computation in the software package MATHEMATICA. Moreover, by means of standard linear-stability analysis, the modulation instability (MI) and the MI gain spectrum is analytically computed for the considered equation. By selecting suitable parametric values, solution profiles are portrayed in order to make the solutions physically relevant. These results yield periodic waves, kink waves, multi-soliton, and their interactions. By comparing the obtained solutions to the previous findings, we can reveal the novelty of the solutions. In addition, all the obtained solutions were verified by substituting them back into the governing equation using the Mathematica software. The acquired results are significant for the study of wave propagation, signal transmission, and applications of super-transmission phenomena.

Bifurcation analysis, chaotic analysis and diverse optical soliton solutions of time-fractional (2+1) dimensional generalized Camassa-Holm Kadomtsev-Petviashvili equation arising in shallow water waves

This paper presents a study on (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation, which is used to describe the behavior of shallow water waves in nonlinear media. The considered equation provides a more accurate description of wave behavior compared to linear wave equations and can account for wave breaking and other nonlinear effects. This model can be used to describe and study the behavior of nonlinear waves such as rogue waves in complex fluid dynamics scenarios. This includes the behavior of waves in stratified fluids, nonlinear dispersive media and wave interactions in fluid flows with varying velocities and densities. The bifurcation analysis of the governing equation has been performed using the planar dynamical system method. The chaotic behavior of the dynamical system has been examined by utilizing various techniques such as time series analysis and the construction of 2D and 3D phase space trajectories. Furthermore, the introduction of a perturbed term has resulted in the observation of chaotic and quasi-periodic behaviors across a range of parameter values. The considered equation has been reduced to ordinary differential equation by performing symmetry reduction. The Kudryashov method has been used to obtain the exact solution of reduced equation. The single soliton solution of governed equation has been obtained by using Hirota method and impact of fractional parameter on the obtained solution has been studied using graphical representation. The extended sinh-Gordon equation expansion method and modified generalized exponential rational function method have been exploited to obtain dark, bright and singular soliton solutions of considered equation. The motivation for this study arises from the need to understand and analyze the complex dynamics of shallow water waves in nonlinear media with a particular focus on the (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation. By performing symmetry reduction and applying various analytical methods, we aim to unravel the intricate behavior and soliton solutions of considered equation, contributing to the broader understanding of nonlinear wave phenomena.

Soliton solutions of generalized time-fractional Boussinesq-like equation via three techniques

Extraction of exact solutions for nonlinear PDEs has become very popular in recent years. The presented work deals with the investigation of time fractional perturbed Boussinesq-like equation in the sense of Beta and M-truncated derivatives. Three well-known methods, namely, extended sinh-Gordon equation expansion method, G′G2−expansion method and (exp(−ϕ(ξ)))-expansion method are used to get the traveling wave solutions of time-fractional perturbed Boussinesq-like equation. New trigonometric, hyperbolic and rational functional solutions are retrieved.

Diverse optical solitons to nonlinear perturbed Schrödinger equation with quadratic-cubic nonlinearity via two efficient approaches

In this study, the nonlinear perturbed Schrödinger equation(NPSE) with nonlinear terms as quadratic-cubic law nonlinearity media with the beta derivative is investigated and this investigated model is considered an icon in the field of optical fibers where it describes the wave function or state function of the optical system. Numerous solutions are extracted by engaging two novel schemes known as generalized -expansion method (GEEM) and rational extended sinh-Gordon equation expansion method (REShGEEM) in distinct forms such as bright, dark, singular and combinations of these solutions. In addition, plane wave, periodic and exponential solutions are also recovered. Using the computer application Wolfram Mathematica 11, the dynamic structure and physical characterization of some observed solutions are vividly depicted by sketching different plots. Comparing our new results with well-known literature is also given which justifies the novelty of this work. Obtained results will hold a significant place in the field of nonlinear optical fibers and suggest that the proposed methods are very influential and effective tools for solving more complex nonlinear partial differential equations in mathematical physics, engineering and nonlinear optics.

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.

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.

New soliton waves and modulation instability analysis for a metamaterials model via the integration schemes

Abstract In this paper, the exact analytical solutions to the generalized Schrödinger equation are investigated. The Schrodinger type equations bearing nonlinearity are the important models that flourished with the wide-ranging arena concerning plasma physics, nonlinear optics, fluid-flow, and the theory of deep-water waves, etc. In this exploration, the soliton and other traveling wave solutions in an appropriate form to the generalized nonlinear Schrodinger equation by means of the extended sinh-Gordon equation expansion method, tan(Γ(ϖ))-expansion method, and the improved cos(Γ(ϖ)) function method are obtained. The suggested model of the nonlinear Schrodinger equation is turned into a differential ordinary equation of a single variable through executing some operations. One soliton, periodic, and singular wave solutions to this important equation in physics are reached. The periodic solutions are expressed in terms of the rational functions. Soliton solutions are obtained from them as a particular case. The obtained solutions are figured out in the profiles of 2D, density, and 3D plots by assigning suitable values of the involved unknown constants. Modulation instability (MI) is employed to discuss the stability of got solutions. These various graphical appearances enable the researchers to understand the underlying mechanisms of intricate phenomena of the leading equation. The individual performances of the employed methods are praiseworthy which deserves further application to unravel any other nonlinear partial differential equations (NLPDEs) arising in various branches of sciences. The proposed methodologies for resolving NLPDEs have been designed to be effectual, unpretentious, expedient, and manageable.

Exact solitons to M-fractional (2+1)-dimensional CNLSE based on three different methods

In this paper, we discuss the exact solitons of M-fractional (2+1)-dimensional Chiral non-linear Schrödinger equation . We use of three distinct schemes: Kudryashov method, extended (G′/G)-Expansion method and extended Sinh-Gordon equation expansion method. Verification of solutions are done by using MATHEMATICA software. It is shown that the obtained solutions are supported by some 2-D and 3-D graphs for different parameters and M-fractional parameters values. The obtained results are newer than the existing results in the literature and can also be fruitful for the development of some different models in future. The methods that are utilized are simple, reliable and useful methods than the other methods. Therefore, these methods may be used to find the exact solutions of other fractional partial nonlinear differential equations.

Exploring the new optical solitons to the time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger system via three different methods

Abstract This current research is about some new optical solitons to the time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger (NLS) system with novel truncated M-fractional derivative. The obtained results may be used in the description of the model in fruitful way. The novel derivative operator is applied to study the aforementioned model. The achieved results are in the form of dark, bright, and combo optical solitons. The achieved solutions are also verified by using the MATHEMATICA software. The obtained solutions are explained with different plots. Modified integration methods, Exp a {{\rm{Exp}}}_{a} function, extended ( G ′ ∕ G ) \left(G^{\prime} /G) -expansion, and extended sinh-Gordon equation expansion method are applied to achieve the results. These exact solitons suggest that these methods are effective, straight forward, and reliable compared to other methods.

Analysis in fiber Bragg gratings with Kerr law nonlinearity for diverse optical soliton solutions by reliable analytical techniques

This paper reveals optical soliton solutions to fiber Bragg gratings (FBGs) with dispersive reflectivity having Kerr law of nonlinear refractive index. Bragg gratings are no doubt a technological spectacle that sustained balance between dispersion and the nonlinear effects that leads to a stable transmission of solitons across intercontinental distances. FBGs as sensor elements are used for measuring numerous engineering parameters such as temperature, strain, pressure, tilt, displacement, acceleration, load. Two recently developed mechanisms such as the unified method and extended sinh-Gordon equation expansion method (ShGEEM) are successfully employed to secure optical pulses in the shapes of the bright, dark, singular, complex combo, periodic, and plane wave solutions. The achieved solutions contain key applications in engineering and physics. These solutions define the wave performance of the governing models. By selecting suitable parametric values, the dynamics of the evaluated results are exemplified by sketching their 2-dimensional, 3-dimensional, and contour profiles to understand the real phenomena for such sort of nonlinear models. The novelty of the gained outcomes is manifested by a detailed comparison with the results that already exist.

RETRACTED: On soliton solutions of a modified nonlinear Schrödinger’s equation of third-order governing in optical fibers

Recently, the development of novel methods for finding solutions to partial differential equations has made great progress in various fields of science. These equations are an important tool in describing many phenomena that occur around us. The research work aims to study a third-order nonlinear Schrödinger’s equation using the modified generalized exponential rational function method, and the extended sinh-Gordon equation expansion method. The common idea between the two methods is that it is first necessary to reduce the equation with partial derivatives to a form of the equation with ordinary derivatives using a new wave definition. Moreover, the retrieved solutions are obtained in a closed form of elementary functions which are important in practical applications. We successfully construct some soliton, singular soliton and singular periodic wave solutions to the nonlinear complex model. Further, numerical simulations are also included via several figures. A significant advantage of both techniques is the ability to introduce a wide range of types of analytical solutions in terms of well-known elementary functions. It is obtained that these methods are also applicable in solving other equations in nonlinear science.

Some new types of optical solitons to the time-fractional new hamiltonian amplitude equation via extended Sinh-Gorden equation expansion method

This research is about some new exact solitons to the new Hamiltonian amplitude equation with novel truncated M-fractional derivatives. Extended Sinh-Gordon equation expansion method (ShGEEM) is applied to obtain solitons to the new Hamiltonian amplitude equation with novel truncated M-fractional derivatives. The obtained results may be used in the description of the model in a fruitful way. The novel derivative operator is applied to study the aforementioned model. The achieved results are in the form of dark, bright, and combo optical solitons. The achieved solutions are also verified by using the Mathematica software. Some of the solutions are drawn in two and three dimensions so that the results can be easily discussed through them.

Optical solitons and stability analysis for the new (3+1)-dimensional nonlinear Schrödinger equation

This paper explores the new [Formula: see text]-dimensional nonlinear Schrödinger equation which is used to model the propagation of ultra-short optical pulses in highly-nonlinear media. This equation is newly derived based on the extended [Formula: see text]-dimensional zero curvature equation. An effective technique, namely, the extended sinh-Gordon equation expansion method is applied to find optical soliton solutions and other solutions for this model. As a result, dark, bright, combined dark–bright, singular, combined singular soliton solutions, and singular periodic wave solutions are obtained. The stability of the model is investigated by using the modulation instability analysis which guarantees that the model is stable and all solutions are stable and exact. Physical explanations of the obtained solutions are presented by using 3D and 2D plots. The reported outcomes are useful in the empirical application of fiber optics.

New optical solitons and modulation instability analysis of generalized coupled nonlinear Schrödinger–KdV system

In this study, the generalized coupled nonlinear Schrödinger–KdV (NLS–KdV) system is investigated to obtain new optical soliton solutions. This system appears as a model for reciprocity between long and short waves in various physical settings. Different kinds of new soliton solutions including dark, bright, combined dark-bright, singular and combined singular soliton solutions are obtained using two effective methods namely, the extended sinh–Gordon equation expansion method and the solitary wave ansatz method. In addition, the modulation instability analysis of the system is presented based on the standard linear-stability analysis. The behaviours of obtained solutions are expressed by 3D graphs.

Investigation of diverse genres exact soliton solutions to the nonlinear dynamical model via three mathematical methods

This work aims to study new exact soliton solutions to the nonlinear Maccari’s system using the new extended direct algebraic method (NEDAM), the unified method, and the extended Sinh-Gordon equation expansion method (ShGEEM). Diverse genres exact soliton solutions are secured in the form of bright, dark, singular and in their combined behavior as bright-dark, dark-singular and periodic wave solutions. The Maccari system is a nonlinear model that represents the dynamics of waves, confined to a small part of space, in different regions such as nonlinear optics, plasma physics, mathematical physics, fluid mechanics, and hydrodynamics, etc. The secured solutions contain vital applications in engineering and physics. These solutions define the wave performance of the governance models. All the obtained solutions verified the considered model in this study. By choosing different parametric values, we established 3D, 2D and contour profiles for some selected solutions to describe the dynamic physical behavior of the acquired solutions. The novelty of the obtained results is discussed with a detailed comparison with the already existing results. The proposed methods are powerful and can be applied alternatively to recover new soliton solutions of different types of partial differential equations.

Exact soliton solutions to the Cahn–Allen equation and Predator–Prey model with truncated M-fractional derivative

The current research is about new exact soliton solutions to the Cahn–Allen equation and Predator–Prey model with the most general fractional derivative operator. The novel truncated M-fractional derivative is applied to study the aforementioned models to secure soliton solutions via the modified integration scheme known as the extended Sinh–Gordon equation expansion method (ShGEEM). This study provides the dark, singular, singular-dark, kink solitons and other solutions with certain conditions. The gained results are also verified with the use of symbolic software. The obtained solutions may be applied in the demonstration of the Cahn–Allen equation and Predator–Prey model in some better way using a fractional derivative. These solutions suggest that this method is more effective, straight forward and reliable as compared to other methods.

Some novel approaches to analyze a nonlinear Schrodinger’s equation with group velocity dispersion: Plasma bright solitons

Taking two efficient analytical techniques, including the generalized exponential rational function method and the extended sinh–Gordon equation expansion method, into account several optical solutions to a variety of nonlinear Schrödinger’s equation with group velocity dispersion are constructed. We are investigated several families of localized structures (bright solitons) for attractive nonlinearities including localized structures in both angular directions in addition to localized structures in one-directional and homogeneous in the other. To provide a better understanding of the obtained results, some numerical simulations for the obtained solutions are carried out. Further, the obtained solutions are represented graphically in order to clarify the vision and deep understanding of the mechanism of all nonlinear phenomena described by these solutions. The acquired results in this survey are new and have not been obtained in previous literature. Moreover, both utilized techniques can be assumed as an important tool for the explanation of some nonlinear physical phenomena such as the modulated envelope localized structures in plasma physics and fluid mechanics. Also, the obtained solutions can help researchers interested in studying the acoustic nonlinear modulated structures in different plasma models especially bright envelope solitons.

Dynamics of optical pulses in fiber optics

In this paper, we pay attention to the nonlinear dynamical behavior of ultra-short pulses in optical fiber. The new Hamiltonian amplitude equation is used as a governing model to analyze the pulses. We secure the ultra-short pulses in the forms of bright, dark, singular, combo and complex soliton solutions by the utilization of three of sound computational integration techniques that are protracted (or extended) Fan-sub equation method (PFSEM), the generalized exponential rational function method (GERFM), extended Sinh-Gordon equation expansion method (ShGEEM). Moreover, Jacobi’s elliptic, trigonometric, and hyperbolic functions solutions are also discussed as well as the constraint conditions of the achieved solutions are also presented. In addition, we discuss the different wave structures by the assistance of logarithmic transformation. The findings demonstrate that the examined equation theoretically contains a large number of soliton solution structures. By selecting appropriate criteria, the actual portrayal of certain obtained results is sorted out graphically in 3D and 2D profiles. The results suggest that the procedures used are concise, direct, and efficient, and that they can be applied to more complex phenomena. The resulting solutions are novel, intriguing, and potentially useful in understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including nonlinear optics. Our new results have been compared to these in the literature.

Some novel optical solutions to the perturbed nonlinear Schrödinger model arising in nano-fibers mechanical systems

This paper aims to compute solitary wave solutions and soliton wave solutions based on the ansatz methods to the perturbed nonlinear Schrödinger equation (NLSE) arising in nano-fibers. The improved [Formula: see text]-expansion method and the rational extended sinh–Gordon equation expansion method are used for the first time to obtain the new optical solitons of this equation. The obtained results give an accuracy interpretation of the propagation of solitons. We held a comparison between our results and those are in the previous work. The outcome indicates that perturbed NLSE arising nano-fibers is used in optical problems. Finally, via symbolic computation, their dynamic structure and physical properties were vividly shown by three-dimensional, density, and [Formula: see text]-curves plots. These solutions have greatly enriched the exact solutions of (2+1)-dimensional perturbed nonlinear Schrödinger equation in the existing literatures.