Extended sinh-Gordon Equation Expansion Research Articles

Multiple soliton and traveling wave solutions for the negative-order-KdV-CBS model

The (3+1)-dimensional new negative-order-KdV-CBS model is investigated in this study. The suggested model combines the Korteweg-de Vries (KdV) and Calogero-Bogoyavlenskii-Schiff (CBS) equations. This research provides multiple soliton solutions and traveling wave solutions for the KdV-CBS model. Multiple exp-function methods have been used for extracting soliton solutions. For this aim, the extended sinh-Gordon equation expansion approach was selected to get traveling wave solutions. The findings are graphically examined by selecting appropriate values for arbitrary parameters.

Explicit solitary wave profiles and stability analysis of biomembranes and nerves

This paper examines the stability analysis and exact solitary wave solutions of the nonlinear partial differential equation known as the Heimburg model. The several types of solitary wave solutions, soliton solutions and Jacobi elliptic doubly periodic function solutions are explored by using the extended Sinh-Gordon equation expansion approach. These investigations exhibit the system’s astounding diversity of waveforms, highlighting its potential applications in nerves and biomembranes. By selecting some appropriate values for the parameters, 3D, 2D, and its corresponding contour graph are plotted to represent the physical relevance of some of the solutions. Additionally, the linearized stability of this system is analyzed. The suggested approach is the finest resource for the analytical investigation of any nonlinear issue that occurs in various scientific fields.

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.

Exact Solutions of M-Fractional Kuralay Equation via Three Analytical Schemes

This article concerns new analytical wave solutions of the Kuralay-II equations (K-IIAE and K-IIBE) with exploration of a new definition of the derivative. This model is used in various fields, like nonlinear optics, ferromagnetic materials and optical fibers. For this purpose, the expa function, the extended sinh-Gordon equation expansion scheme, and the generalized Kudryashov schemes were utilized. The resulting solutions are dark, bright, dark-bright, periodic, singular and other kinds of solitons. These results are obtained and also verified by the Mathematica tool. Some of the solutions are explained with 2-D, 3-D and contour plots using the Mathematica tool. The solutions obtained succede the present solutions in the literature. For the first time, the effect of the fractional derivative on the solutions is also shown graphically for this model. The analytical wave solutions are highly desirable as they offer insights into the underlying physics or mathematics of a system and provide a framework for further analysis. The results obtained can also be fruitful for the development of models in the future. The schemes used in this research are effective, easy to apply, and reliably handle other fractional non-linear partial differential equations.

Analytical wave solutions of an electronically and biologically important model via two efficient schemes

We search for analytical wave solutions of an electronically and biologically important model named as the Fitzhugh–Nagumo model with truncated M-fractional derivative, in which the expa function and extended sinh-Gordon equation expansion (EShGEE) schemes are utilized. The solutions obtained include dark, bright, dark-bright, periodic and other kinds of solitons. These analytical wave solutions are gained and verified with the use of Mathematica software. These solutions do not exist in literature. Some of the solutions are demonstrated by 2D, 3D and contour graphs. This model is mostly used in circuit theory, transmission of nerve impulses, and population genetics. Finally, both the schemes are more applicable, reliable and significant to deal with the fractional nonlinear partial differential equations.

Optical solitons of improved perturbed nonlinear Schrödinger equation with cubic-quintic-septic and triple-power laws in optical metamaterials

Purpose. This paper aims to extract optical solitons of improved perturbed nonlinear Schrödinger equations (IP-NLSE) with cubic-quintic-septic (CQS) and a triple-power law (TP-law) using the new Kudryashov and the extended sinh-Gordon equation expansion (eShGEE) methods. Methodology. First, we apply a wave transformation to the studied equations to generate the nonlinear ordinary differential equation (NLODE) form. Next, by computing the balancing constant in the NLODE form, we use the new Kudryashov and eShGEE methods to obtain the equation’s solution in the NLODE form. We get an algebraic equation system on the NLODE by replacing the suggested solution function and its derivatives in the NLODE form. With the help of the solutions of the system, we are able to determine the appropriate solution sets for unknown parameters. Substituting these sets and wave transforms into the proposed solution functions by the new Kudryashov and eShGEE methods, we get the solutions for the problems under investigation. Findings. We have successfully obtained soliton solutions for the considered equations and plotted 3D and 2D graphs of the derived solution functions. In addition to obtaining the soliton solutions, we present some graphical investigation of the impact of the parameters in the considered equations. Originality. To our best knowledge, the improved perturbed nonlinear Schrödinger equations with CQS and a triple-power law have not been studied before. It is also innovative to examine how the equation’s parameters affect the soliton’s behavior. In this regard, the study’s findings are novel, and it is anticipated that they will advance research in the area.

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.

Optical solitons to the beta-fractional density dependent diffusion-reaction model via three different techniques

This research is concerned with some modernistic optical solitons of beta-fractional density-dependent diffusion-reaction model. The collected optical solutions can be executed in exposing this model in a prominent form. The obtained results include the trigonometric, hyperbolic trigonometric, rational and exponential functions. Verification of achieved results is also done by using Mathematica tools. Three distinct techniques named as [Formula: see text] function, extended sinh-Gordon equation expansion (EShGEE) and modified extended [Formula: see text] function expansion techniques are employed to protect the results. The achieved results are also illustrated by 2-D, 3-D and contour plots. The gained results can also be fruitful for the development of model in future.

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.

Solitary wave behavior of (2+1)-dimensional Chaffee-Infante equation.

The behavior of gas diffusion in a homogeneous medium is described by the (2+1)-dimensional Chaffee-Infante equation. In this work, the solitary wave behavior of the (2+1)-dimensional Chaffee-Infante equation is studied with the help of extended sinh-Gordon equation expansion technique. Bright, dark, periodic, kink, anti-kink and singular traveling wave patterns are observed for suitable choice of parameters. The 3D graphs, 2D plots and contour plots are included to understand the dynamics of the obtained solutions. The obtained results depict that the extended sinh-Gordon equation expansion technique provides an efficient tool for solving other equations that occur in different branches of science and technology.

Bifurcation Analysis and Bounded Optical Soliton Solutions of the Biswas-Arshed Model

We investigate the bounded travelling wave solutions of the Biswas-Arshed model (BAM) including the low group velocity dispersion and excluding the self-phase modulation. We integrate the nonlinear structure of the model to obtain bounded optical solitons which pass through the optical fibers in the non-Kerr media. The bifurcation technique of the dynamical system is used to achieve the parameter bifurcation sets and split the parameter space into various areas which correspond to different phase portraits. All bounded optical solitons and bounded periodic wave solutions are identified and derived conforming to each region of these phase portraits. We also apply the extended sinh-Gordon equation expansion and the generalized Kudryashov integral schemes to obtain additional bounded optical soliton solutions of the BAM nonlinearity. We present more bounded optical shock waves, the bright-dark solitary wave, and optical rogue waves for the structure model via these schemes in different aspects.

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.

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.