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A new fractional-order derivative operator is presented and applied to nonlinear Sasa-Satsuma equation having Kerr nonlinearity law with third-order dispersion. Two novel families of singular, dark and bright soliton solutions for the fractional-order nonlinear Sasa-Satsuma equation were obtained by considering the modified simple equation method and the F-expansion method. Behavior of these soliton solutions are illustrated in 3d graphs, contour plots and 2d plots. Furthermore, periodic singular solutions of the equation are obtained by both analytical techniques. Optical type soliton solutions of the nonlinear Schrödinger equation is generated as well. To show out the behavior of the modulation gain spectra with the effects of the fractional derivative order, the linear stability technic was used. The effects of the fractional derivative order have been investigated in normal and anomalous dispersion regime associated to the Kerr nonlinearity effects. MI gain spectra were depicted by choosing adequate parameters of the fractional nonlinear Sasa-Satsuma equation. [ABSTRACT FROM AUTHOR]

In this paper a new definition of a local fractional derivative of order α is introduced and applied to the study of the fractional nonlinear Schrödinger equation (FNLSE) with third-order dispersion and with Kerr and power laws of nonlinear refractive index. The analytical soliton solutions correspond to bright, dark and singular solitons obtained by different analytical methods. We found new optical soliton solutions with some constraints conditions that arise between the parameters of the NLSE. Typical behavior of the acquired solitons is depicted in some interesting simulations. [ABSTRACT FROM AUTHOR]

This article deals with new soliton solutions of the generalized nonlinear Schrodinger equation with variable coefficients by the direct algebraic method. Once the variables of this technique are considered as special values. we could achieve the solitary waves that are unique from those attained by the other methods. It can be inferred that the solution methodology in conjunction with symbolic computing eligible to solve nonlinear partial differential equations precisely. [ABSTRACT FROM AUTHOR]

Based on two algorithm integrations, such as the exp (- Φ (ξ)) -expansion method and the hyperbolic function method, we build dark, bright and trigonometric function solution to the Klein–Gordon equations with M-fractional derivative of order α . By adopting the travelling-wave transformation, the constraint condition between the model coefficients and the travelling-wave frequency coefficient for the existence of soliton solutions is also obtained. Moreover, miscellaneous soliton solutions obtained is depicted in 3D and 2D. [ABSTRACT FROM AUTHOR]

A new local fractional-order derivative operator is introduced and the Lakshmanan–Porsezian–Daniel (LPD) model is interpreted via this operator. New analytical solutions to the LPD equation is presented by Jacobi elliptic functions and an anzätz method. The complex-valued LPD equation includes a nonlinear term which is considered from three different cases: Kerr, parabolic and anti-cubic law of nonlinearities. For each case, dark, bright, singular optical soliton solutions related with optical fibers are presented. Simulations representing behavior of these solutions for different parameter values are provided. [ABSTRACT FROM AUTHOR]

Generally, metamaterials are any material with a special properties that are not in nature. Among them, the optical metamaterial is a type of electromagnetic metamaterial, that interacts with light or visible wavelengths. The mathematical modeling of this interaction is usually introduce by perturbed Schrödinger equation. The aim of this paper is to study the perturbed Schrödinger equation model of optical metamaterials for obtaining their solitary wave solutions. To obtain the solitary wave solutions of governing equation, we firstly reduced the equation to an ordinary differential equation (ODE), then multiplication of this equation with the derivative of the unknown function leads to a well known ODE. Secondly, integrating this resultant equation for some particular choices of constant coefficients gives many solutions in different classes. Many of these solutions are generalized forms of the solutions reported in the literature. Elliptic and triangular solutions are constructed in various forms besides bright and singular type soliton solutions. [ABSTRACT FROM AUTHOR]

The Kerr law and dual power law Schrödinger equations for obtaining optical soliton solutions are studied. In this paper, we investigate these equations via generalized logistic equation method. These various kinds of solutions are favorable for explaining diverse nonlinear physical phenomena. The outcomes are useful in describing the diffusion of optical solitons. [ABSTRACT FROM AUTHOR]

In this paper and for the first time, we describe and introduce a new extended direct algebraic method which is a new method for solving nonlinear partial differential equations arising in nonlinear optics and nonlinear science. By applying this method, we have constructed new solitary wave solutions for the unstable Schrödinger equation. A large family of traveling wave type exact solutions covering exponential, generalized trigonometric, rational and generalized hyperbolic functions to this equation is determined. The solutions are expressed in explicit forms. [ABSTRACT FROM AUTHOR]

In the field of nonlinear optics, both fractional and classical-order nonlinear Schrödinger (NS) equations are investigated. However, the fractional-order NS equation has gained widespread acceptance due to its higher compatibility. The space-time fractional nonlinear Schrödinger equation enfolding beta derivative has a wide range of applications in nonlinear optics, quantum computing, Bose-Einstein condensates, wave propagation in complex media, quantum mechanics, and engineering, where understanding wave propagation and nonlinear interactions are diametrical. In this article, the improved Bernoulli sub-equation function (IBSEF) procedure has been used to establish optical soliton solutions in the form of trigonometric, exponential, and hyperbolic functions comprising substantive parameters. These soliton solutions have different shapes, including kink, periodic soliton, singular kink, breathing soliton, and other types. The physical features of the solitons are revealed through three-, two-, contour, and density graphs. The research findings confirm that the IBSEF scheme is effective, straightforward, and applicable for ascertaining soliton solutions in various nonlinear fractional-order models in the fields of physics and communication engineering. [ABSTRACT FROM AUTHOR]

In this paper, solitons solutions of higher-order dispersive cubic–quintic Schrödinger equationincluding third-order as well as fourth-order derivatives with respect to time, that describes the dynamics of ultrashort pulses in optical fibers are investigated in detail. In this respect,a solution procedure in the locality of applied mathematics called the hyperbolic function method is appliedusing multi-linear variable separation approach (MLVSA). As an outcome, a bunch of soliton solutions isderived in conjunction with plotting dark and periodic wave solutions. The credibility of the results is examined by setting each solution back into its governing equation. Through portraits, different forms of wave solutions are depicted. Moreover, the restrictions on the parameters are also given for the existence of the obtained solutions. [ABSTRACT FROM AUTHOR]

In this paper, the full nonlinearity form of the space–time fractional Fokas–Lenells equation is scrutinized to get new analytical solutions. To achieve this, a convenient method is applied namely, a new Kudryashov method. The achieved solutions show pulse propagation in different wave patterns such as singular bell shape, peakon, singular anti-bell shape, dark, anti-bell and bright solutions. These physical characteristics are studied thoroughly by the graphical representation, which shows the fruitfulness and functionality of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we implemented extended e x p - φ ξ -expansion method for some exact solutions of (3 + 1)-dimensional nonlinear Schrödinger equation (NLSE) and coupled nonlinear Schrodinger's equation. The solutions we obtained are hyperbolic, trigonometric and exponential solutions. We observed that these solutions provided the equations through Mathematica 11.2. Apart from that, we have shown the graphics performance of some of the solutions found. This method has been used recently to obtain exact traveling wave solutions of nonlinear partial differential equations. The results achieved in this study have been confirmed with computational software Maple or Mathematica by placing them back into NLFPDEs and found them correct. We posited that the approach is updated to be more pragmatic, efficacious, and credible and that we pursue more generalized precise solutions for traveling waves, like the solitary wave solutions. [ABSTRACT FROM AUTHOR]

This paper presents an investigation into solutions of two types of nonlinear Schrödinger equations of logarithmic form. This model is of particular interest to researchers due to its vast application in the field of the implication of nanofibers. Through a comprehensive analysis, it was found that the solutions take the form of Gaussian-type solitary solutions. These solutions possess a variety of new structures due to the presence of many arbitrary constants and time-dependent functions in their general expressions. By fixing these constants and functions appropriately, Gaussian solitary waves for nonlinear perturbed Schrödinger equations can be represented, which could be beneficial in understanding the dynamics of these equations in nanooptical fibers. The results obtained from this study may provide a useful tool for further research into nonlinear Schrödinger equations and can be extended to fit more applications. [ABSTRACT FROM AUTHOR]

Solving partial differential equations has always been one of the significant mathematical tools for modeling applied phenomena in various fields. In this paper, our main motivation is to find the traveling wave solutions of unstable nonlinear Schrödinger equation (UNLSE) which describes the two layer baroclinic instability, and two lossless symmetric stream plasma instability, and also disturbance of time period in both stable and unstable media. The methods which are used to endure these solutions are extended rational sine-cosine/sinh-cosh. New solutions are constructed in the different forms such as hyperbolic and trigonometric. The achieved results present that proposed techniques are consequential for exploring several types of nonlinear partial differential equations (NLPDEs) in applied sciences and solutions are presented in the form of novel periodic, dark, bright, and periodically bright solitons. These methods are highly effective, robust, and provide an alternative approach for establishing new soliton solutions for various types of partial differential equations (PDEs) used in mathematical physics. [ABSTRACT FROM AUTHOR]

We are concerned with new form optical solitons to the non-linear Schrödinger equation (NLSE) by generalized anti-cubic non-linearity. Some new optical solitons containing dark, singular and periodic solitons in chirp-free and chirped form are constructed via modified extended tanh expansion method. These results are very useful in the field of optics. The obtained results are verified and also explain by drawing different graphs. Parametric conditions for the chirp-free and chirped shape soliton solutions are also mentioned. Computational results indicate the efficiency of the aforesaid analytical expansion scheme. [ABSTRACT FROM AUTHOR]

In this paper, we considered the nonlinear Schrodinger equation modeling cubic optical solitons in a polarization-preserving fiber with Kerr law. Using the improved t a n (φ / 2) -expansion method, a powerful and effective method, we constructed exact solutions including the hyperbolic function solution, the trigonometric function solution, the exponential solution and the rational solution with free parameters. The geometrical shapes for some of the obtained results are depicted for various choices of the free parameters that appear in the results. The obtained solutions are entirely new and can be considered as generalization of the existing results in the ordinary derivative case. [ABSTRACT FROM AUTHOR]

In this paper, the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity is studied by adopting four mathematical methods namely the Modified Kudryashov method, the extended Tanh–Coth method, the modified simple equation method and soliton ansatz method. As a results, a set of various types of solitons that contains dark, singular, dark–singular, bright optical solitons and other form of optical soliton solutions are obtained. Firstly, we solve the perturbed nonlinear Schrödinger equation by considering the dual power law parameter using the first three integration methods and secondly we set this parameter equal to one in order, to solve the problem with the fourth method. The used methods in this paper present various applications in fields of nonlinear wave equations. Comparing our new results with well-known in literature are also given. Moreover, the graphical representations of the modulus of some optical soliton solutions and their 2-D profile are also depicted. [ABSTRACT FROM AUTHOR]

This paper secures dark and singular resonant optical solitons that is studied with dual-power law nonlinearity and fractional temporal evolution. Khalil's conformable fractional derivative is put into perspective to retrieve these soliton solutions. The parameter restrictions for the existence of such solitons are also indicated. [ABSTRACT FROM AUTHOR]

The present study examines the nonlinear time-fractional model in the sense of a solitonic structure. A non-linear Schrödinger equation has applications in light scattering, indirect optical pulses as well as planer waves and to Bose-Einstein condensates enclosed in an anisotropic-shaped cigar, in a mean-field state, etc. A new extended direct algebraic method is utilized to get the soliton solutions with modified M-truncated and Atangana–Baleanu fractional operators which have Mittag-Leffler kernel. The obtained solutions contain new families of functions such as trigonometric, hyperbolic, rational, and exponential functions. The graphical 2D, 3D, contour, and also 3D spherical presentation pictorial the analysis with the feasible parametric values. On the evidence of the acquired solutions, it can be presumed that this technique is more effective and generalized to obtain solutions of many other non-linear partial differential equations that appear in different scientific disciplines. [ABSTRACT FROM AUTHOR]

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Correction to: Optical solitons of (3 + 1) dimensional and coupled nonlinear Schrodinger equations Correction to: https://doi.org/10.1007/s11082-022-03613-yhttps://doi.org/10.1007/s11082-022-0... The article with title "Optical solitons of (3 + 1) dimensional and coupled nonlinear Schrodinger equations" has been published twice with https://doi.org/10.1007/s11082-022-03613-y and https://doi.org/10.1007/s11082-022-03599-7. [Extracted from the article]

In this work, we study the exact traveling wave solutions of (2 + 1) -dimensional Chiral nonlinear Schrödinger equation with the aid of generalized auxiliary equation method. The aforementioned model is used as a governing equation to discuss the wave in quantum field theory. The suggested technique is direct, effective, powerful, and offers constraint conditions to ensure the existence of solutions. The solutions obtained are bright solitons, dark solitons, singular solitons, mixed solitons, periodic waves, exponential, rational, and complex solutions that are relevant in various applications of applied science. Finally, some solutions are depicted in two and three dimensional to better understand the behavior of the considered model. [ABSTRACT FROM AUTHOR]

• The Schrödinger equation for polynomial law medium with new local fractional derivative. • The fractional sub-equation method is used. • The obtained results are given by 2D and 3D figures. • The bright, dark, combined optical solitons are obtained. Novel solutions for the nonlinear dynamics of Schrödinger equation for polynomial law medium with third-order dispersion (TOD), fourth-order dispersion (FOD), and self-steepening are investigated based in a novel local fractional derivative of order α and the Jacobi elliptic function method which are combined into a novel fractional sub-equation method. The Jacobi elliptic function method will provide different types of analytical solutions and not only soliton type solutions for the propagation of ultra-short optical signals through a polynomial law medium. Solutions are illustrated in 3-D graphs, contour plots and 2-D plots under the obtained constraint conditions. [ABSTRACT FROM AUTHOR]

Presented herein is an exact examination on the resonant nonlinear Schrödinger equation (RNLSE) with Kerr law nonlinearity considering inter-modal dispersion and spatio-temporal. To this end, the ordinary differential equation (ODE) is derived from the traveling wave transformation. Then, the new extended direct algebraic method (NEDAM) is utilized in the axial direction to obtain the results of this study. Subsequently, a bunch of optical soliton solutions in conjunction with dark, bright, and dark-bright soliton solutions are plotted. As a result, it is verified the methodology has an effective mathematical gadget to solve nonlinear partial differential equations (NPDEs). [ABSTRACT FROM AUTHOR]

In this paper, we examined the effects of the nonlinear parameters and driven amplitude on the training envelope in discrete cubic-quintic nonlinear Schrödinger equation with arbitrarily high-order nonlinearities. From the numerical simulation, we exhibited the generation of the train of pulses and dark soliton when the driven amplitude is above the threshold. For a specific time of propagation, we illustrate how the variation of the driven amplitude and the nonlinear cubic-quintic parameters can generate instability in the forbidden gap. It emerges that the model of the discrete nonlinear Schrödinger equation with arbitrarily high-order nonlinearities can be used to generate the train of waves despite the fact that it is not integrable in the continuum limit approximation. We also displayed the effects of the cubic-quintic terms on the modulation instability growth rate. The obtained results will open new features to the train of pulses and dark soliton in the nonlinear fibers optics. • We studied discrete cubic-quintic nonlinear Schrödinger equation with arbitrarily high-order nonlinearities. • We examined the effects of the nonlinear parameters and driven amplitude on the training envelope in this nonlinear model. • We illustrated the effects of the cubic-quintic terms on the modulation instability growth rate. • The obtained solutions are depicted by 2D and 3D plots. [ABSTRACT FROM AUTHOR]