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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]

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, 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]

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]

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]

• 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]