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Fractional differential equations are being used to define numerous physical phenomena instead of conventional ordinary or partial differential equations. The secret is to get more generalization of the analysis, hence its solutions. The applications vary between electrical circuit modeling, oscillating circuits, shallow water behavior, viscous fluids, solution transportation, control theory and ultrafast optics. Sine–Gordon Expansion (SGE) Method is being employed hereafter to fully investigate the space–time conformable fractional Vakhnenko–Parkes equation and its modified version. Based on SGE method, the fractional models are transformed into an equivalent ordinary differential equation. The solutions are very important in studying the behavior of ultrafast optics in fibers or waveguides and the wave propagation throughout fiber optics. The solutions are formatted as solitons and are illustrated graphically. The illustrations show that the traveling waves inside fiber optics behave like solitons with traveling peak values according to the wave velocity. One other mode of propagation is to travel in kink shapes. Additionally, the ultrafast optical waves may propagate in shockwaves mode if the waves propagate at ultra-high speeds. [ABSTRACT FROM AUTHOR]

In this article, based on the concepts of the extended tanh expansion method, the general nonlinear magneto-optic waveguides that maintain parabolic-nonlocal law of refractive index is discussed. Then, the physical properties and structures of these obtained solutions are analyzed by means of some graphics. The obtained results can be used in describing the substantial understanding of the studious structures as well as other related non-linear physical structures. [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]

The present paper aims to investigate the coupled nonlinear Schrödinger equation in magneto-optic waveguides having anti-cubic (AC) law nonlinearity. The solitons secured to magneto-optic waveguides with AC law nonlinearity are extremely useful to fiber-optic transmission technology. Three constructive techniques, namely, the (G ′ / G) -expansion method, the modified simple equation method, and the extended tanh method are utilized to find the exact soliton solutions of this model. Consequently, dark, singular, combined dark-singular and periodic soliton solutions are obtained. The behaviours of soliton solutions are presented by 3D and 2D plots. [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 principal purpose of this research is to utilize the exp-function approach for achieving exact solutions of nonlinear optical fibers, including fractional order in the sense of conformable derivatives. Owing to the algorithm of symbolic computational, Some soliton solutions are obtained, including solitary soliton, singular kink-type soliton, and periodic solitons. These results are generated and developed by using the exp-function. To the greatest of our knowledge, the soliton solutions given in this study could be highly beneficial in comprehending physical phenomena. [ABSTRACT FROM AUTHOR]

This paper investigates (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system equations using the improved Sadar subequation method. It uncovers analytical solutions, including dark solitons, bright solitons, and periodic waves. The dynamic behavior of these solutions is illustrated through 2D and 3D plots by adjusting parameters. The results highlight the effectiveness and simplicity of the proposed methods, providing a versatile approach to obtaining various traveling wave solutions. These findings are expected to advance the shallow water theory of ideal fluids. [ABSTRACT FROM AUTHOR]

The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is investigated in this study. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is an extension of Calogero–Bogoyavlenskii–Schiff equation that describes the movement of Riemann waves along y-axis while long waves moves along the x-axis. The dynamics of Riemann waves is one of the most significant applications including tsunami in rivers, internal waves in oceans and magento-sound waves in plasmas. Finding new precise solutions with the assistance of a relatively new extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique is the primary objective of this effort. The suggested techniques are important tools in the fields of mathematical physics. Successful extraction of hyperbolic, rational, and trigonometric function solutions are achieved by using the proposed analytical methods. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is studied for the first time using extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique in this work and novel solutions are observed. 3D plots, contour plots and 2D plots are used to depict the dynamics of the extracted solutions. [ABSTRACT FROM AUTHOR]

The propagation of optical solitons via nonlinear metamaterials with cubic-quintic nonlinearity, detuning intermodal dispersion, self steepening effect, and nonlinear third and fourth-order dispersions is the focus of this study. To find the optical solitons and other solutions, the extended sinh-Gordon equation expansion method is applied to the aforementioned model. As a result, dark, bright, combined dark–bright, singular, combined singular soliton, and singular periodic wave solutions are obtained. To our best knowledge, the application of the method to the model, and the acquired combined soliton solutions are novel. To understand the nonlinear propagation theory of solitons in metamaterials, the reported outcomes can be enriched by the soliton theory. [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]

This article is concerned with analyzing exact optical soliton solutions to Kudryashov's model. To this end, the sine-Gordon expansion and Kudryashov methods are exerted and a bunch of new optical soliton solutions is acquired. The implementation and algebraic methods are discussed in detail. Finally, singular and bright-dark soliton solutions emerge from the scheme and the existing criteria for such solutions are elucidated. [ABSTRACT FROM AUTHOR]

The exact solutions of the (2 + 1) dimensional Broer–Kaup–Kupershmidt (BKK) system which has been recommended to model the nonlinear and dispersive long gravity waves traveling along with the two horizontal directions in the shallow water of uniform depth were obtained. Firstly, the given system was reduced to an ordinary differential equation (ODE) with the help of the wave transformations. Then, the reduced ODE was solved with the help of two methods which are called the modified (G ′ / G) -expansion method and new extended generalized Kudryashov method. We checked the results with the Maple software and plotted 3D, contour and 2D plots of some obtained solutions. As a result, we obtained exact solutions that are different from each other and have not been obtained before. Results can enhance the nonlinear dynamical behavior of a given system and demonstrate the effectiveness of the employed methodology. Results will be beneficial to a large number of engineering model specialists and useful for understanding the wave motions. [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]

In this work, we investigate diverse analytical solutions and modulation instability of the nonlinear Schrödinger equation with an anti-cubic nonlinear term. We use the traveling wave transformation and the New Generalized Extended Direct Algebraic Method to perform a variety of exact traveling wave soliton-like solutions. The obtained solutions are among other trigonometric function solutions, complex soliton solutions, and rational solutions. A particular behavior has been depicted in self-focusing and defocusing nonlinearity where dark soliton changes to super rogue waves and breather soliton is obtained. We then use the linearizing scheme to get the growth rate of modulation instability. For a specific value of the excitation wave number and a specific time of simulation, diverse waveforms of the modulated waves are obtained. We have demonstrated that the combined dispersion and nonlinear terms, as well as the anti-cubic nonlinear term, can develop modulation instability. To confirm the predictions based on the analytical results, we use the direct numerical simulation, from which we have displayed a modulation wave of the growth rate modulation instability. It results equally that the anti-cubic nonlinearity is an important object to control the propagation of the solitonic waves in the nonlinear systems. [ABSTRACT FROM AUTHOR]

Searching for soliton solutions of nonlinear partial differential equations is one of the most interesting and important areas of science in the field of nonlinear phenomena. Soliton is a localized wave with exponential wings or is a localized wave with an infinite support. In this work, we study two extensions of (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Based on the simplified Hirota's method and the Cole-Hopf transformation method, new multiple front wave solutions are obtained for both versions. [ABSTRACT FROM AUTHOR]

In this paper, we studied a nonlinear wave equation that models the propagation of optical solutions in a weakly nonlocal and parabolic competing nonlinear medium. The exact traveling wave type solutions formulated in hyperbolic functions, rational and trigonometric functions multiplied by some exponential functions to the governing equation are determined explicitly by the extended form of the Kudryashov method. We have examined the behavior of the modulation instability (MI) growth rate. To substantiate the stability of the obtained dark and bell-shaped solitons, we use the split-step Fourier method. In addition, the conservation laws describing significant physical concepts of this equation are examined. Compared to the obtained results with Younis et al. (J Nonlinear Opt Phys Mater 24(04):1550049, 2015), Zhou et al. (Optik 124(22):5683–5686, 2013; Proc Rom Acad Ser A 16(2):152–159, 2015) and Akinyemi et al. (Optik 230:166281, 2021), we have shown the propagation of the solitonic waves which sometime tilt from right to left with stable shape. [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, a variety of optical soliton solutions are derived for a nonlinear generalized equation with variable coefficients. At first, a computational approach is used to obtain solutions for the proposed model for a particular case. After, a generalized approach is considered to obtain other type of solutions given in a more general form. From the model considered here, the classical perturbed Fokas-Lenells equation is obtained and new optical soliton solutions for this last case are presented. Finally, some conclusions are given. [ABSTRACT FROM AUTHOR]

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]