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Dark and bright with singular solitons shall be yielded to Fokas–Lenells equation which describes soliton dynamics in optical fibers. The two integration schemes that are applied in this context are the modified simple equation method and the trial equation method. Additional solutions, besides, optical solitons, are recovered. [ABSTRACT FROM AUTHOR]

The notion of a convergence complete symmetric space is introduced in [6]. In this paper we shall show that every symmetric space admits a weakly S-convergence complete symmetric space. As applications of convergence completeness, we present some fixed point results for self-maps defined on a symmetric space. [ABSTRACT FROM AUTHOR]

In this paper, highly dispersive optical solitons are obtained with the perturbed complex Ginzburg--Landau equation, incorporating the Kerr law of nonlinearity, by the complete discriminant classification approach. A variety of solutions emerge from this scheme that include solitons, periodic solutions and doubly periodic solutions. The numerical sketches support the analytical findings. [ABSTRACT FROM AUTHOR]

This paper uses the classical Lie symmetry method for optical solitons to study the Chen-Lee-Liu equation. We are able to establish symmetries that convert the model into a set of ordinary differential equations and obtain solutions of the reduced equations through various methods. [ABSTRACT FROM AUTHOR]

In the current paper, a full spectrum of 1-soliton solutions to the concatenation model with the power-law of self-phase modulation has been recovered. The method of undetermined coefficients has permitted us to solve this problem successfully. The parameter constraints naturally emerge from the derivation and are also listed, guaranteeing these solitons' existence. It has been proved that dark solitons and singular solitons of a specific type would exist only when the power-law parameter collapses to unity. [ABSTRACT FROM AUTHOR]

The current paper recovers cubic–quartic optical solitons in fiber Bragg gratings having polynomial law of nonlinear refractive index structures. Lie symmetry analysis is carried out, starting with the basic analysis. Then, it is followed through with improved Kudryashov and generalized Arnous schemes. The parameter constraints are also identified for the existence of such solitons. Numerical surface plots support the adopted applied analysis. [ABSTRACT FROM AUTHOR]

Propagating chirped soliton solutions for the Chen–Lee–Liu equation also called the derivative nonlinear Schrödinger II equation are investigated by application of the ansatz method. The model incorporating self-steepening term has many applications in nonlinear optical fibers and plasma physics. A nonlinear differential equation describing the evolution of the wave amplitude in the nonlinear media is derived by means of the coupled amplitude-phase formulation. Special exact chirped soliton solution that takes the shape of “ W ” is determined for the first time in presence of all physical effects. It is shown that the nonlinear chirp associated with this type of solitons is crucially dependent on the wave intensity and related to self-steepening and group velocity dispersion parameters. Parametric conditions on system parameters for the existence of the chirped soliton structure are also presented. This soliton solution exists due to a balance among group velocity dispersion and self-steepening effect solely. [ABSTRACT FROM AUTHOR]

We consider the evolution of ultrashort pulses in metamaterials wherein the pulse propagation is governed by the generalized nonlinear Schrödinger equation with higher-order effects such as pseudo-quintic nonlinearity and self-steepening effect. We present three new types of exact singular soliton-like solutions for the model in the presence of the self-steepening term, whose phase chirping varies as a function of the pulse intensity. The obtained results indicate that the parameters of solutions are directly related the self-steepening parameter. Parametric conditions on physical parameters for the existence of the novel structures are also presented. Particular cases are discussed. [ABSTRACT FROM AUTHOR]

A higher-order nonlinear Schrödinger equation with the third-order dispersion, the cubic–quintic nonlinearities, derivative non-Kerr nonlinear terms, Kerr dispersion, and stimulated inelastic scattering is considered. The model can be used to describe the subpicosecond or femtosecond optical pulse propagation in highly nonlinear optical fibers. We find the exact gray solitary wave solution (i.e., a dark localized structure with nonzero minimum in intensity) on a continuous-wave background for the equation by adopting an appropriate ansatz solution. The propagation characteristics of such solitary pulses are determined in the presence of all higher-order effects. We also find black solitary wave solutions for the higher-order evolution equation under specific conditions for its coefficients. The parametric conditions on fiber parameters for the existence of both types are given. [ABSTRACT FROM AUTHOR]

This paper derives bright 1-soliton solution to the resonant nonlinear Schrödinger's equation that is studied with anti-cubic nonlinearity. The semi-inverse variational principle is adopted to secure analytical solution to the model. The parametric restrictions that emerge for the existence of solitons are also presented. [ABSTRACT FROM AUTHOR]

In the present investigation, we employed the Jacobi elliptic function (JEF) method to invoke the perturbed nonlinear Schrodinger equation with self-steepening (SS), self-phase modulation (SPM), and group velocity dispersion (GVD), which govern ¨ the propagation of solitonic pulses in optical fibres. The proposed algorithm proves the existence of the family of solitons in optical fibers. Consequently, chirped and chirp free W-shaped bright, dark soliton solutions are obtained from dn(ξ ), cn(ξ ) and sn(ξ ) functions. The final results are displayed in three-dimensional plots with specific physical values of GVD, SPM and SS for an optical fiber. [ABSTRACT FROM AUTHOR]

In the present investigation, we employed the Jacobi elliptic function (JEF) method to invoke the perturbed nonlinear Schrödinger equation with self-steepening (SS), self-phase modulation (SPM), and group velocity dispersion (GVD), which govern the propagation of solitonic pulses in optical fibres. The proposed algorithm proves the existence of the family of solitons in optical fibers. Consequently, chirped and chirp free W-shaped bright, dark soliton solutions are obtained from dn(!), cn(!) and sn(!) functions. The final results are displayed in three-dimensional plots with specific physical values of GVD, SPM and SS for an optical fiber. [ABSTRACT FROM AUTHOR]

This paper implements two of Kudryashov's approaches to extract optical soliton solutions to the concatenation model that is a conjunction of the nonlinear Schrödinger's equation, Lakshmanan-Porsezian-Daniel model, and the Sasa-Satsuma equation. A full spectrum of soliton solutions emerged along with the parameter constraints that are all comprehensively presented. [ABSTRACT FROM AUTHOR]

We derive stationary optical solitons for the case when the nonlinear refractive index of Kudryashov's quintuple form is coupled with a nonlocal type of self-phase modulation in the presence of nonlinear chromatic dispersion. An enhanced Kudryashov's approach has made this derivation of soliton solutions possible for the case of generalized temporal evolution. [ABSTRACT FROM AUTHOR]

This paper obtains soliton solutions to Sasa–Satsuma equation, which is studied in birefringent fibers with multiplicative noise. The extended auxiliary equation approach reveals solutions in terms of Jacobi's elliptic functions from which bright, dark and singular solitons emerge when the limiting operation is applied to the modulus of ellipticity, with the limit approaching zero or unity. [ABSTRACT FROM AUTHOR]

This paper obtains exact bright, dark and singular dispersive optical soliton solutions to DWDM system that is studied with spatio-temporal dispersion in addition to group-velocity dispersion. The governing model is Schrödinger–Hirota equation. The method of undetermined coefficients is employed to secure these soliton solutions that appear with necessary constraints for their existence. [ABSTRACT FROM AUTHOR]

This paper applied modified simple equation method to extract soliton solutions to birefringent fibers, in presence of perturbation terms. Both Kerr law and parabolic law of nonlinearity are studied. Dark and singular optical soliton solutions are listed for these nonlinearities. The existence of these solitons is guaranteed with appropriate constraints that are also presented. [ABSTRACT FROM AUTHOR]

We consider an extended nonlinear Schrödinger equation with competing quadratic-cubic nonlinearity. The model appears as an approximate form of a relatively dense quasi-one-dimensional Bose–Einstein condensate with repulsive contact interactions between atoms and a long-range dipole–dipole attraction between them. Exact analytical soliton solutions for the model are derived by utilizing the method of undetermined coefficients. The solutions comprise bright, dark, and singular solitons. Furthermore, we present a new structure which describes the evolution of two different type solitons in the form of W -shaped soliton solution and a bright soliton solution on a continuous-wave background. The constraint conditions for the existence of these solutions are also exhibited. The numerical evolution of the obtained solutions is also presented. [ABSTRACT FROM AUTHOR]

We investigate a higher-order nonlinear Schrödinger equation including the intermodal dispersion, self-steepening, delayed nonlinear response effect, and different contributions to the nonlinear dispersion. The model describes ultra-short (femtosecond) optical soliton propagation in metamaterials with Kerr law nonlinearity. By adopting a complex amplitude ansatz composed of the product of bright and dark solitary waves, we present exact dipole soliton solution for the model under certain parametric conditions. This type of soliton pulses characteristically exists due to a balance among nonlinear dispersion effects of different nature. Numerical results and discussions are also presented. [ABSTRACT FROM AUTHOR]

The current work recovers a perturbed bright 1-soliton solution to the governing model that maintains generalized quadratic–cubic nonlinearity. The perturbation terms appear with full intensity. The semi-inverse variational principle secures this soliton when all of the known approaches fail, in this regard. [ABSTRACT FROM AUTHOR]

We provide a detailed description of a numerical approach that makes use of the shifted Chebyshev polynomials of the sixth kind to approximate the solution of some fractional order differential equations. Specifically, we choose the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) to describe this method. We write our approximate solution in the product form, which consists of unknown coefficients and shifted Chebyshev polynomials. To compute the numerical values of coefficients, we use the initial and boundary conditions and the collocation technique to create a system of equations whose number matches the unknowns. We test the applicability and accuracy of this numerical approach using two examples. [ABSTRACT FROM AUTHOR]

Abstract: Alexandroff spaces include finite spaces and have a wide range of applications in many areas such as computer graphics and image analysis. We give results on small inductive dimension of Alexandroff -spaces. Particularly, we characterize the small inductive dimension through connected tuples and study standard properties of Dimension Theory using this characterization. Also, we investigate the relations between the small inductive dimension, the large inductive dimension, and the covering dimension in the class of all Alexandroff -spaces. We finally put questions on dimension of Alexandroff -spaces. [Copyright &y& Elsevier]

We use the homotopy perturbation method for solving the fractional nonlinear two-point boundary value problem. The obtained results by the homotopy perturbation method are then compared with the Adomian decomposition method. We solve the fractional Bratu-type problem as an illustrative example. [ABSTRACT FROM AUTHOR]

This paper employs two integration schemes to extract dark and singular optical soliton solutions to the coupled Fokas-Lenells equation that is studied in the context of polarization-mode dispersion fibers. The existence criteria for these solitons are also presented. [ABSTRACT FROM AUTHOR]

This paper reveals bright, dark and singular solitons in fiber Bragg gratings with dispersive reflectivity having parabolic form of nonlinearity. The method of undetermined coefficients yielded such soliton solutions. The existence criteria of such solitons are also enumerated that are listed as constraint conditions. [ABSTRACT FROM AUTHOR]

Optical soliton solutions are recovered for fibers having anti-cubic nonlinear law. Bright, dark and singular optical soliton solutions are retrieved by the aid of three forms of integrability tools. The existence criteria for such solitons are also presented. [ABSTRACT FROM AUTHOR]

In this work, symmetry reduction for nonlinear Schrödinger's equation with anti-cubic nonlinearity is determined by using the invariance of equations under Lie group of transformations. Using the similarity transformations the equation is reduced into system of ordinary differential equations. Corresponding to reduced ordinary differential equations, some exact solutions are obtained. These yields bright solitons, dark solitons, singular solitons and other explicit solutions. It is also pointed out that some of the results reported earlier are particular forms of current solutions. [ABSTRACT FROM AUTHOR]

This paper secures bright optical solitons in presence of third and fourth order dispersion terms while group velocity dispersion term is absent. The perturbation terms that are included appear with full nonlinearity. The semi-inverse variational principle is implemented to secure such solutions. [ABSTRACT FROM AUTHOR]

Abstract We investigate the propagation of self-similar optical solitons on a continuous-wave background through a non-centrosymmetric waveguide with second- and third-order nonlinearities. The generalized inhomogeneous nonlinear Schrödinger equation with quadratic–cubic nonlinearities and gain or loss is used to describe the beam propagation through the waveguide. Three new types of exact self-similar localized solutions propagating on a continuous-wave background are found under certain parametric conditions. The self-similar wave solutions comprise bright solitons, W-shaped, and kink solitons. The dynamic behaviors of these solitons are analyzed for a periodically distributed dispersion system. Highlights • Self-similar optical solitons with continuous-wave background are studied. • The self-similar wave solutions comprise bright solitons, W-shaped, and kink solitons. • The dynamic behaviors of these solitons are analyzed. [ABSTRACT FROM AUTHOR]

This work is about highly dispersive optical solitons with Kerr and power law nonlinearity. Bright, dark and singular solitons are obtained from undetermined coefficients. This method proves that highly dispersive optical solitons are obtainable only with inter-modal and group velocity dispersions for both nonlinear forms. [ABSTRACT FROM AUTHOR]

Optical soliton solutions are retrieved for Lakshmanan–Porsezian–Daniel equation by employing Riccati equation approach. The soiton solutions that are presented are generalized versions of solutions that have been reported in the past. The constraint conditions guarantee the existence of such solitons. [ABSTRACT FROM AUTHOR]

This paper retrieves soliton solutions to Kaup–Newell's equation in optical fibers. Two fundamental integration schemes, namely the csch-function method and traveling wave hypothesis, obtain singular and bright soliton solutions respectively. The existence criteria for these solitons are also presented. [ABSTRACT FROM AUTHOR]

This paper obtains bright, dark and bright–dark combo cubic–quartic optical solitons when group velocity dispersion is negligible or totally absent. The model is studied with fractional temporal evolution by F -expansion scheme with Kerr and parabolic laws of nonlinearity. The results can be used to achieve slow-light dynamics thus enabling the control of Internet traffic flow. [ABSTRACT FROM AUTHOR]

This paper retrieves bright, dark and singular solitons of both types for Bragg gratings with dispersive reflectivity. The method of undetermined coefficients is the integration algorithm adopted in this paper. Several constraints guarantee the existence of such solitons. [ABSTRACT FROM AUTHOR]

Abstract This paper employs modified extended direct algebraic method to recover bright, dark and singular solitons for resonant nonlinear Schrödinger's equation that is studied with dual-power law media. Singular periodic solutions also emerge as a byproduct of this integration scheme. [ABSTRACT FROM AUTHOR]

Abstract This paper conducts numerical studies of optical solitons with Lakshmanan–Porsezian–Daniel model having spatio-temporal dispersion. A newly proposed version of the well-known Adomian's decomposition method is applied to secure numerical simulations of the recently reported analytical soliton solutions. The focus of this paper stays with bright soliton solutions to the model. [ABSTRACT FROM AUTHOR]

Abstract This paper retrieves dark and singular optical solitons in birefringent fibers that is modeled by Kundu–Eckhaus equation. The most fundamental principle of undetermined coefficients is the extraction algorithm applied to the model. These solitons appear with integrability criteria that are also presented. [ABSTRACT FROM AUTHOR]

Abstract In this paper, to build soliton solutions of the governing equation, describing a propagation of sub-picosecond pulses with cubic-quintic nonlinearity and fourth-order dispersion in optical fibers, we use three methods: auxiliary equation, sine-cosine method and csch function method. The results obtained are 1-soliton solutions that propagate in the fiber optic and LHMs. [ABSTRACT FROM AUTHOR]

Abstract This paper retrieves optical solitons in birefringent fibers having non-local nonlinearity in presence of four-wave mixing. Phase-matching conditions have been implemented to display these solitons. Bright, dark and singular soliton solutions are revealed. [ABSTRACT FROM AUTHOR]

Abstract This paper studies solitons in optical couplers that are made up from metamaterials. Twin core as well as multiple-core couplers are considered. The study is conducted by the aid of exp(− Φ(ξ))-expansion scheme for four forms of nonlinearity and they are Kerr law, power law, parabolic law and dual-power law. Singular and combo-soliton solutions are recovered from this algorithm. [ABSTRACT FROM AUTHOR]

Abstract This paper obtains soliton solutions to perturbed Fokas-Lenells equation by the aid of exp-function approach. The perturbation terms are considered with full nonlinearity. Singular and combo-soliton solutions are presented along with their existence criteria. [ABSTRACT FROM AUTHOR]

Abstract The multiplier approach has been adopted to derive the conservation laws for optical solitons with non-local form of nonlinearity. The four conserved quantities are obtained from the 1-soliton solution that has already been reported earlier. The results are in terms of Gauss' hypergeometric functions whose convergence criteria are compatible with the soliton structure. [ABSTRACT FROM AUTHOR]

Abstract This paper employs two strategic integration algorithms to extract soliton solutions to Kundu-Eckhaus equation in the context of birefringent fibers. The two schemes are G ′/ G 2-expansion approach and sine-cosine method. Bright, singular and dark-singular combo solitons are retrieved. With reversed constraint conditions, singular-periodic solutions also emerge, although they are not applicable to the photonics field. [ABSTRACT FROM AUTHOR]

Abstract This paper implements two efficient integration algorithms to retrieve soliton solutions that appear with weak nonlocal nonlinearity. These are trial equation method and modified simple equation scheme. Bright, dark and singular soliton solutions are recovered and their existence criteria are also presented. [ABSTRACT FROM AUTHOR]

Abstract This paper retrieves mean free velocity of optical Gaussons that travel with stochastic perturbation in addition to bandpass filters and multi-photon absorption. The Langevin equation is derived and analyzed to arrive at the velocity in the limiting case. [ABSTRACT FROM AUTHOR]

Abstract This paper retrieves chirped singular and bright-singular combo optical soliton solutions to the Chen–Lee–Liu equation. Three forms of independent intregration schemes are implemented to the model. They are csch function scheme, tanh–coth method and finally the modified simple equation approach. The integrability criteria for these are also presented. [ABSTRACT FROM AUTHOR]

Abstract This work studies fractional temporal evolution of oblique resonant optical solitons in (3+1)-dimensions with Kerr- and parabolic-law nonlinearities. The generalized exp(− Φ (ξ))-expansion method along with the Khalil's conformable fractional derivatives is implemented to locate several forms of oblique resonant solitons. It is observed that obliqueness significantly modified resonant wave dynamics. The obtained results are very useful for understanding the dynamics of obliquely propagating resonant optical solitons and optical bullets. [ABSTRACT FROM AUTHOR]

Considering the self-steepening effect in a metamaterial (MM) can significantly change its behaviour. We study the propagation of ultrashort pulses in nonlinear MMs that is governed by a generalized nonlinear Schrö dinger equation with higher order effects such as pseudo-quintic nonlinearity and self-steepening effect. A class of chirped quasi-soliton solutions is obtained in the presence of the self-steepening term, and some of which are derived for the first time. The solutions comprise chirped bright quasi-solitons on a constant and zero background, kink and anti-kink quasi-solitons, and double-kink quasi-solitons. It is found that the nonlinear chirp associated with each of these waves is directly proportional to the intensity and its amplitude can be controlled by selecting the self-steepening and dispersion coefficients. Particular cases of chirp-free quasi-solitons are discussed. The conditions on MM parameters for the formation of these structures are also presented. The obtained results are important to explore much richer localized light pulses in MMs. [ABSTRACT FROM AUTHOR]

Abstract We investigate the Kaup-Newell equation that represents one of the forms of derivative nonlinear Schrödinger equation. The model applies to the description of sub-pico-second pulse propagation through an optical fiber. A special complex envelope traveling-wave method is applied to find a nonlinear equation with a fifth-degree nonlinear term describing the dynamics of field amplitude in the nonlinear media. It is shown that the phase associated to the obtained pulses has a nontrivial form and possesses two intensity dependent chirping terms in addition to the simplest linear contribution. A class of soliton solutions of the bright, dark and singular type are derived for the first time. The requirements concerning the optical material parameters for the existence of these chirped structures are also discussed. [ABSTRACT FROM AUTHOR]

Abstract We investigate the existence and stability properties of nonlinearly chirped solitary waves on a continuous-wave background in nonlinear metamaterials with higher-order effects such as pseudo-quintic nonlinearity and self-steepening effect. Novel classes of chirped gray solitary pulses (dark pulses with nonzero minimum intensity) are derived by employing the traveling-wave method. The conditions for the formation of these localized pulses as well as the nonlinear chirp associated to them are presented. The stability of the obtained structures is also demonstrated numerically under finite initial perturbations. The results show that the envelope pulses obtained here represent new types of extremely robust nonlinear chirped structures in highly nonlinear metamaterials. Highlights • The existence and stability properties of nonlinearly chirped solitary waves are studied. • The nonlinear metamaterials with higher-order effects are considered. • Novel classes of chirped gray solitary pulses are derived. [ABSTRACT FROM AUTHOR]