Kerr Law Nonlinearity Research Articles

Exact Solutions and Dynamic Properties of the Perturbed Nonlinear Schrödinger Equation with Conformable Fractional Derivatives Arising in Nanooptical Fibers

The main idea of this paper is to investigate the exact solutions and dynamic properties of a space-time fractional perturbed nonlinear Schrödinger equation involving Kerr law nonlinearity with conformable fractional derivatives. Firstly, by the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then a conserved quantity, namely, the Hamiltonian, is constructed, and the qualitative analysis of this system is conducted via this quantity by classifying the equilibrium points. Moreover, the existences of the soliton and periodic solution are established via the bifurcation method. Furthermore, all exact traveling wave solutions are constructed to illustrate our results explicitly by the complete discrimination system for the polynomial method.

Investigation of new solitons in nematic liquid crystals with Kerr and non-Kerr law nonlinearities

Nematicons became a key topic of interest in liquid crystal technology in recent years. This paper contributes in understanding the fantastic features of nematicons in optics and further disciplines. This piece of research investigates nematicons for obtaining various exact solutions for Kerr and non-Kerr law nonlinearities with the help of the Kudryashov’s approach and the tanh–coth technique. The acquired outcomes involve rational, periodic and hyperbolic solutions as well as their combo-type solutions for all the four cases of nonlinearity. A comparative study is conducted to show the novelty of present results with results already existing in the literature. The constraint conditions obtained ensured that the existence of these solutions is extraordinarily favorable to further investigate the dynamics of nematicons for various kinds of nonlinearity. The dynamics of the few of the obtained solutions are also discussed by 3D plots.

Comment on “A comparative study of the optical solitons for the fractional complex Ginzburg–Landau equation using different fractional differential operators, Optik 256 (2022) 168626”

In a recently published article, G. Akram et al proposed a new extended ϕ6-model expansion method to construct optical solitons for the fractional complex Ginzburg–Landau equation with Kerr law nonlinearity using different fractional differential operators. In this article, we point out that there is additional constraint condition in the proposed extended ϕ6-model expansion method. Neglecting the additional constraint condition will cause mistakes.

Dispersive optical solitons in birefringent fibers for (2+1)-dimensional NLSE with Kerr law nonlinearity and spatio-temporal dispersion having multiplicative white noise via Itô calculus

Object:The current work handles for the first time, dispersive optical solitons in birefringent fibers for the (2+1)-dimensional NLSE with Kerr law nonlinearity and spatio-temporal dispersion having multiplicative white noise in the Itô sense. Methods:Two integration technologies via the unified Riccati equation algorithm and the extended auxiliary equation approach are applied. Results:Combo soliton solutions, dark solitons, bright solitons, and singular soliton solutions are presented.

New Path of Popularized Homogeneous Balance Method and Travelling Wave Solutions of a Nonlinear Klein-Gordon Equation

The aim of this paper is to obtain a set of traveling wave solutions for klein –Gorden equation with kerr law non-linearity. More precisely, we apply a new path of popularized homogeneous balance (HB) method in terms of using linear auxiliary equations to find the results of non-linear klein-Gorden equation, which is a fundamental approach to determine competent solutions. The solutions are achieved as the integration of exponential, hyperbolic, trigonometric and rational functions. Besides, some of the solutions are demonstrated by the3D graphics.

Bifurcations and multiple optical solitons for the dual-mode nonlinear Schrödinger equation with Kerr law nonlinearity

The main attention of this work focuses on dynamical behavior and optical solitons for the dual-mode nonlinear Schrödinger equation with Kerr law nonlinearity. Here, we analytically constructed dark soliton solutions and singular soliton solutions through the bifurcation method of dynamical system. What is more, some other soliton solutions which include rational function solutions and many families of Jacobian elliptic solutions have been derived with the help of the complete discriminant method and symbolic computation. It is notable that we consider the more general case of α≠β of Eq. (1.1) compared with Jaradat et al. (2018). The obtained solutions may improve or complement previous results. These soliton solutions may motivate us to explore new complex physical phenomena in this system.

Perturbation of dispersive optical solitons with Schrödinger–Hirota equation with Kerr law and spatio-temporal dispersion

Objective:The principal purpose of this paper is to examine the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity which governs the propagation of dispersive pulses in optical fibers by proposing and using a direct algebraic form of the enhanced modified extended tanh expansion method for the first time. Our aim is not only restricted to obtaining different and more soliton solutions by proposed method for the first time in this study, but also includes examining the effect of the coefficients of self-steepening and nonlinear dispersion terms to the soliton propagation in the investigated problem. Methodology:Utilizing a traveling wave transformation, the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity can be transformed into an nonlinear ordinary differential equation (NODE). Then, the NODE is convert into a set of algebraic equations by taking account into the Riccati differential equation. Solving the set of algebraic equations, we acquire the analytical soliton solutions of the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity. In the proposed method, the modified extended tanh function method is enhanced by presenting more solutions of Riccati differential equations with the direct algebraic form, is utilized. Results:The more solutions have been established to the literature with new significant physical properties of the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity. We indicated that the presented method are effective, easily computable, and reliable in solving such nonlinear problems. Moreover, we demonstrate the dynamical behaviors and physical significance of some soliton solutions at appropriate values of parameters. Originality:A variety of soliton solutions to the perturbed Schrödinger–Hirota equation with Kerr law non-linearity by the direct algebraic form of enhanced modified extended tanh expansion method have been acquired. These solutions are dark–bright, trigonometric, hyperbolic, periodic, and singular soliton solutions. 3D, contour and 2D plots of some obtained solutions have been demonstrated to interpret the physical meaning of the equation. For some parameter values in the equation, the behavior of soliton solutions has been examined. The constraint conditions are established to confirm the existence of valid solutions. The obtained results can be effective in interpreting the physical meaning of this nonlinear system. We have seen that the proposed direct algebraic form of the enhanced modified extended tanh expansion method is a powerful mathematical technique which can be utilized to acquire the analytical solutions to different complex nonlinear mathematical models.

On the analytical optical soliton solutions of perturbed Radhakrishnan–Kundu–Lakshmanan model with Kerr law nonlinearity

On the analytical optical soliton solutions of perturbed Radhakrishnan–Kundu–Lakshmanan model with Kerr law nonlinearity

Optical solitons in polarization preserving fibers for perturbed resonant NLSE with Kerr law nonlinearity and Bohm potential having multiplicative white noise via Itô calculus

Object:We discuss for the first time, the perturbed resonant nonlinear Schrödinger equation (NLSE) with Kerr law nonlinearity and Bohm potential having a multiplicative white noise. Methods:Three integration schemes via modified simple equation method, sine–cosine method and Jacobi elliptic equation method are used. Results:We retrieve soliton solutions to the model, which are described using parametric constraints. Jacobi elliptic function solutions, bright, dark, singular soliton solutions are presented.

Abundant novel nematicon soliton wave solutions in liquid crystals with Kerr law nonlinearity

Nematicons in liquid crystals with Kerr law nonlinearity are studied using three recent analytical techniques. An essential goal of this study is to discover a variety of nonlinear system solution types, such as singular and solitary wave solutions in both bright and dark states. It also seeks to highlight its answers’ uniqueness by comparing and contrasting with previously published results achieved via other approaches. Two-dimensional, three-dimensional, contour, polar, and spherical graphics have been used to demonstrate specific solutions. Their performance shows the effectiveness of imposed methods in solving nonlinear evolution equations. Many different domains, including optics, fluid dynamics, gaseous plasma, and acoustics, rely on molecules’ characteristics in liquid crystals.

New wave approach to the conformable resonant nonlinear Schödinger’s equation with Kerr-law nonlinearity

New wave approach to the conformable resonant nonlinear Schödinger’s equation with Kerr-law nonlinearity

An investigation of fractional complex Ginzburg–Landau equation with Kerr law nonlinearity in the sense of conformable, beta and M-truncated derivatives

An investigation of fractional complex Ginzburg–Landau equation with Kerr law nonlinearity in the sense of conformable, beta and M-truncated derivatives

New optical behaviours of the time-fractional Radhakrishnan-Kundu-Lakshmanan model with Kerr law nonlinearity arise in optical fibers

New optical behaviours of the time-fractional Radhakrishnan-Kundu-Lakshmanan model with Kerr law nonlinearity arise in optical fibers

Pure-cubic optical soliton perturbation with full nonlinearity by a new generalized approach

This work introduces a new generalized integration scheme to construct the pure-cubic optical solitons in a polarization-preserving fiber with Kerr law nonlinearity. First, Lie symmetry analysis is performed to find the similarity transformation. Then, the exact solutions of the governing equation are derived via a purposed new generalized integration scheme. In particular, we have derived the pure-cubic dark, singular and combo dark-singular soliton solutions. The existence criteria of such solutions are also provided.

RETRACTED: Abundant solitary wave solutions to a perturbed Schrödinger equation with Kerr law nonlinearity via a novel approach

The main purpose of the present paper is to introduce a reliable method, for the first time, in solving differential equations with partial derivatives. The significant idea behind this method is the modification of a well-known method. The main base functions used in the solution’s structures are Jacobi elliptic functions which are known functions and have many applications in practice. In order to achieve these results, the perturbed Schrödinger equation with Kerr law nonlinearity is considered. This nonlinear equation illustrates the propagation of optical solitons in nonlinear optical fibers. Thereupon, several analytical solutions corresponding to this model are obtained through employing an improved generalized exponential rational function method. Then we present a new version of it that is presented for the first time in the literature. Using these methods, several wave solutions related to the considered model are obtained. Moreover, several numerical simulations relevant to the resulting solutions are performed in the paper. As a notable features of the proposed method, the obtained solutions are characterized by the Jacobi elliptic functions. These solutions contain an index that for some specific choices is reduced to standard functions such as trigonometric and hyperbolic. The obtained results and solutions can be used for investigating the mechanism of several nonlinear phenomena laboratory and space plasmas. All necessary calculations are performed via symbolic packages of Wolfram Mathematica.

Highly dispersive optical solitons in birefringent fibers having Kerr law of refractive index by Laplace–Adomian decomposition

This paper revisits highly dispersive optical solitons in birefringent fibers, with Kerr law nonlinearity, from a numerical perspective. The Laplace–Adomian decomposition scheme reveals the spectral display of bright and dark optical solitons that are with differential group delay. The error measure is also displayed that is stunningly low.

Dynamics of new optical solutions of fractional perturbed Schrödinger equation with Kerr law nonlinearity using a mathematical method

Dynamics of new optical solutions of fractional perturbed Schrödinger equation with Kerr law nonlinearity using a mathematical method

Optical solitons for conformable space-time fractional nonlinear model

In search of the exact solutions of nonlinear partial differential equations in solitons form has become most popular to understand the internal features of physical phenomena. In this paper, we discovered various type of solitons solutions for the conformable space-time nonlinear Schrodinger equation (CSTNLSE) with Kerr law nonlinearity. To seek such solutions, we applied two proposed methods which are Sardar-subequation method and new extended hyperbolic function method. In this way several types of solitons obtained for example bright, dark, periodic singular, combined dark-bright, singular, and combined singular solitons. Some of the acquired solutions are interpreted graphically. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. It is revealed that the proposed methods offer a straightforward and mathematical tool for solving nonlinear conformable space-time nonlinear Schrodinger equation. These results support in attaining nonlinear optical fibers in the future.

A comparative study of the optical solitons for the fractional complex Ginzburg–Landau equation using different fractional differential operators

This paper presents a study of optical solitons for the fractional complex Ginzburg–Landau equation (CGLE) with Kerr law nonlinearity using different fractional differential operators. The soliton solutions of the equation have been investigated with conformable, β and M-truncated derivatives. A Jacobi elliptic function finder technique, namely, new extended ϕ6-model expansion method has been utilized for the construction of the solutions. A variety of soliton solutions to the complex Ginzburg–Landau equation have been obtained. The comparison of some of the retrieved solutions has also been graphically illustrated using the three fractional differential operators.

Pseudo-spectral approach for extracting optical solitons of the complex Ginzburg Landau equation with six nonlinearity forms

In this paper, a compelling pseudo-spectral approach namely split-step Fourier transform (SSFT) is utilized for the first time to report the complex Ginzburg Landau (CGL) equation. In this regard, six nonlinear forms of this notable equation are associated with this model, which are the kerr law nonlinearity, power law nonlinearity, quadratic-cubic nonlinearity, cubic-quintic nonlinearity, dual power nonlinearity, and polynomial law nonlinearity. Since opting for an analytical solution is complicated and only provided for a limited set of soliton solutions, the numerical solution might render a more generalized aspect in addressing versatile solutions for a wide range of arbitrary input pulses. Therefore, the SSFT scheme is efficiently employed to extract a plethora of optical solitons solutions such as kink waves, bright, dark, bright-dark, singular, singular periodic solitons. The obtained results, via MATLAB, are in total agreement with the findings explored form the exact analytical solutions. As a result, this numerical technique may provide a creditable mathematical tool to solve a wide range of other nonlinear evolution partial differential equations in the mathematical physics field.