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Most recently, Baber et al. (Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability, Mod. Phys. Lett. B (2024) 2450341, doi:10.1142/S021798492450341X2450341-1) claimed to have obtained exact solutions of a two-dimensional nonlinear Schrödinger equation with an instantaneous Kerr nonlinearity that governs beam movement within a multimode optical fiber featuring a parabolic index shape, using the Jacobi Elliptic Function Expansion. Additionally, they have analyzed the modulation instability for the underlying model. It is the aim of this comment to point out that all results found in that work and published in the journal Mod. Phys. Lett. B (doi:10.1142/S021798492450341X2450341-1) are all erroneous, from the exact solutions of the considered equation to the modulation instability of the underlying model.

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations.

In this paper, we study the Schrödinger–Hirota equation in birefringent fibers incorporating cubic–quartic dispersion. The model studied comes with a cubic–quintic nonlinear structure. The governing model introduced in this study is novel and original, and the obtained solitons have not been reported before. The Schrödinger–Hirota equation is an important modified structure within the higher-order NLSE that garnered significant attention in the fields of mathematics and physics, especially in the context of the transmission of solitons in water waves and optical fibers. To investigate optical solitons for this model, three widely applied in various research studies are implemented for our governing model. These methods are the Unified Riccati expansion method, the addendum to Kudryashov’s method, and the addendum to the modified Sub-ODE method. A variety of solutions have emerged with this governing model, including bright, dark, singular solitons and the hybrid of these solitons. Also, Jacobi and Weierstrass doubly periodic-type solutions are recovered, which, with selected parameters, these solutions give solitons. Furthermore, for more illustration of the obtained solutions, a detailed discussion with a graphical presentation is presented.

This research utilizes the new auxiliary equation method to investigate solitary wave patterns in the integrable Zhanbota-IIA equation, a model relevant to phenomena like tidal waves and tsunamis. This method simplifies complex nonlinear-coupled partial differential equations into ordinary differential equation through a traveling wave transformation. Previous studies have not achieved such solutions. The main aim is to enhance comprehension of the integrable Zhanbota-IIA equation behavior under various conditions. Prior to this study, there does not exist any study in which such kinds of solutions are discussed by using the new auxiliary equation method. The new auxiliary equation method is utilized to derive analytical solutions for the underlying model equation. These solutions are expressed in the form of hyperbolic, trigonometric, exponential, and rational functions such as dark, bright, periodic, dark–bright, smooth topological with high peaks, dark-singular, bright singular and periodic singular. The key advantage of this method compared to others lies in its capacity to yield more comprehensive solutions with certain flexible parameters. To illustrate the dynamic structures of these solutions, 3D and 2D graphs are employed with suitable parameter values. This study explores soliton behavior and its telecommunication applications, integrating mathematics, computer science, and physics for technological advancements.

The $(1+1)$-dimensional fractional coupled nonlinear Schrödinger equations (FCNLSEs) are investigated in this work. This model is important as it involves applications in allowing soliton wavelength division multiplexing and pulse propagation in two-mode optical fibers. This means it is important for the study of the dynamics of solitons in dispersive, inhomogeneous, nonlinear media, and, therefore, it is a tool of prime value for the development of advanced optical communication technologies. Beta space-time fractional derivative, which is important in dielectric polarization and electromagnetic systems, is used to evaluate the FCNLSE. We capitalize a modified version of the Sardar sub-equation method for the purpose of getting different soliton solutions. Solutions take the form of optical solitons, including dark, bright, periodic, peakons, compactons, and kink soliton solutions, formulated using exponential, trigonometric, and hyperbolic functions. To make our understanding of the physical meaning of these solutions richer, we apply advanced plotting techniques, examples of which are the three-dimensional (3D), two-dimensional (2D), contour, and density plots. Such plots provide a colorful overview of the dynamic behavior of the solutions obtained. The solutions do not only pinpoint the fact that the result is an effective and accurate method application, but they also set a high standard in further probing meaning in the numerous physical phenomena. These new solutions go beyond earlier studies in the literature by incorporating beta derivatives as a modeling tool for FCNLSE. The methodology applied here functions seamlessly and can be extended to address numerous advanced models in contemporary areas of science and engineering. The anticipated usefulness of the derived solutions in various scientific domains, particularly optics, is expected to make a valuable contribution to the fiber communication system in future research endeavors.

This study introduces an innovative examination of the stochastic Kaup–Newell equation within the realm of fiber Bragg grating, signifying a pioneering effort in this area. The study uses two main methods for integration: the $(\frac{G^{\prime }}{G})$-expansion method and the enhanced direct algebraic method to find different one-step solutions to the Kaup–Newell equation. The outcomes of this research are multifaceted, revealing an array of optical soliton solutions such as dark, singular, bell-shaped, kink-shaped, bright, and straddled solitons. These findings contribute a novel perspective on the behavior of optical solitons in fiber Bragg gratings, which are essential for optical communications and signal processing. Additionally, applying numerical schemes to these analytically obtained solutions offers a detailed visualization of their characteristics and behaviors, enhancing our understanding of their potential in practical scenarios. This work extends the range of soliton solution types for application and paves the way for subsequent explorations in the field of nonlinear optical systems.

This study delves into the perturbed stochastic nonlinear Schrödinger equation in an optical coupler for metamaterials, introducing a parabolic nonlocal law for refractive index nonlinearity, eighth-order dispersion, and multiplicative white noise. An enhanced version of Kudryashov’s method, combined with the direct algebraic method, is utilized to examine stochastic effects in nonlinear optical systems. The findings reveal diverse solution types such as dark and singular solitons, Weierstrass and Jacobi doubly periodic solutions, and straddled solitons, showcasing the complex dynamics driven by nonlocal nonlinearity and higher-order dispersion under stochastic conditions. This pioneering work on the nonlinear Schrödinger equation with a parabolic nonlocal nonlinearity and eighth-order dispersion investigates the impact of random fluctuations on nonlinear dynamics, advancing our understanding of light propagation at higher frequencies. The study aims to inspire further research into the behavior of light in unique materials and the development of more efficient light-based devices, leveraging the insights gained to innovate advanced communication tools and other optical devices, highlighting the response of these materials to random disturbances.

The symplectic integrator of the Gauss–Legendre type is tested on the nonlinear Schrödinger equation. Preservation of high integrals (up to 10 or more) and quasiperiodic motion have been detected for dynamics on both stable soliton and homoclinic manifolds, which indicate applicability of symplectic integrators for adequate simulation of integrable equation. The tested integrator is applied to the problem of long-time stability of the solitons in higher-derivative nonlinear Schrödinger equation. The slow logarithmic-type depletion of the soliton amplitude with time has been detected.

For describing wave propagation in an inhomogeneous erbium-doped nonlinear fiber with higher-order dispersion and self-steepening, an inhomogeneous coupled Hirota–Maxwell–Bloch system is considered with the aid of symbolic computation. Through Painlevé singularity structure analysis, the integrable condition of such a system is analyzed. Via the Painlevé-integrable condition, the Lax pair is explicitly constructed based on the Ablowitz–Kaup–Newell–Segur scheme. Furthermore, the analytic soliton-like solutions are obtained via the Darboux transformation which makes it exercisable to generate the multi-soliton solutions in a recursive manner. Through the graphical analysis of some obtained analytic one- and two-soliton-like solutions, our concerns are mainly on the envelope soliton excitation, the propagation features of optical solitons and their interaction behaviors in actual fiber communication. Finally, the conservation laws for the system are also presented.

This paper investigates the ($2+1$)-dimensional Heisenberg ferromagnetic spin chain (HMF) model. The model describes the nonlinear spin dynamics of HMF. By adopting the modified $F$-Expansion and projective Riccati equation methods, we report the dark, combined dark-bright and envelope optical solitons, complexitons singular solutions of the equation along with the conditions that must be satisfied for solitons to exist. The physical structure of the obtained solutions are shown by graphic illustration in order to give a better understanding on the dynamics of optical solitons.

In this paper, we have presented the analytical analysis of integrable perturbed nonlinear Schrödinger (PNLSE) equation with third-order dispersion (TOD) in Kerr media using our newly proposed technique, extended modified auxiliary equation mapping method. By the implementation of this technique, we have obtained a variety of some new families and more general form of exact traveling wave solutions including triangular-type solutions, periodic and doubly periodic-like solutions, combined soliton-like solutions, kink and anti-kink type soliton-like solutions using three parameters which is the key difference of our newly proposed method. PNLSE is a well-known governing model to study the propagation of optical solitons in nonlinear optical fibers and other telecommunication networks with a type of Kerr law nonlinearity. This particular type of nonlinearity originates when a light wave in an optical fiber is subjected to nonlinear responses. For graphical representation of our newly found results, we have presented them with detailed dynamical physical representation using Mathematica 10.4 to explain in a more efficient manner the behavior of different physical structures of solutions. Also, the computational work and efficiency of the method demonstrate the reliability, straightforwardness, and simplicity of the technique for solving other nonlinear complicated partial differential equations.

In this work, a new method has been applied for finding solutions to the Kundu–Mukherjee–Naskar (KMN) equation in (2+1) dimensions. These optical solutions were derived with the aid of a new method named the Exp-Function method. The method is used for solving the nonlinear governing equation, which contains high nonlinear term. This method is considered as an alternative technique for obtaining both analytical and approximate solutions for a different type of PDE with several applications in science and engineering. The efficiency of the proposed technique has been shown by a few test problems. By comparing the results obtained in this paper with other researchers it can be noticed that this method is a promising technique for finding solutions to other similar problems.

In this research article, our aim is to construct new optical soliton solutions for nonlinear complex Ginzburg–Landau equation with the help of modified mathematical technique. In this work, we studied both laws of nonlinearity (Kerr and power laws). The obtained solutions represent dark and bright solitons, singular and combined bright-dark solitons, traveling wave, and periodic solitary wave. The determined solutions provide help in the development of optical fibers, soliton dynamics, and nonlinear optics. The constructed solitonic solutions prove that the applicable technique is more reliable, efficient, fruitful and powerful to investigate higher order complex nonlinear partial differential equations (PDEs) involved in mathematical physics, quantum plasma, geophysics, mechanics, fiber optics, field of engineering, and many other kinds of applied sciences.

In today’s life, time is money, so everyone wants rapid communications in one’s life. The optical fibers are the most prominent sources for rapid communications that are described by the nonlinear Schrödinger equations. So, it is very important to investigate the optical solitons solutions. In this paper, the solitons of optical metamaterials along anticubic law of nonlinearity are retrieved. To obtain the required solutions of the aforesaid model, the modified extended tanh expansion scheme is applied successfully. As a result, a series of different kinds of solitons solutions, such as dark and singular solitons and periodic solutions, are secured and verified by the aid of MATHEMATICA. The numerical simulations of certain solutions are made to analyze the influence of the parameters involved.

In this paper, we will investigate a famous model of nonlinear sciences namely $(2+1)$-dimensional nonlinear spin dynamics of Heisenberg ferromagnetic spin chains (HFSC) model for the evaluation of optical travelling waves by implementing unified method (UM). We will extract dark as well as bright solitary waves along with elliptic waves. We will draw conserved quantities of our governing model by utilizing dilation symmetry. At the end, the integrability of nonlinear spin dynamics of HFSC model with the help of Painlevé test will also be studied in this paper.

In order to obtain some novel analytical solutions for fractional improved nonlinear Schrodinger equation with perturbation, the polynomial expansion, simplest equation and extended sine-Gordon expansion schemes are utilized. A collection of different types of solitons are reported with a physical and important perspective, containing optical dark solitons. The methods reveal the solutions in a structure of rapidly converging series. This research highlights additional important features of the methods being considered. In addition, the modulation instability analysis is carried out to discuss the stability analysis of the solutions reached, and the movement role of the waves is examined, which affirms that all the established solutions are exact and stable.

In this work, we demonstrate the extraction of some optical soliton solutions of the Chen–Lee–Liu equation (CLLE) with applications in optical fibers. A novel method is presented which is called the new extended direct algebraic method (EDAM). This method is based on converting the nonlinear CLLE equation into an ordinary type equation through a wave transformation. We acquire new solutions like dark, bright, combined dark-bright, combined bright-singular and periodic singular soliton solutions by using this effective technique. These acquired soliton solutions are new with some new physical properties. Some graphical representations for these solutions are provided which prove that the new method is effective and can be extended to some similar problems.

In this research paper, a simple integration scheme is executed to secure new dark and singular soliton solutions for the highly dispersive nonlinear Schrödinger’s equation having Kudryashov’s arbitrary form with generalized nonlocal laws and sextic-power law refractive index.

In this paper, the newly derived solutions for the optical soliton of Kerr law nonlinearity form of Biswas–Arshed model are investigated. The exact solutions are extracted by deploying two different novel methods namely, $\left(m+\left(\frac{G^{\prime }}{G}\right)\right)$-expansion method and Riccati–Bernoulli sub-ODE method. Furthermore, in different conditions, the resultants show different wave solutions like singular, kink, anti-kink, periodic, rational, exponential and dark soliton solutions. Also, the dynamics of the attained solutions are presented graphically.

The main consideration of this paper is to discuss cubic optical solitons in a polarization-preserving fiber modeled by nonlinear Schrödinger equation (NLSE). We extract the solutions in the forms of hyperbolic, trigonometric including a class of solitary wave solutions like dark, bright–dark, singular, singular periodic, multiple-optical soliton and mixed complex soliton solutions. A recently developed integration tool known as new extended direct algebraic method (NEDAM) is applied to analyze the governing model. Moreover, the studied equation is discussed with two types of nonlinearity. The constraint conditions are explicitly presented for the resulting solutions. The accomplished results show that the applied computational system is direct, productive, reliable and can be carried out in more complicated phenomena.