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In this study, the generalized nonlinear Schrödinger equation (GNLSE) is examined. This model takes into account the group velocity dispersion, dual-power law nonlinearity and self-frequency shift. This model is used to mimic the wave propagation over nano optical fibers. Studying is conducted by applying the modified extended direct algebraic method (MEDAM). Based on this method, various types of solutions, such as bright solitons, dark solitons, singular solitons, singular periodic solutions, exponential solutions and Weierstrass elliptic solutions are extracted. The extracted solutions confirmed the efficacy and strength of the current technique. Moreover, the graphs for some solutions are presented to demonstrate their physical nature. [ABSTRACT FROM AUTHOR]

Many mathematical models describe the optical solitons propagation through optical fibers such as nonlinear Schrödinger's equation. In this work, we study the generalized derivative resonant nonlinear Schrödinger equation that encompasses cubic–quintic nonlinearity. Studying is conducted by implementing the improved modified extended tanh-function method which denotes various types of solutions. These solutions including bright solitons, dark solitons, triangular solutions, hyperbolic solutions, Jacobi elliptic functions solutions, exponential solutions and Weierstrass elliptic doubly periodic solutions. The extracted solutions confirmed the efficacy and strength of the current technique. To illustrate the physical characteristics of the established solutions, some selected solutions are visually depicted. [ABSTRACT FROM AUTHOR]

In this work, the perturbed nonlinear Schrödinger's equation (NLSE) with self-phase modulation (SPM) of Kudryashov's sextic power law which describes the soliton's propagation in optical fiber is investigated. New exact solutions will be introduced including doubly periodic solutions in the form of Weirstrass and hyperbolic functions by using the simple equation method. There have been several different kinds of optical soliton solutions proposed, including dark, bright, kink, and anti-kink soliton solutions. To further understand some of its physical characteristics, the resulting optical soliton solutions are illustrated graphically. [ABSTRACT FROM AUTHOR]

The behavior of optical soliton solutions in optical fiber theory is described by the Schrödinger–Hirota equation, a nonlinear partial differential equation with multiple applications in science and engineering. In this work, a highly effective approach for solving the Schrödinger–Hirota equation is utilized based on the modified generalized Riccati equation mapping approach. The remarkable achievement of obtaining over one hundred plus newly-formed distinct analytical soliton solutions for the governing Schrödinger–Hirota equation is accomplished through the employment of this technique. The technique employed is characterized by its power, simplicity, and adaptiveness, enabling the extraction of various forms of soliton solutions into a unified framework. As a consequence of the applications, traveling waves, bell-shaped solitons, kink waves, and combo-multisoliton wave profiles are generated. To provide valuable insights into the behavior and characteristics of the solutions, two-dimensional, three-dimensional, and corresponding contour graphs are plotted for selected solutions. These findings contribute to the improvement of theoretical knowledge regarding the Schrödinger–Hirota equation and lay the groundwork for practical implementations. The abundant soliton solutions discovered through this method have the potential to find utility in a variety of fields, including electronics, quantum physics, and optical fibers. [ABSTRACT FROM AUTHOR]

In this study, we will extract new impressive visions for the optical soliton solution and other novel set of the rational soliton solutions to the perturbed Biswas-Milovic equation with Kudryashov's law of refractive index (PBMEWKL) that describes pulses propagation of different categories in optical fiber. Three various schemas are enforcement to extract these new visions. The three proposed techniques that are employed for these purposes are the solitary wave ansatz method (SWAM) the Paul-Painlevé approach method (PPAM) and the Riccati-Bernoulli Sub ODE method (RBSODM). Many new types of soliton solutions have been configured like bright soliton dark soliton combination between bright and dark soliton solutions, W-like shape soliton M-like shape soliton and other rational soliton solutions. The solutions imbedded in this article will play a principal role to understand the exact dynamics of soliton arising in nonlinear optics. The effective of all used algorithms that are examined previously for many other models, usually achieves very good results. Several emblematic 2D and 3D graphs are drawn under specific parameters, which can identify the morphology of optical waves more intuitively obvious. Moreover, we will establish a comparison not only for our achieved results via these three various paths but also with all who studied this model via other techniques. [ABSTRACT FROM AUTHOR]

This research is concerned with some modernistic optical solitons of beta-fractional density-dependent diffusion-reaction model. The collected optical solutions can be executed in exposing this model in a prominent form. The obtained results include the trigonometric, hyperbolic trigonometric, rational and exponential functions. Verification of achieved results is also done by using Mathematica tools. Three distinct techniques named as exp a function, extended sinh-Gordon equation expansion (EShGEE) and modified extended tanh function expansion techniques are employed to protect the results. The achieved results are also illustrated by 2-D, 3-D and contour plots. The gained results can also be fruitful for the development of model in future. [ABSTRACT FROM AUTHOR]

Due to the evolution of data centric applications demand for high speed data transfer and more bandwidth is increasing continuously. The unavailability of components like tunable wavelength converters (TWCs) restrict the transfer of parallel information using wavelength division multiplexing (WDM), therefore in the present scenario optical orthogonal frequency division multiplexing can be used. Moreover in optical communication narrow Gaussian pulses are transmitted, which spread with distance and leads to the broadening of the pulse and pulse peak power goes down and thus limits the system. In this paper a Soliton based optical communication system is proposed and its comparison with Gaussian pulse is presented and it has been found that soliton pulse has lesser bit error rate in comparison to Gaussian pulses. [ABSTRACT FROM AUTHOR]

We consider the coupled propagation of an optical field and its second harmonic in a quadratic nonlinear medium governed by a coupled system of Schrodinger equations. We prove the existence of ring‐profiled optical vortex solitons appearing as solutions to a constrained minimization problem and as solutions to a min–max problem. In the case of the constrained minimization problem, solutions are shown to be positive with undetermined wave propagation constants, but in the min–max approach, the wave propagation constants can be prescribed. The quadratic nonlinearity introduces some interesting properties not commonly observed in other coupled systems in the context of nonlinear optics, such as the system not accepting any semi‐trivial solutions, meaning that optical solitons cannot be observed when, say, one of the beams is off. Additionally, the second harmonic always remains positive. [ABSTRACT FROM AUTHOR]

This article explores the generalized Gerdjikov–Ivanov equation describing the propagation of pulses in optical fiber. The equation studied has a variety of applications, for instance, in photonic crystal fibers. In contrast to the classical Gerdjikov–Ivanov equation, the solution of the Cauchy problem for the studied equation cannot be found by the inverse scattering problem method. In this regard, analytical solutions for the generalized Gerdjikov–Ivanov equation are found using traveling-wave variables. Phase portraits of an ordinary differential equation corresponding to the partial differential equation under consideration are constructed. Three conservation laws for the generalized equation corresponding to power conservation, moment and energy are found by the method of direct transformations. Conservative densities corresponding to optical solitons of the generalized Gerdjikov–Ivanov equation are provided. The conservative quantities obtained have not been presented before in the literature, to the best of our knowledge. [ABSTRACT FROM AUTHOR]

Solitons have been a subject of intense intellectual curiosity and an avenue for a plethora of applications since they are first observed in water waves. Optical solitons, in particular, have seen tremendous progress, starting from their first theoretical predictions in optical fiber in 1973. On the 50th anniversary of this seminal work from Haseagawa and Tappert, a review is provided on the progress in an important class of optical solitons, the Bragg soliton. Bragg solitons are optical solitons which exist in structures periodic in one dimension. The history of Bragg solitons is described, including the most salient advancements made both in theory and experiments. Recent progress made in integrated photonic devices and their associated design advantages are highlighted, with particular emphasis on how these have facilitated the realization of Bragg soliton dynamics on a chip and their optical processing functions. A discussion and future outlook for this burgeoning field provides a perspective on how Bragg solitons may impact the field of photonics and future applications. [ABSTRACT FROM AUTHOR]

The Kundu-Mukherjee-Naskar equation holds significant relevance as a nonlinear model for investigating intricate wave phenomena in fluid and optical systems. This study uncovers new optical soliton solutions for the KMN equation by employing analytical techniques that utilize combined elliptic Jacobian functions. The solutions exhibit mixtures of distinct Jacobian elliptic functions, offering novel insights not explored in prior KMN equation research. Visual representations in the form of 2D ContourPlots elucidate the physical behaviors and properties of these newly discovered solution forms. The utilization of symbolic computations facilitated the analytical derivation of these solutions, offering a deeper understanding of the nonlinear wave dynamics governed by the KMN equation. These employed techniques showcase the potential for future analytical advancements in unraveling the complex soliton landscape of the multifaceted KMN model. The findings provide valuable insights into the intricacies of soliton behavior within this nonlinear system, offering new perspectives for analysis and exploration in areas such as fiber optic communications, ocean waves, and fluid mechanics. Maple symbolic packages have enabled us to derive analytical results. [ABSTRACT FROM AUTHOR]

In this work, we realised the soliton solutions of nonlinear Schrödinger equation (NLSE) that arise from optical fibers, we considered the modified Sardar sub-equation method (MSSEM) to find solitary solutions analytically. The stability of the retrieved soliton solutions realised from the NLSE are investigated. We demonstrate the soliton solutions that are stable and can last for a very long time without losing its form or energy under specific circumstances and those soliton solutions that are unstable. The MSSEM is a frequently employed technique in research for addressing specific mathematical modeling or physical phenomena problems. Its selection in this specific study might stem from its proven efficacy in handling the particular problem under investigation. The decision to utilize MSSEM could be driven by several considerations, including its precision, computationally efficient, effectiveness, greater accuracy and capability to manage intricate systems. Finally, our method offers greater flexibility in modeling various physical phenomena, which makes it particularly useful in applications in diverse fields such as quantum mechanics and nonlinear optics. The findings have ramifications for the architecture of optical fiber communications and offer significant new insights into the behavior of solitons in optical systems. The NLSE has proven to be an effective tool for understanding wave behavior in fiber optics. Its applications have helped engineers and scientists optimize the design of optical fibers and predict the behavior of various conditions. Moreover, our study provides insights into the fundamental properties of solitary solutions in the NLSEs and their practical implications in physical systems. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE), one of the most fundamental physical models for understanding the fluctuations of optical soliton development, plays a significant role to demonstrate the dynamics of optical fibers. Therefore, the wave propagation in nonlinear dispersive medium is a subject of considerable interest due to the wide range of possibilities for ultrafast data processing and light pulses in communications. In this article, the coupled space-time fractional NLSE is investigated which is used to describe the non-relativistic quantum mechanical behaviour. A collection of comprehensive soliton structures are developed to study the dynamics of the governing model with the aid of some efficient analytical strategies. These solutions incorporate dark soliton and trigonometric function solutions, singular solition, dark singular solition plane wave, singular solition, opposite singular solition, smooth, bell shaped, periodic, bright, anti kink, singular bell and traveling wave with darkness. The presence of some attained solutions are flourish in 3D graphs with various fractional orders by using Mathematica. The results which we obtained reveal that the suggested approaches are more convenient and productive techniques to depict the dynamics of numerous complex wave structure in contemporary areas of science and technology. [ABSTRACT FROM AUTHOR]

The temporal behavior and formation characteristics have been investigated for the first time in nonlinear optical media exhibiting photorefractive effect due to simultaneous linear and quadratic electro-optic effect. The space charge field which is a function of time is derived and hence leads to the time dependent wave equation satisfied by light beams in such novel photorefractive materials. The soliton width evolution and hence the evolution of the nonlinearity with time is investigated for different values of peak intensity ratio. The dark soliton formation does not show quasi steady state behaviour and only steady state solitons are formed. There is interplay between the linear and quadratic electro-optic effect that affects the formation of dark solitons. Finally, the effect of crystal temperature on the formation of dark solitons is studied. In a photorefractive crystal, the temperature modification results in consequent changes in dark irradiance and dielectric constant which has a profound effect on the time evolution of these dark solitons. A relevant example of 0.67PMN-0.33PT crystal has been considered for an illustration of our results. [ABSTRACT FROM AUTHOR]

In this paper, the perturbed resonant nonlinear Schrödinger equation with dual-power law nonlinearity describing the pulse phenomena in nonlinear optics is investigated. By utilizing the complete discrimination system for polynomial method and the trial equation method, a variety of exact solutions of this equation have been acquired, including rational solutions, triangular function periodic solutions, solitary wave solutions, elliptic function double periodic solutions, as well as the corresponding nonlinear chirps. Besides, several representative two-dimensional graphs are plotted at specific parameters, which can supply a more comprehensive physical interpretation of the complex nonlinear model based on the intuitive identification of the morphology of optical waves. [ABSTRACT FROM AUTHOR]

This paper seeks to introduce the wave structures and dynamics properties of the perturbed Chen–Lee–Liu equation (PCLLE). Using the traveling wave transformation, we derive the corresponding traveling wave system from the original equation and construct a conserved quantity named as Hamiltonian. Subsequently, we establish periodic solutions and the existence of soliton using the bifurcation method. The bifurcation method is a mathematical technique used to study how the qualitative behavior of a system changes as one or more parameters of the system are varied. It involves analyzing the system's equilibrium points and studying how they behave as the parameters are changed. Finally, we construct the exact traveling wave solutions using the complete discriminant system (CDS) of polynomial method (CDSPM) to explicitly validate our findings. [ABSTRACT FROM AUTHOR]

We demonstrate interesting dynamics of soliton pairs and clusters in the (2 + 1)-dimensional nonlinear Schrödinger equation with self-focusing Kerr nonlinearity and linear potentials. We display regular oscillation and rotation of such multidimensional solitary structures in our model, achieved by the proper choice of simple potentials. By utilizing linear stability analysis, we establish that these soliton pairs and clusters are completely robust, and the solitons do not experience any distortion during the perturbed propagation, even when the potentials evolve along the propagation direction. Note also that both the oscillating and rotating periods of soliton pairs and clusters can be easily controlled by the parameters of external potentials. [ABSTRACT FROM AUTHOR]

The nonlinear Fokas–Lenells system is a popular model which can be wildly applied in the fields of heat pulses in solids, signal processing, circulation system of chemical industry, nonlinear optics, fluid mechanics and so on. The main purpose of this paper focuses on dynamical behavior and optical solitons in optical fibers with the nonlinear Fokas–Lenells equation. Here, we analytically constructed dark solitons, periodic wave solutions and bright solitons for the Fokas–Lenells model through the bifurcation theory of planar dynamical systems. Moreover, some other traveling wave solutions are also derived by means of the complete discriminant system technique and symbolic computation. Finally, numerical results are also given to illustrate the physical properties of the obtained optical soliton solutions. [ABSTRACT FROM AUTHOR]

Cavity Solitons are short optical pulses that propagate indefinitely in driven optical resonators. They are sustained by a double balance which prevents them from spreading or declining. This complex mechanism makes them "dissipative structures", a general term coined by Ilya Prigogine to describe patterns emerging from noise in nonlinear systems. In the past decade, cavity solitons have attracted a lot of attention both for their fundamental interest and many potential applications. This article aims at introducing the reader to these fascinating objects. [ABSTRACT FROM AUTHOR]

Fifty years ago, the concept of solitons in optical fibres was proposed numerically by F. Tappert and A. Hasegawa from Bell Labs. Seven years later in 1980, experiments confirmed that dispersion and the Kerr nonlinearity could combine to yield a coherent ultrashort pulse able to propagate without distortion. The implications of this discovery go well beyond the sector of telecommunications as initially suggested, and solitons now play a major role in modern ultrafast nonlinear photonics. [ABSTRACT FROM AUTHOR]

A passively mode-locked Er-doped fiber laser employing ethylene glycol as a saturable absorber (SA) has been proposed and experimentally validated. The ethylene glycol-SA is fabricated by filling the space between the end facets of two optical patch cords inserted into a cannula with ethylene glycol. Utilizing such an SA with a modulation depth of 9.6 %, stable mode-locking operation are manifested in the laser cavity, including the production of bright pulse and bright-dark pulse pair. The generated bright pulse has a center wavelength at 1562.28 nm, a 3 dB bandwidth of 0.21 nm, a repetition frequency of 20.24 MHz, and a signal-to-noise ratio (SNR) of 57 dB. Through precisely manipulating the polarization state and the pump power in the laser cavity, the bright-dark pulse pairs can be successfully attained at the same repetition frequency. This is the first demonstration of a passively mode-locked fiber laser utilizing ethylene glycol as the saturable absorber. Due to their superior optical absorption characteristics, low cost, simplicity in fabrication and non-volatile nature, ethylene glycol-SAs have enormous potential in the field of nonlinear optics and ultrafast photonics.

The primary focus of this study was on exploring the optical soliton solutions and chaotic patterns in magneto-optic waveguides through the coupled stochastic Schrödinger–Hirota equation with multiplicative white noise. Firstly, by means of traveling wave transformations and homogeneous balance principle, the coupled stochastic Schrödinger–Hirota equation in magneto-optic waveguides is transformed into ordinary differential equation. By selecting some suitable parameters, phase diagrams are plotted with the help of the mathematical software Maple. Secondly, the optical soliton solutions of the coupled stochastic Schrödinger–Hirota equation corresponding to phase orbits can be easily deduced through the method of dynamical systems. In addition, chaotic behavior for the coupled stochastic Schrödinger–Hirota system with perturbation term has been discussed in detail. Finally, the two-dimensional and three-dimensional graphs of the stochastic Schrödinger–Hirota equation are drawn, which further explain the propagation of the coupled stochastic Schrödinger–Hirota equation in nonlinear optics.

The exploration of higher-order and multicomponent extensions of the nonlinear Schrödinger equation holds significant importance across diverse applications, notably within the field of optics. Among these equations, the integrable Sasa–Satsuma equation stands out for its captivating soliton solutions. The Sasa–Satsuma equation serves as a mathematical representation for describing the propagation of femtosecond light pulses through optical fibers. The breather, multiwave, combined dark–bright, singular, dark, bright and periodic singular optical soliton solutions are derived in this paper by using the 1φ(ϑ),φ′(ϑ)φ(ϑ) and multivariate generalized exponential rational integral function approach. Furthermore, the paper emphasizes the essential fiber parameters essential for the emergence of these structures and offers visual representations of chosen solutions to elucidate their physical characteristics. The novelty of this work lies in the inaugural application of these methods to the Sasa–Satsuma equation and gives a significant advancement in the understanding of optical phenomena. This study not only opens avenues for further research but also introduces a fresh perspective on the Sasa–Satsuma model in the field of nonlinear physics with its applications in optical systems.

In this paper, we consider the nonlinear Schrödinger equation (NLSE) with nonlocal nonlinearity having nonlinear chromatic dispersion and Kudryashov’s generalized quintuple-power. For the suggested model, dark solitons, and singular solitons are investigated using the modified extended mapping approach. Furthermore, rational solutions, exponential solutions, hyperbolic wave solutions, singular periodic solutions, and Jacobi elliptic function solutions are shown. We impose certain limits on the parameters in order to guarantee the existence of the obtained soliton solutions. Furthermore, a few chosen solutions are also displayed graphically to illustrate the physical characteristics of the precise solutions. Furthermore, we use the idea of modulation instability analysis to discuss the stability of the derived wave solutions.

In this paper, the improved modified extended Tanh function technique is used to find analytical solutions to the cubic-quartic Lakshmanan-Porsezian-Daniel model which describes the pulse propagation in optical fiber. Using the traveling wave theory, the nonlinear partial differential form of the cubic-quartic Lakshmanan-Porsezian-Daniel equation is transformed into an ordinary differential equation. Then applying the proposed methods bright, dark, bright-dark, periodic, singular, and Jacobian elliptic solutions were derived as analytical solutions. Besides, hyperbolic, rational, trigonometric, and exponential functions were also obtained. In order to understand the obtained solutions, graphical simulations were depicted by 3D surfaces and 2D graphs.

The study of soliton theory plays a crucial role in the telecommunication industry’s utilization of nonlinear optics. The principal area of research in the field of optical solitons revolves around optical fibers, metamaterials, metasurfaces, magneto-optic waveguides, and other related technologies. Therefore, the examination of these solitons received significant attention from scholars in recent years. Optical solitons refer to electromagnetic waves that are confined within nonlinear dispersive medium, wherein the balance between dispersion and nonlinearity effects enables the intensity to remain constant. In this manuscript, the dynamical behavior of Kudryashov’s equation is discussed with the assistance of truncated M-fractional derivative. Kudryashov’s equation is a mathematical model utilized to characterize complex phenomena in telecommunications and transmission technology. It describes the propagation of nonlinear pulses in optical fibers. Different forms of soliton solutions like bright, dark, singular and combo solitions have been extracted. Moreover, hyperbolic, periodic and Jacobi elliptic function solutions are recovered. Two recently modern integration tools like Φ6-expansion method and modified generalized exponential rational function method have been adopted to recover the solutions. The used methods not only provides previously extracted solutions but also secures new solutions. In order to visually depict the output, a variety of graphs featuring distinct shapes are produced in response to appropriate parameter values. The results obtained from this research indicate that the chosen methodologies are effective in improving the understanding of nonlinear dynamical phenomena. A large number of engineers who employ engineering models are expected to find this research interesting. The results demonstrate that the selected methods are practical, straightforward to implement, and applicable to intricate systems across numerous domains, with a specific emphasis on the field of optical fibers. The findings indicate that the system may contain a considerable number of soliton structures.

As a result of oversimplified anti-cubic non-linearity, we are interested in form optical solitons to the nonlinear Schrödinger equation (NLSE). The modified G′G2 creates new solitons that include periodic type solitons in chirped forms. These findings are extremely beneficial to the study of optics. The acquired outcomes are confirmed and explained using various graphs. Computational results show that the preceding analytical expansion scheme is efficient. We present some sufficient conditions for the presence of random quasi-periodic measures for various partial differential equations and sufficient conditions for the presence of quasi-periodic measurements and satisfaction of the NLSE equation. It is discovered that a saddle point bifurcation causes the transition from periodic to quasi-periodic actions in a sensitive area. Further investigation reveals favorable conditions for the multidimensional bifurcation of increasingly complex solutions. Our results provide insight into the complex interplay between periodic and quasi-periodic behaviors and further our understanding of the complex dynamics displayed by these PDE‘s. The outcomes that are being given broaden the field's current understanding and provide opportunities for more investigation into the intricate mathematical structures that underlie partial differential equation solutions.

This document is a correction notice for an article titled "On some novel bright, dark and optical solitons to the cubic-quintic nonlinear non-paraxial pulse propagation model" published in the journal Optical & Quantum Electronics. The authors apologize for a typesetting error in the equations in the original publication. The corrected equations are provided for four different cases, along with corresponding graphs. The document includes mathematical formulas and does not provide any additional information beyond the correction. [Extracted from the article]

In this research, a dynamical (3 + 1)-dimensional nonlinear Schrödinger model (NLSM) is under consideration which is used to model the propagation of ultra-short optical pulses in highly-nonlinear media. This aforementioned model has been widely applied in Bose–Einstein condensates, nonlinear optical fiber communication plasma physics, hydrodynamics, elastic media and other diversified areas of mathematical physics and engineering. This study has two main objectives. Firstly, to obtain some novel solutions in the shape of singular, dark, periodic and plane wave solutions that are innovative and not previously found in the literature. Secondly, an amended extended tanh-function method with modulation instability (MI) is used for this model. As far as we know, this has never been investigated in this manner. The 2-D, 3-D and contour plots have been defined using the required parametric values to support the results of physical compatibility. According to the examined results, the method used in this study to retrieve consisting and conventional solutions is efficient and more immediate in computing and can be thought of as a useful tool in resolving more difficult problems that arise in the field of engineering, physics, mathematics and fiber optics.

In this work, we consider a concatenation model that is a concatenated version of the familiar nonlinear Schrödinger’s equation, Lakshmanan–Porsezian–Daniel equation and the Sasa–Satsuma equation. With the help of the improved modified extended tanh-function method (IMETFM), new families of solutions are extracted for the proposed model. Bright solitons, dark solitons, combo bright-dark solitons, singular solitons, periodic solutions, exponential solutions, rational solutions and Jacobi elliptic solutions are obtained. The IMETFM enables to recover a wide spectrum of solutions to the governing model. The extracted solutions confirmed the efficacy and strength of the current technique. Moreover, the graphs for some solutions are presented to demonstrate their physical nature.

This study investigates optical solitons solutions within the framework of the two new (3 + 1)-dimensional integrable wave equations by employing the modified Sardar sub-equation method. The importance of these suggested models span diverse fields including geophysics, seismology, medical imaging, photonics and sensor development. The derived solutions, meticulously verified using Mathematica, encompass a rich array of mathematical functions including trigonometric, hyperbolic, and exponential functions. Visualization techniques, such as 3D plots, 2D plots, density graphs, contour graphs and polar graphs, effectively illustrate the diverse behaviors exhibited by these soliton solutions. These behaviors include lump, periodic, kink, dark, bright, peakons, cuspons, compactons soliton waves and other complex phenomena. Moreover the modulation instability of the consider model is examined, and corresponding conditions are established. The soliton solutions highlight the efficiency and effectiveness of these method in discovering traveling wave solutions, presenting a valuable tool for addressing a range of nonlinear evolution equations arising in diverse scientific fields such as hydrodynamics, optical fiber communications, plasma physics, ocean engineering, nonlinear dynamics, and condensed matter physics.

Optical soliton solutions in birefringent fibers and other analytic solutions for Sasa-Satsuma equation with Kerr law nonlinearity are investigated using extended F-expansion method. Bright, dark, singular, combo bright-singular, combo dark-singular and combo singular soliton solutions are extracted. Also, periodic, rational and Jacobi elliptic functions solutions are derived. Finally, we offer some 3D, 2D and contour plots to illustrate the nature of the provided solutions.

This investigation of the current work presents the dispersive concatenation model that integrates the spatio-temporal dispersion and the effect of multiplicative white noise in the Itô sense. The study aims to secure stochastic optical soliton solutions within this framework. To achieve this purpose, we implemented the enhanced Kudryashov technique and the new projective Riccati equations strategy. Initially, the traveling wave transform hypothesis was applied to convert the stochastic model into its nonlinear ordinary differential equation form, encompassing real and imaginary components with emerging the related parametric restrictions. Subsequently, we introduced the enhanced Kudryashov and the new projective Riccati equations techniques successfully employed in analyzing the generated nonlinear ordinary differential equation. By utilizing the appropriate solution sets, which encompass variables with unknown values and necessary transformations, we have derived many stochastic optical soliton-type solutions for the model. Detailed discussions for the dynamics of some selected soliton solutions are presented with the supporting of the 3D and 2D graphs that illustrate the white noise effect on these solitons.

In this paper, we explore optical nonlinear waves in a birefringent fiber by investigating the dynamics of a coupled nonlinear Schrödinger system with four-wave mixing factors. Utilizing the Jacobi elliptic function approach, we achieve this goal by building cusp-like singular soliton solutions in this optical system. Depending on the characteristics of the focusing and defocusing optical medium, we investigate four different physical possibilities. We identify numerous cusp-type singular soliton profiles in these settings, namely cusp-like bright, dark, kink, and antikink solitons. Group velocity dispersion, spatio-temporal dispersion, and four-wave mixing effects are further evaluated in terms of their influence on the constructed singular soliton solutions. We show that the intensity of the singular soliton profiles can be suppressed or enhanced by altering the dispersion values. We also witness interesting transitions: when four-wave mixing parameters are tuned, cusp-like kink-type solitons can become cusp-like antikink solitons and vice versa. These novel findings about singular solitons in birefringent fibers have great promise for experimental uses in the realm of optics, as well as for advancing our knowledge of four wave mixing and nonlinear propagation.

The study examines the propagation of solitary waves in optical fibers controlled by the Radhakrishnan–Kundu–Lakshmanan model (RKL) equation, which includes a cubic–quintic non-Kerr refractive index. The first step in this study involves utilizing the extended algebraic method to derive solitonic solutions for the given equation. These solutions are subsequently graphically represented for a variety of parameter values. The proposed strategy allows for the discovery of a range of solutions.The controlling model is incorporated into a planar dynamical system via the Galilean transformation, resulting in the transformation of the governing model. The effects of both the quintic nonlinearity and self-phase modulation on the perturbed and unperturbed systems are explored using the effective potential and related phase portrait. The governing model is thus transformed into a planar dynamical system. Furthermore, this study harnessed a variety of numerical techniques. These methods encompassed time series analysis, phase portraits examination, and meticulous sensitivity inspections. Additionally, we employed a Poincaré section to assess the equation’s responsiveness to external perturbations. The outcomes reveal a fascinating observation: the perturbed system’s behavior shifts from a quasi-periodic state to a chaotic one, contingent upon the amplitude and frequency of the external perturbation. Moreover, we conducted multistability analysis for specific sets of physical parameters, uncovering the equation’s tendency to exhibit multistability in such scenarios.

The Fokas-Lenells equation (FLE) including the M-truncated derivative or beta derivative is examined. Using the modified mapping method, new elliptic, hyperbolic, rational, and trigonometric solutions are created. Also, we extend some previous results. Since the FLE has various applications in telecommunication modes, quantum field theory, quantum mechanics, and complex system theory, the solutions produced may be used to interpret a broad variety of important physical process. We present some of 3D and 2D diagrams to illustrate how M-truncated derivative and the beta derivative influence the exact solutions of the FLE. We demonstrate that when the derivative order decreases, the beta derivative pushes the surface to the left, whereas the M-truncated derivative pushes the surface to the right. [ABSTRACT FROM AUTHOR]

In optical fiber theory, the nonlinear Schrödinger equation (NLSE) is one of the most important physical models for describing the transmission of optical soliton growth. Due to the large range of possibilities for superfast signal processing and light pulses in communications, the wave propagation in nonlinear fibers is currently a topic of substantial curiosity. In this study, the third order generalized NLSE is investigated analytically by using the extended hyperbolic function technique, the improved F-expansion method, and the Adomian decomposition method, which has great importance in applied physics especially in the study optical fibers. The obtained solutions are newly made also they have the form of optical solitons, singular bell shaped, multi-bell shaped, and singular periodic solitons wave behavior and have applications in the optical fibers transmission, physics and many other scientific fields. The stability along with the modulation instability (MI) of the governing model has also been examined to validate the results. The nonlinear model properties have been illustrated using 3D, 2D, and contour plots with the appropriate set of parameters, which is demonstrated to visualize the physical structures more productively. Finally, it is concluded that similar strategies can also be implemented to study many contemporary models. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) is one of the most important physical model in optical fiber theory for explaining changes in optical soliton growth. Due to the large range of opportunities for ultrafast signal routing systems and brief pulses of light in communications, optical soliton transmission in nonlinear fibers is currently a topic of significant interest for researchers. In this article, we study the dynamics of the generalised unstable NLSE which specified the time evolution of perturbations in barely steady or unstable media. A set of efficient analytical approaches are implemented to develop various interesting travelling wave structures such as bright, dark, singular, optical, bell shaped and periodic solitons solutions. Additionally, the stability of the obtained results is validated. Finally, the modulation instability of the governing model has also been investigated to present change effects that occurs as a result of the irregular refractive indices in an optical fiber, leading to distinct behaviour patterns. The methods as reported are applicable in clarity, effectiveness, and simplicity, establishing their applicability to various sets of dynamic and static nonlinear equations concerning evolutionary phenomena in computational physics, as well as other practical domains and numerous areas of research. [ABSTRACT FROM AUTHOR]

The central objective of this study is to examine how the Lie symmetry technique and conservation law theories can be employed to analyze fractional partial differential equations with a mixed derivative ∂ t 2 β (u xx) , specifically considering the Riemann–Liouville time-fractional and integer-order x-derivatives. In the beginning, we shed light on the prolongation formula that encompasses the infinitesimal generator, particularly in the presence of mixed derivatives. Following that, through the identification of Lie symmetries, we introduce a comprehensive formula for deriving conservation laws, incorporating the notion of nonlinear self-adjointness. Utilizing the Lie symmetries and an alternative reduction technique, we achieve a novel reduction that eliminates the need for the Erd e ´ lyi-Kober operator. Additionally, family of wave solutions have been obtained corresponding to the longitudinal wave equation in magneto-electro-elastic structures and bifurcation analysis and chaotic analysis has been done for the governing model via Galilean transformation and phase plane portraits have been used to depict the changes in the behavior of the model by changing the values of the potential parameter. [ABSTRACT FROM AUTHOR]

In this study, using the Kudryashov R function method and the generalized Kudryashov method, the (1 + 1) -dimensional resonant nonlinear Schrödinger equation with dual-power law nonlinearity has been effectively examined for finding optical soliton solutions. The Kudryashov R function technique has numerous advantages that make symbolic computing considerably simpler, particularly while dealing with strongly dispersive nonlinear equations. The generalised Kudryashov method is noteworthy due to its capacity to address a wide range of complex nonlinear ordinary differential equations (NLODEs) observed in diverse engineering, scientific and mathematical fields. Newly generated nonlinear Schrödinger equation is demonstrated by the resonant nonlinear Schrödinger equation used to describe nonlinear optical phenomena. In order to achieve the goal, the governing model was first transformed into a NLODE, and then the solution sets and solution functions were derived based on the definitions of the suggested approaches. Solitons are localized wave forms that retain their shape and stability as they propagate across optical fiber. Optical solitons are characterized by their ability to maintain their shape and amplitude during propagation, even when encountering other solitons. Using the suggested approaches, the singular, dark, kink-antikink and bright soliton solutions from the governing equation have been extracted. Under the appropriate selection of parameter values, 3D, 2D and contour graphs are shown to illustrate the physical characteristics of the obtained results. All generated solutions are demonstrated to be analytically stable by the analysis of modulation instability, which also reveals the stability and movement of the waves. These solutions have extensive implications in the field of telecommunications and nonlinear fiber optics and assist in understanding the physical phenomena underlying the equation. These methods are new and standardised, and they can be applied to solve a variety of mathematical and physical problems. [ABSTRACT FROM AUTHOR]

In this paper, using the Lagrangian approach, we study the modulational instability of the one-dimensional optical solitons with a nonlinear Schrödinger equation in pure quartic media along with competing cubic and quintic nonlinearities. By adding weak perturbation to the plane wave solution, we deduced the dispersion relation which accommodates the impacts of fourth-order dispersion strength, cubic and quintic nonlinearities. Following that, by employing a time-dependent variational approach, we find the stable and unstable regions of the optical solitons by tuning the fourth-order dispersion for competing cubic and quintic nonlinear strengths. We have also been able to identify the region of modulational instability growth rate. We complemented our analytical results with full numerical simulations and good agreement has been found. [ABSTRACT FROM AUTHOR]

The work aims to explore the exponential rational function technique and the generalized Kudryashov procedure to investigate one of the most important equations which is the equation of cold bosonic atoms in a zig-zag optical lattice. The considered equation is reduced to a supreme equation by using the continuum approximation which describes the soliton's dynamics with the implication pulse variables. The adopted techniques are accurate, simple, straightforward, and succinct to compute. [ABSTRACT FROM AUTHOR]

The objective of this work is to investigate the perturbed Gerdjikov–Ivanov (GI) equation along spatio-temporal dispersion which explains the dynamics of soliton dispersion and evolution of propagation distance in optical fibers, photonic crystal fibers (PCF), and metamaterials. The algorithms, namely hyperbolic extended function method and generalized Kudryashov's method, are constructed to obtain the new soliton solutions. The dark, bright, periodic, and singular solitons are derived of the considered equation with the appropriate choice of parameters. These results are novel, confirm the stability of optical solitons, and have not been studied earlier. The explanation of evaluated results is given by sketching the various graphs in 3D, contour and 2D plots by using Maple 18. Graphical simulations divulge that varying the wave velocity affects the dynamical behaviors of the model. In summary, this research adds to our knowledge on how the perturbed GI equation with spatio-temporal dispersion behaves. The obtained soliton solutions and the methods offer computational tools for further analysis in this field. This work represents an advancement in our understanding of soliton dynamics and their applications in photonic systems. [ABSTRACT FROM AUTHOR]

This paper aims to reveal the effects of the fourth-order dispersion and parabolic law which comes from self-phase modulation on the soliton behavior of the cubic-quartic nonlinear Schrödinger equation (CQ-NLSE) by using the modified new Kudryashov method. First, applying the complex wave transformation, the nonlinear ordinary differential form (NODE) has been obtained. Then, the modified new Kudryashov method (mNKM) has been expressed and applied. In the next step, linear algebraic system has been gained and solved. Then analytical soliton solution of the CQ-NLSE has been derived and checked for accuracy so that it satisfies the main equation. For the obtained solution functions, bright and singular soliton solutions have been gained and their graphical presentations have been made. The effects of both the fourth-order dispersion parameter and the parabolic law nonlinearity on the soliton dynamics have been examined and the necessary comments have been made. To our best knowledge, no such study has been reported for the equation examined. [ABSTRACT FROM AUTHOR]

The study focuses on the combination of three mathematical models, namely nonlinear Schrödinger's equation, Lakshmanan–Porsezian–Daniel model, and Sasa–Satsuma model, which is called the concatenation model. The study investigates optical solitons and other exact solutions using a modified extended mapping approach. The solutions derived include bright solitons, dark solitons, combo bright-dark soliton solutions, singular solitons, periodic and singular periodic solutions, exponential and rational solutions, and Jacobi elliptic solutions. The study considers constraints on the parameters to ensure the existence of the obtained soliton solutions. Selected solutions are depicted graphically to demonstrate their physical nature. [ABSTRACT FROM AUTHOR]

We have found a new class of three-dimensional vector laser solitons within the framework of the quasi-optical approximation (parabolic equation) for a medium with a fast response of nonlinear amplification and absorption by numerical modeling. These solitons have different polarization singularities. The range of parameters in which solitons are stable is studied, and scenarios of destabilization when leaving this range are indicated. [ABSTRACT FROM AUTHOR]

In this study, an effort has been made to obtain optical soliton solutions of the (1+1)-dimensional Biswas-Milovic equation which was introduced by Biswas and Milovic in 2010, having Kerr law and parabolic-law with weak non-local nonlinearity in the presence of spatio-temporal dispersion, which is one of the important models for nonlinear optics. Although, the model is an equation that has been studied by many researchers, the fact that the form to be examined has not been studied before. As a general algorithm, first converting the model to nonlinear ordinary differential form with a complex wave transformation, then obtaining candidate optical soliton solutions by utilizing the new Kudryashov technique, determining the ones that satisfy the main equation from these solutions as the exact solution, and in order to better understand the obtained solutions by making graphical presentation and providing the necessary comments constitute the main framework of the article. [ABSTRACT FROM AUTHOR]

In the present study, we focus on obtaining the optical soliton solutions of the Drinfeld-Sokolov-Satsuma-Hirota (DSSH) equation using Kudryhashov methods. First, a wave transformation is applied to the DSSH equation, resulting in a nonlinear ordinary differential equation (NLODE). A balance number is deduced through the balancing of this NLODE. Subsequently, candidate solutions, inclusive of their respective derivatives and the wave transformation, are inserted into the DSSH equation. This results in an equation in polynomial form. Terms with equal power are grouped together in the new equation, and their coefficients are set to zero, leading to a system of algebraic equations. Solving this algebraic system facilitates the identification of undetermined parameters within the candidate solutions, thus yielding the solutions for the DSSH equation. Visualization of these solutions is executed through contour plots as well as two- and three-dimensional graphical representations. The contributions of this study are of substantial relevance for subsequent research in nonlinear science. [ABSTRACT FROM AUTHOR]