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In this paper, the stochastic 2-dimensional nonlinear Schrödinger equation with multiplicative Brownian motion is investigated. This equation deals with beam propagation in a Graded-Index multimode optical fiber with a parabolic index profile. The spectral, temporal, and spatial features of ultrashort light pulses can be controlled in novel ways by this kind of nonlinear multimode optical fiber, which is gaining popularity. The bifurcation analysis, chaotic and sensitivity behavior of the wave solutions are examined using the qualitative theory for planar dynamical systems. The generalized exponential rational functional method is used to get the exact optical soliton solutions under the effect of noise. This method provides us the rational, exponential, dark, bright, combined dark-bright, singular, and solitary waveform solutions. Moreover, we use the MATHEMATICA11.1 tools to plot the 3-dimensional, 2-dimensional, and their corresponding contour diagrams to show the effects of multiplicative time noise on the optical solitons for the GRIN multimode optical fiber. Finally, we will show the multiplicative Brownian motion stabilizes the solutions of 2D-nonlinear Schrödinger equation of Graded-Index multimode optical fiber a round zero. [ABSTRACT FROM AUTHOR]

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov's with sextic‐power law refractive index and generalized nonlocal laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell‐shaped soliton solutions, and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model's solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams. [ABSTRACT FROM AUTHOR]

The coupled higher-order nonlinear Schrödinger equations (CHNLSEs) are investigated in this paper. This model mimic the actuation of the solitons through multimode fibers. The technique of improved modified extended tanh function is used to conduct the study. Many exact solutions are achieved like bright solitons, dark solitons and singular solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, hyperbolic and Jacobi elliptic solutions are raised. Using 3D and 2D graphs, the physical behavior of the produced solutions is illustrated. [ABSTRACT FROM AUTHOR]

ABSTRACT This study focuses on the manipulation of soliton parameters within optical waveguides characterized by a dual‐power law refractive index, drawing similarities with published papers. The precise management of solitons holds significant importance in the field of fiber‐optical communication. While previous research has primarily concentrated on modifying soliton attributes such as amplitude and width, comparatively less attention has been dedicated to controlling the wave speed of solitons. In order to bridge this gap, the researchers adopted the employment of the nonlinear Schrödinger equation with time‐dependent dispersion and two power‐law nonlinearities, mirroring similar approaches found in published literature. By employing this methodology, the researchers successfully achieved active control over the wave speed of solitons in non‐uniform optical waveguides. To effectively handle the inherent complexities of the nonlinear system, two well‐established techniques, namely, the generalized Kudryashov method and the simple equation method, were utilized to derive solutions. These techniques have been previously employed and demonstrated in published papers, further enhancing the validity and reliability of the research findings. We discuss the traveling wave solutions by finding the fixed points of the dynamical planar system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Optical solitons have long been of interest both from a fundamental perspective and because of their application potential. Both cubic (Kerr) and quadratic nonlinearities can lead to soliton formation, but quadratic solitons can practically benefit from stronger nonlinearity and achieve substantial wavelength conversion. However, despite their rich physics, quadratic cavity solitons have been used only for broadband frequency comb generation, especially in the mid-infrared. Here, we show that the formation dynamics of mid-infrared quadratic cavity solitons, specifically temporal simultons in optical parametric oscillators, can be effectively leveraged to enhance molecular sensing. We demonstrate significant sensitivity enhancement while circumventing constraints of traditional cavity enhancement mechanisms. We perform experiments sensing CO2 using cavity simultons around 4 μm and achieve an enhancement of 6000. Additionally, we demonstrate large sensitivity at high concentrations of CO2, beyond what can be achieved using an equivalent high-finesse linear cavity by orders of magnitude. Our results highlight a path for utilizing quadratic cavity nonlinear dynamics and solitons for molecular sensing beyond what can be achieved using linear methods. Optical molecular sensors benefit many applications from fundamental science to industry. In this work, the authors leverage the formation dynamics of cavity solitons to achieve molecular sensing in the mid-infrared spectral region. [ABSTRACT FROM AUTHOR]

This study delves into the soliton solutions of the (1+1)-dimensional perturbed complex Ginzburg–Landau equation characterized by Kerr law, under the condition of absent chromatic dispersion. The complex Ginzburg–Landau equation, a basic equation in nonlinear science, describes various complex systems including optical fibers, Bose–Einstein condensates, and fluid dynamics. By applying the Auxiliary equation scheme, we explore the bright and kink soliton solutions and modulation instability of the considered model by incorporating Kerr nonlinearity, which describes the intensity-dependent refractive index in optical media. Furthermore, detailed 3D-surface and 2D graphs of obtained soliton solutions have been added. Additionally, graphs illustrating the impacts of various parameters for the considered model on soliton solutions are presented in this study. Our findings provide deeper inspirations into the structures of soliton formation and propagation in nonlinear media, contributing to the theoretical understanding and potential implementations in optical communication systems and beyond. This work underscores the critical role of nonlinearity in forming solitons and offers a framework for further exploration of perturbed nonlinear systems devoid of chromatic dispersion effects. [ABSTRACT FROM AUTHOR]

This study delves into utilizing the Kudryashov auxiliary equation method on the time-fractional (3+1)-dimensional Sasa–Satsuma equation, pivotal for transmitting ultra-fast pulses in optical fibers. The main aim is to generate a variety of optical solitons for the time-fractional nonlinear (3+1)-dimensional Sasa–Satsuma model, including dark, bright, singular, and straddled solitons. These solitons are derived from exponential and hyperbolic function-type solutions, presenting novel findings not previously reported. The study demonstrates the effectiveness of the implemented method in deriving analytical solutions for fractional differential equations. The (3+1)-dimensional Sasa–Satsuma model's potential in optical fiber transmission is highlighted, offering significant insights into generating optical solitons and their applications. [ABSTRACT FROM AUTHOR]

This paper applies the extended direct algebraic method (EDAM) for the first time to obtain the soliton solutions of the fractional Sasa–Satsuma equation (FSSE) with the descriptions of two fractional derivatives. To evaluate the optical soliton and some other solutions of the FSSE, the presented research applies the fractional beta and conformable derivatives. As a useful result of this research, several types of solitons have been identified, such as the single soliton, M-shaped, W-shaped, combined dark-bright, smooth dark, sharp dark, dark-bright, anti-bell shaped, and periodic. Among the nonlinear optics fields expanding at the fastest rate are optical solitons. A fundamental equation in the theory of optical fibers and telecommunications is the fractional Sasa–Satsuma equation. The transmission of femtosecond pulses in optical fibers as well as the interaction and propagation of ultrashort pulses in the sub-picosecond or femtosecond region are influenced by this equation. Additionally, the physical properties of the completed solutions are illustrated via contour, 3D, and 2D graphs. The EDAM is a practical and effective technique that offers a variety of avenues for future investigation. The fresh versions of the findings presented here have not been published in the full body of literature before. [ABSTRACT FROM AUTHOR]

In the present research work, we construct and examine the self-similarity of optical solitons by employing the Riccati Modified Extended Simple Equation Method (RMESEM) within the framework of non-integrable Coupled Nonlinear Helmholtz Equations (CNHEs). This system models the transmission of optical solitons and coupled wave packets in nonlinear optical fibers and describes transverse effects in nonlinear fiber optics. Initially, a complex transformation is used to convert the model into a single Nonlinear Ordinary Differential Equation (NODE), from which hyperbolic, exponential, rational, trigonometric, and rational hyperbolic solutions are produced. In order to better understand the physical dynamics, we offer several 3D, contour, and 2D illustrations for the independent selections of physical parameter values. These illustrations highlight the graphic behaviour of some optical solitons and demonstrate that, under certain constraint conditions, acquired optical solitons lose their stability when they approach an axis and display periodic-axial perturbations, which lead to the generation of optical fractals. As a framework, the generated optical solitons have several useful applications in the field of telecommunications. Furthermore, our suggested RMESEM demonstrates its use by broadening the spectrum of optical soliton solutions, offering important insights into the dynamics of the CNHEs, and suggesting possible applications in the management of nonlinear models. [ABSTRACT FROM AUTHOR]

The different types of optical soliton solutions are investigated for the 2-dimensional nonlinear Schrödinger equation with an instantaneous Kerr nonlinearity. This equation contains the beam propagation in a multimode optical fiber with a parabolic index profile. Different soliton solutions are examined, such as Breather waves, Lump waves, Mixed types, Periodic cross kink, Multiwave, M-shape, M-shape rational one, two kinks, and rogue waves. To obtain these optical solitons, we apply the Hirota bilinear transformation approach. These results are spectral, temporal, and spatial features of ultrashort light pulses that can be controlled in novel ways by this kind of nonlinear multimode optical fiber, which is gaining popularity. Moreover, the 3-dimensional, 2-dimensional, and contour plots are drawn for some solutions that show their physical interactions by selecting different parameters. These results are very helpful for further study of dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we employ the auxiliary equation method (AEM) to show the leverage of the nearest neighbors hopping and the boson–boson interaction in a zig zag optical lattice on the stressed optical solitons (os) and the Modulation Instability (MI) gain of cold bosonic atoms. The pedestal of the work is set on the continuum approximation of a cold bosonic atoms in a zig zag optical lattice which has demonstrated in detailed in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589]. The results collected have enclosed bright, dark optical solitons and the combined bright-dark and bright-bright optical solitons. Compared the acquired results with Tala-Tebue et al. [Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44], further the os including U-shaped and W-shaped profile have been drop in and the powerful of the nearest neighbors hopping and the boson–boson interaction has been exposed across the spatiotemporal and plot evolution of the bright and dark os. Therewith, the MI bands have been stressed and it is observed some new attitude of the MI gain spectra under the induction of the model parameters. The collected results are new in literature in our knowledge and will improve the previous results acquired in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589; Zhao XH, Tian B, Liu DY, et al. Dark solitons interaction for a (2+1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Superlattices Microstruct 100(2016):587–595; Uddin MF, Hafez MG, Hammouch Z, et al. Periodic and rogue waves for Heisenberg models of ferromagnetic spin chains with fractional beta derivative evolution and obliqueness. Waves Random Complex Media 2020. https://doi.org/10.1080/17455030.2020.1722331; Inc M, Aliyu AI, Yusuf A. Optical solitons to the nonlinear Shrdingers equation with spatio-temporal dispersion using complex amplitude ansatz. J Mod Opt 2017;64(21):2273–2280]. [ABSTRACT FROM AUTHOR]

The generalized mK(m,n) model is used to portray the movement of nonlinear waves by applying four mathematical methods with assistance of Mathematica software. These methods are called enhanced simple equation (SE) method, Exp(−Ψ(ξ))-expansion method, (G′∕G)-expansion method and modified F-expansion method. For physical behavior, few solutions are plotted graphically in the form of two-dimensional (2D) and three-dimensional (3D). The techniques and soliton solutions that are developed provide computational tools for additional research in this area. With this work, we have gained a better knowledge of soliton motion and how it is used in dynamical systems and other related technologies. Hence, this work will help physicists envision some new ideas and hypotheses in nonlinear science. [ABSTRACT FROM AUTHOR]

We propose a simple optical model, which consists of two Kerr nonlinear parallel planar waveguides sandwiching a one-dimensional photonic crystal for demonstrating the phenomenon of spontaneous symmetry breaking in optics. We study the shape and stability of soliton pairs propagating in two planar waveguides and the spontaneous symmetry breaking of the power distribution of solitons in the two waveguides. The results of numerical simulation show that there is supercritical type of spontaneous symmetry breaking in this system. More optical power P and smaller refractive index of photonic crystals C 0 make soliton solutions more prone to spontaneous symmetry breaking bifurcation. Asymmetric soliton solutions and symmetric soliton solutions correspond to different dispersion relations. Under the given total power P, for the symmetric soliton solutions, the propagation constant k increases linearly with C 0 , and for asymmetric soliton solutions, the propagation constant k decreases nonlinearly with C 0 . The minimum point of k (C 0) is just the demarkation between the symmetric soliton solutions and the asymmetric soliton solutions. If C 0 is fixed, the minimum point of the Hamiltonian of the system, E (P) , is also the critical point of the two different kinds of soliton solutions. We also study stable and unstable distributions of two kinds of soliton solutions. [ABSTRACT FROM AUTHOR]

Period-doubling bifurcation, as an intermediate state between order and chaos, is ubiquitous in all disciplines of nonlinear science. However, previous experimental observations of period doubling in ultrafast fiber lasers are mainly restricted to self-sustained steady state, controllable manipulation and dynamic switching between period doubling and other intriguing dynamical states are still largely unexplored. Here, we propose to expand the vision of dissipative soliton periodic doubling, which we illustrate experimentally by reporting original spontaneous, collisional, and controllable spectral period doubling in a polarization-maintaining ultrafast fiber laser. Specifically, the spontaneous period doubling can be observed in both single- and double-pulses. The mechanism of the switchable state and periodic doubling was revealed by numerical simulation. Moreover, state transformation of individual solitons can be resolved during the collision of triple solitons involving stationary, oscillating, and period doubling. Further, controllable deterministic switching between period doubling and other dynamical states, as well as exemplifying the application of period-doubling-based digital encoding, is achieved under programmable pump modulation. Our results open a new window for unveiling complex Hopf bifurcation in dissipative systems and bring useful insights into nonlinear science and applications. [ABSTRACT FROM AUTHOR]

In this study, we aimed to obtain optical soliton solutions of the perturbed Fokas–Lenells equation having the parabolic law of self-phase modulation in the presence of spatio-temporal dispersion. The Fokas–Lenells equation is significant in nonlinear optics, particularly for modeling femto-second pulse propagation. By implementing the enhanced Kudryashov method, we derive dark, bright, and kink optical solitons for the investigated model. Visualizing the obtained results supported by 3D and 2D simulations and examining the effects of the parameters within the model on the soliton behavior constitute the other branches of the study. Considering that the interaction of various self-phase modulation forms with the chromatic dispersion term is different and its effects on soliton behavior are different, the parabolic law nonlinearity form is examined in the study and the results obtained are aimed at contributing to studies on both the model and nonlinear optics. [ABSTRACT FROM AUTHOR]

The Schrödinger equation depends on the physical circumstance, which describes the state function of a quantum‐mechanical system and gives a characterization of a system evolving with time. The essential focus of proposed research is to observe the solution for fractional generalized nonlinear Schrödinger (FGNS) equation using q‐homotopy analysis transform method (q‐HATM). The fractional order derivative is taken in the Atangana‐Baleanu (AB) sense. The physical behaviours of achieved solution for FGNS equation are discussed and sketch out graphically. The existence of the solution for the FGNS equation is presented through theorems 4.1 to 4.3. The proposed numerical simulations confirm the advantages of the AB derivative through q‐HATM. Few numerical experiments were carried out to validate the proposed method. Moreover, numerical simulations are carried out to verify efficiency and robustness of the derived results by considering two cases. [ABSTRACT FROM AUTHOR]

This study investigates the effects of multiplicative white noise on optical soliton perturbations with the Gerdjikov-Ivanov equation. Using novel analytical techniques, which include the enhanced direct algebraic method and the new projective Riccati equation method, we study the dynamics of soliton solutions. Our investigation uncovers a wide range of soliton solutions, encompassing bright, dark, singular, and straddled solitons. Additionally, we obtain solutions that are represented using Jacobi and Weierstrass doubly elliptic functions. These solutions can be reduced to soliton types under certain conditions. The motivation for this study arises from the crucial role that the Gerdjikov-Ivanov equation plays in a wide range of applications, highlighting its importance in the analysis of nonlinear wave propagation. The originality of our contribution lies in providing a novel structural framework for studying the Gerdjikov-Ivanov equation in birefringent fibers with multiplicative white noise, an area that has not been previously examined. The primary advantage of this study is its potential to significantly enhance our understanding and predictive capabilities regarding the dynamics of nonlinear wave propagation in complex optical media, thereby opening up new avenues for research and application in the field of photonics and optical telecommunications. The validity of our findings is supported by graphical plots that vividly depict the dynamics of certain solitons, offering insights into the impact of white noise on their behavior. This study not only enhances the theoretical comprehension of the Gerdjikov-Ivanov equation but also contributes new analytical techniques and solutions to the existing literature, thus facilitating future research on the topic. [ABSTRACT FROM AUTHOR]

In this paper, we address a newly nonlinear Schrödinger equation (NLSE) in (2 + 1) -dimensions influenced by cubic–quintic–septic law nonlinearity and spatial dissipations effects. We properly construct the dark and bright optical soliton solutions for our governing model. Furthermore, periodic, other singular solutions, multiwave, homoclinic breather, M-shaped rational and their interaction with kink waves of various structures will be derived. In addition to this, some 3D, 2D and contour profiles will be added to anticipate the wave dynamics. [ABSTRACT FROM AUTHOR]

This review embarks on a captivating odyssey of tracing the birth of light from the Big Bang to its intricate interplay with materials. It delves into the fundamental truth that nonlinearity is ubiquitous, and induces fascinating spatiotemporal structures, chaos, and complexity in the medium. After a brief exploration of waves and the effect of nonlinearity in diverse domains, the review article focuses on the field of photonics. This comprehensive review dives into the captivating physics of solitons. This study explores the formation of solitons in optical fibers due to specific nonlinear effects within the material, such as the Kerr effect, the fundamental behaviour of solitons in integrable models, diverse interactions, and the formation of intricate soliton molecules, soliton complexes, and soliton crystals within the dissipative optical systems. We analyse key research on optical solitons and highlight the control of optical solitons for advancements in communication systems, signal processing, optical computing, quantum technologies, etc. Through a meticulous research survey, we find that there is a limited understanding of weak soliton interactions. Further, more theoretical models to be investigated for exploring anisotropy of material and optomechanical interplay. Bridging these gaps will definitely propel future soliton research.Article Highlights: Dance of Light and Matter: This review explores how light and matter interact, from their origins in the Big Bang to how they form special light waves called solitons in fibers. Solitons: Beyond Light Waves: We delve into solitons, self-reinforcing light waves, and their unique behaviors in different models. Along with studies on several temporal and spatial solitons it also includes recent studies on soliton molecules, soliton complexes, and soliton crystals with potential applications. Taming Light's Potential: The review highlights key areas for future research to unlock the full potential of solitons. Highlights the need to study the effect of weak interactions in multi-soliton dynamics in complex systems, tells the importance of exploring their behavior in new materials, and integrating them with nanophotonic devices. This work proposes the concept of 'photobot,' intelligent soliton molecules that we believe will be achievable in the near future. [ABSTRACT FROM AUTHOR]

This article studied the new traveling wave solutions of the cascaded model with higherorder dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise. In the process of exploring traveling wave solutions, a clever combination of the polynomial complete discriminant system was used to discover more forms of traveling wave solutions for this equation. In order to better observe and analyze the propagation characteristics of traveling wave solutions, we used Maple and Matlab software to provide two-dimensional and three-dimensional visualization displays of the equation solutions. Meanwhile, we also analyzed the internal mechanism of nonlinear partial differential equations using planar dynamical systems. The research results indicated that there are differences in the results of different forms of soliton solutions affected by external random factors, which provided more beneficial references for people to better understand the cascaded model with higher-order dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise, and helped people to more comprehensively understand the propagation characteristics of optical solitons. The solution method in this article was also applicable to the study of other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

We investigated the dynamics of highly dispersive nonlinear gap solitons in optical fibers with dispersive reflectivity, utilizing a conformable fractional derivative model. The modified extended direct algebraic method was employed to obtain various soliton solutions, including bright solitons and singular solitons, as well as hyperbolic and trigonometric solutions. The key findings demonstrated that the fractional derivative parameter (α) can effectively control the wave propagation, causing a shift in the wave signal while maintaining the same amplitude. This is a novel contribution, as the ability to control soliton properties through the conformable derivative is explored for the first time in this work. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. This research provides insights into the potential applications of fractional calculus in the design and optimization of photonic devices for optical communication systems. [ABSTRACT FROM AUTHOR]

The coupled higher-order nonlinear Schrödinger equations (CHNLSEs) are investigated in this paper. This model mimic the actuation of the solitons through multimode fibers. The technique of improved modified extended tanh function is used to conduct the study. Many exact solutions are achieved like bright solitons, dark solitons and singular solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, hyperbolic and Jacobi elliptic solutions are raised. Using 3D and 2D graphs, the physical behavior of the produced solutions is illustrated.

AbstractIn this article, truncated M-fractional coupled nonlinear Schrodinger equation (NLSE) with quadratic–cubic nonlinearity is under observation. The studied model is composed of chromatic dispersion, magneto-optic parameter and inter-modal dispersion. The NLSE is the most significant physical model to explain the fluctuations of optical soliton proliferation. The NLSEs have become more popular because of the clarity with which they explain a wide range of complex physical phenomena and the depth with which they display dynamical patterns via localized wave solutions. Optical soliton propagation in magneto-optic is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. The optical solitons are secured in the forms of bright, dark, singular and combo solitons. In addition, hyperbolic, periodic and exponential function solutions have been recovered. The modified Sardar subequation and enhanced modified extended tanh-expansion approaches recently developed integration tools are adopted in this study for securing the solutions. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. The effect of parameters have been observed by allotting suitable values and sketching the different shapes of the graphs.

In this paper, the optical soliton and solitary wave solutions of the (2 + 1) -dimensional Mel'nikov equation are investigated using the Kudryashov R function technique. The Kudryashov R function approach has various features that significantly facilitate symbolic computing, particularly for highly dispersive nonlinear equations. In computations, this approach has the benefit of not requiring the use of a certain function form. This approach gives an algorithm that is straightforward, efficient, and simple for finding solitary wave solutions. In addition, this approach is very influential and reliable when it comes to discovering hyperbolic function solutions of nonlinear equations. Many new hyperbolic function solutions have been obtained from the governing equation by using this technique. In addition, numerous types of soliton solutions describing various structures of optical solitons are retrieved. Using this method, breather, W-shaped, bell shaped, and bright soliton solutions have been generated from the governing equation. From the obtained results, it can be asserted that the applied approach may be a useful tool for addressing more highly nonlinear problems in various fields. By choosing particular values for the relevant parameters, the dynamic features of some breather, W-shaped, bell shaped and bright soliton solutions to the (2 + 1) -dimensional Mel'nikov equation have been displayed in 3D, 2D and contour graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study the Radhakrishnan–Kundu–Lakshmanan equation (RKLE) that plays a key role in modeling the nonlinear propagation of the light pulses. Two novel methods namely Wang's Bäcklund transformation-based method and Wang's direct mapping method are employed to construct the diverse exact optical soliton solutions. Abundant exact solutions like the bright soliton solution, dark soliton solutions, singular soliton solutions, algebraic solitary wave solutions and the singular periodic wave solutions expressed in the form of the rational function type, double-exp function type, Sin-Cos function type, Tan function type, Cot function type, Sinh-Cosh function type, Tanh function type, Coth function type, Sech function type, Csch function type, Sec function type and Csc function type are obtained. The dynamics of the different exact solutions are presented through the 3D plot, contour and 2D curve, and the corresponding physical interpretations are elaborated in detail. It is confirmed that the proposed methods are effective and powerful, and can be adopted to construct the optical solitons of the other PDEs arising in the optical physics. [ABSTRACT FROM AUTHOR]

In this paper, the extended simplest equation method is utilized to construct the novel exact optical solitons solutions of the perturbed time fractional Chen–Lee–Liu equation with conformable fractional derivative. The acquired optical solitons and other solutions are expressed via the rational functions, the trigonometric functions, and the hyperbolic functions; in addition to guaranteeing the existence of the acquired results, the constrain conditions are provided. Furthermore, to elucidate the magnitude of the proposed equation, various obtained solutions are plotted via two-dimensional (2D) and three-dimensional (3D) graphs using appropriate values of parameters. It is observed that the present technique is a powerful tool and efficient for finding the analytical solution for nonlinear differential equations of integer and fractional orders. [ABSTRACT FROM AUTHOR]

We demonstrate that higher-order vortex solitons (with larger vortex charges m) in the nonlinear Schrödinger equation with cubic (defocusing Kerr) nonlinearity can be stabilized by the action of a flat-bottom potential. Such a model can also support the common fundamental (vorticityless) solitons, including the Gaussian-like and flat-top solitons. The effective radii of the fundamental and vortex solitons can be modified by tuning the radius of the flat-bottom part of the potential. We display the contours and phases of vortex solitons with vorticity numbers up to m = 12 . Interestingly, the central holes of vortex solitons increase with the increase of vortex charges, and the central portions of these higher-order vortices become flat. We investigate the existence and stability of both the fundamental and vortex solitons, for different values of the relevant model parameters: the radius of the flat bottom, initial beam power, propagation constant, and the strength of nonlinearity. We also consider the propagation of perturbed initial beams, as well as the propagation in longitudinally modulated flat-bottom potentials. Such a modulated propagation allows for an easy soliton management. We find that the fundamental solitons are completely stable, while the higher-order vortex solitons are prone to the modulation instability, degenerating into m simple vortices that fly away from the inside to the outside of solitons. Eventually, an initial higher-order vortex beam turns into a fundamental (chargeless) soliton. [ABSTRACT FROM AUTHOR]

The stochastic nonlinear Kodama (SNLK) equation forced in the Stratonovich sense by multiplicative noise is considered here. New elliptic, hyperbolic, trigonometric, and rational stochastic solutions are acquired using (G ′/ G)-expansion method and mapping method. Because the SNLK equation is extensively used in is extensively used in nonlinear optics, fluid dynamics, and plasma physics, the obtained solutions may be used to study a broad range of relevant physical phenomena. In order to interpret the effects of multiplicative noise, the dynamic performances of the various obtained solutions are displayed utilizing 3D and 2D graphs. We infer that multiplicative noise impacts and stabilizes the solutions of SNLK equation. [ABSTRACT FROM AUTHOR]

The stochastic generalized nonlinear Schrödinger equation (SGNLSE) of third order in the Stratonovich sense is examined here. New elliptic, hyperbolic, trigonometric, and rational stochastic solutions are obtained using a modified mapping method. Because the GNLSE is extensively used in nonlinear optical phenomena, optical fiber communication systems, communication and heat pulse propagation in materials, the obtained solutions may be used to study a broad range of relevant physical phenomena. In order to interpret the effects of multiplicative noise, the dynamic performances of the various obtained solutions are displayed utilizing 3-D and 2-D graphs. We infer that multiplicative noise impacts and stabilizes the behavior of SGNLSE solutions.

This research explored optical soliton solutions for the (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation (gFKMNE), which is a nonlinear model for explaining pulse transmission in communication structures and optical fibers. Two enhanced variants of $ (\frac{G'}{G}) $-expansion method were employed, namely, extended $ (\frac{G'}{G}) $-expansion method and the generalized $ (r+\frac{G'}{G}) $-expansion method, based on the wave transformation of the model into integer-order nonlinear ordinary differential equations (NODEs). By assuming a series-form solution for the resultant NODEs, these strategic methods further translated them into a system of nonlinear algebraic equations. Solving these equations provided optical soliton solutions for gFKMNE using the Maple-13 tool. Through 3D and contour visuals, it was revealed that the constructed soliton solutions are periodically arranged in the optical medium, forming dark soliton lattices. These dark soliton lattices are significant in several domains, such as optical signal processing, optical communications, and nonlinear optics.

The cubic–quartic perturbed Biswas–Milovic equation, which contains Kudryashov's nonlinear form and two generalized nonlocal laws, has been explored qualitatively and quantitatively, as demonstrated in the present work. The research methods used include the complete discrimination system for polynomial method and the trial equation method. The results show that the Hamiltonian has the conservation property, and the global phase diagrams obtained via the bifurcation method reveal the existence of periodic and soliton solutions. Furthermore, we fully classify all the single traveling wave solutions to substantiate our findings, covering singular solutions, solitons, and Jacobian elliptic function solutions. We analyze their topological stabilities and present two‐dimensional graphs of solutions. We also delve deeper into the dynamic system by incorporating the perturbation item to explore the chaotic phenomena associated with the equation. These outcomes are valuable for studying the propagation of high‐order dispersive optical solitons and have potential applications in optimizing optical communication systems to improve efficiency. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this study, we obtained optical soliton solutions of the perturbed nonlinear Schrödinger–Hirota equation with generalized anti‐cubic law nonlinearity in the presence of spatio‐temporal dispersion. This equation models the propagation of optical pulses in fiber optic cables. Due to the anti‐cubic nonlinearity, perturbation, and spatio‐temporal dispersion present in the model, it provides more accurate results for high‐speed and long‐distance transmissions. Given the significant developments in the field of optics, studies on complex equations such as this model are of great importance. With the increase in real‐life applications, obtaining solutions to optical equations has become crucial. In this study, we used the improved F‐expansion method to derive the optical soliton solutions for the relevant model. This technique allows for obtaining various solutions through the Jacobi elliptic auxiliary functions it employs. The obtained solutions consist of trigonometric and hyperbolic functions. As a result of the application, 10 solutions were obtained, and 2D and 3D graphics of these solutions are included. These graphs illustrate the motion directions of optical solitons and the effect of the nonlinearity parameter p$$ p $$ and spatio‐temporal dispersion parameter β$$ \beta $$ on soliton behavior. No restrictions were encountered during the study. Finally, the originality of the study lies in the first application of this technique to the relevant model and in examining the effect of the parameters p$$ p $$ and β$$ \beta $$ on this model. [ABSTRACT FROM AUTHOR]

The stochastic generalized nonlinear Schrödinger equation (SGNLSE) of third order in the Stratonovich sense is examined here. New elliptic, hyperbolic, trigonometric, and rational stochastic solutions are obtained using a modified mapping method. Because the GNLSE is extensively used in nonlinear optical phenomena, optical fiber communication systems, communication and heat pulse propagation in materials, the obtained solutions may be used to study a broad range of relevant physical phenomena. In order to interpret the effects of multiplicative noise, the dynamic performances of the various obtained solutions are displayed utilizing 3-D and 2-D graphs. We infer that multiplicative noise impacts and stabilizes the behavior of SGNLSE solutions. [ABSTRACT FROM AUTHOR]

The stochastic Schrödinger–Hirota (SSH) equation perturbed by multiplicative noise is taken into consideration here. The mapping approach and the (G′/G)-expansion method are used to generate new elliptic, hyperbolic, rational, and trigonometric stochastic solutions. Since the SSH equation is extensively used in optical fiber, the generated solutions can be applied to explore a wide variety of significant physical phenomena. To explain the impacts of multiplicative noise, the dynamic behavior of the various solutions is illustrated using 3D and 2D graphs. We conclude that the solutions of the SSH equation are affected by and stabilized by multiplicative noise. [ABSTRACT FROM AUTHOR]

Optical solitons are controlled and modified in a four level N-type atomic medium under the effect of cross Kerr nonlinearity. Significant control over optical solitons are reported with variation of driving fields and system parameters. The single and mixed types optical solitons are controlled for the first time with strength of applied driving fields in the medium. Single dark and bright optical solitons are achieved by fixing Rabi frequencies, detuning and phase of applied driving fields. The multi-peakon and periodic type optical solitons are also modified with strength of applied driving fields. Further more the mixed-types of optical solitons including dark-peakon, breather-periodic, multi-peakon-breather and multi-peakon-breather-periodic types optical solitons are investigated with the variation of control fields Rabi frequencies, detunings and decay rates. The modified single and mixed types optical solitons have potential application in soliton radar technology and sensors as well as communication technologies. [ABSTRACT FROM AUTHOR]

This paper leverages the Ginzburg–Landau equation, a fundamental model for elucidating the dynamics of dissipative optical solitons, in conjunction with the Markov Chain Monte Carlo method and the Lilliefors normality test, to precisely identify key parameters such as dispersion coefficient, nonlinear coefficient, and gain coefficient. Our comprehensive analysis yields not only posterior mean estimates and confidence intervals for these parameters but also their corresponding posterior probability density functions. These findings offer valuable statistical insights and theoretical underpinnings for the design and optimization of fiber lasers, with implications for enhancing the performance of fiber-optic communication systems. [ABSTRACT FROM AUTHOR]

The exact soliton solutions of perturbed Gerdjikov-Ivanov equation with both constant and variable coefficients are constructed in this work through the application of the modified F-expansion method. This method is used for the very first time on the considered equation. As a result, the traveling wave solutions including soliton solutions are obtained. These soliton solutions enhance understanding of nonlinear phenomena in the fields of photonic crystal fibers, fiber optics and nonlinear fiber optics. The explanation of the obtained results is provided by drawing some 2-dimensional and 3-dimensional graphs using Mathematica. Furthermore, this study discusses the modulation instability for the considered model. The considered method is found to be effective, simple and useful to study the perturbed Gerdjikov-Ivanov equation. [ABSTRACT FROM AUTHOR]

In this paper, the Kundu–Eckhaus equation which describes the propagation of ultrashort pulses or femtosecond light pulses in nonlinear optics is investigated using the new mapping method. As a result, different structures of chirped solitary and periodic waves were yielded. The dynamic of the soliton and kink soliton chirp waves were illustrated from which we can conclude that the sign of the Raman parameter identifies the chirp wave function direction. [ABSTRACT FROM AUTHOR]

For systems modeled by the resonant nonlinear Schrödinger equation, we aim to investigate the soliton dynamics and the stability of the system. We extend the equation to a generalized form (GRD-NLSE) and utilize the modified F-expansion method to derive various soliton solutions, including bright, kink, and singular solitons for both (1 + 1)-dimensional and (2 + 1)-dimensional models. Additionally, we graphically illustrate soliton behavior. To analyze system stability, we perform modulation instability and soliton stability analyses. Our findings provide valuable guidance for experimental studies and observations of solitons in GRD-NLSE systems. [ABSTRACT FROM AUTHOR]

This article explores a wide range of envelope soliton pulses and their behavior during propagation through a birefringent optical fiber. The propagation of light in such fibers is governed by two coupled non-linear Schrödinger equations (CNLSEs), which account for both coherent and incoherent non-linear couplings. To investigate these solitons, an improved modified extended tanh function scheme (IMETFS) is employed. The study focuses on elliptical core optical fibers and examines the significance of various optical solitons in the presence of group-velocity dispersion and third-order nonlinearity. The findings illustrate that birefringent optical fibers can generate a diverse array of nonlinear periodic waves and localized pulses. The different types of soliton solutions identified include singular, bright, rational, exponential, Jacobi elliptic functions (JEFs), Weierstrass elliptic doubly periodic solutions, and singular periodic solutions. Comparisons with existing literature highlight the novelty and substantive contributions of the discovered wave solutions to current research in nonlinear optics. The success of this approach suggests its potential applicability in addressing nonlinear challenges across different fields, particularly in soliton theory, where similar models are frequently encountered. Additionally, the article presents visual representations of these solutions through 3D and 2D plots, aiding in the understanding of their behaviors. [ABSTRACT FROM AUTHOR]

The paper introduce a novel auxiliary equation method for the successful derivation of traveling wave solutions for the non-linear Kairat-X (K-X) equation. Along with other novel results, soliton, singular, triangular periodic, and doubly periodic topological solutions are among the solutions obtained. The study revisits the concept of optical solitary waves, enhancing our understanding of the model. Previous studies have already derived analytical solutions using diverse approaches, contributing to the discovery of new soliton solutions within this framework. These solutions are characterized through three-dimensional, contour plot, and two-dimensional profile analyses. Additionally, the impact of time on the propagation of wave patterns is explored. The outcomes show how well our suggested approach works to solve non-linear evolution equations by producing fresh, more thorough solutions, making it a powerful mathematical tool for doing so. Through this article, we elucidate how leveraging NAEM with the Kairat-X equation can lead to optimized optical systems, improved data transmission rates, and the evolution of nonlinear optics towards more efficient and reliable communication technologies. [ABSTRACT FROM AUTHOR]

In the current work, we investigate the optical solitons and exact solutions for the conformable time fractional generalized Radhakrishnan–Kundu–Lakshmanan (GRKL)equation with the polynomial law of nonlinearity which represent the mathematical model of pulse propagation in a nonlinear medium in optical fibers. The modified extended direct algebraic method is presented to get the exact solutions for this model. Many varieties of traveling wave solutions are established like, solitons (bright, dark, singular, combo singular-dark), combo periodic solutions, periodic solutions, Jacobi elliptic solutions, exponential solutions and Weierstrass elliptic solutions. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. [ABSTRACT FROM AUTHOR]

This research explored optical soliton solutions for the (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation (gFKMNE), which is a nonlinear model for explaining pulse transmission in communication structures and optical fibers. Two enhanced variants of (G'/G)- expansion method were employed, namely, extended (G'/G)-expansion method and the generalized (r + G'/G)-expansion method, based on the wave transformation of the model into integer-order nonlinear ordinary differential equations (NODEs). By assuming a series-form solution for the resultant NODEs, these strategic methods further translated them into a system of nonlinear algebraic equations. Solving these equations provided optical soliton solutions for gFKMNE using the Maple-13 tool. Through 3D and contour visuals, it was revealed that the constructed soliton solutions are periodically arranged in the optical medium, forming dark soliton lattices. These dark soliton lattices are significant in several domains, such as optical signal processing, optical communications, and nonlinear optics. [ABSTRACT FROM AUTHOR]

This paper attains cubic-quartic optical solitons of the Biswas–Milovic equation with dual power law and Kerr law nonlinearities. Dark, bright, singular and combo-singular solitons solutions are obtained with the F-expansion method. The solutions gained in our this work are newer than the obtained solutions using different method before this work in the literature. In order to better convey the physical significance of the solutions, we also include 2D and 3D graphics. The computational software MATHEMATICA is the helping tool for visualizing and analyzing the derived solutions. [ABSTRACT FROM AUTHOR]

The perturbed nonlinear Schrödinger equation is employed to characterize the dynamics of optical wave propagation when confronted with dissipation (or gain) and nonlinear dispersion that vary with both time and space. This equation serves as a fundamental model for investigating pulse dynamics within optical fibers and has application to nanofiber applications. This study successfully discovers optical solitons within this framework using the unified solver, Jacobi elliptic function, and simplest equation methods. We extract solutions using hyperbolic, trigonometric, and rational functions, including multi-solitons, dark, singular, bright, and periodic singular solitons. This study thoroughly compares our results with existing literature to provide novelty and significance of our findings. We have incorporated a detailed comparison between the methods employed in our study, which highlights their importance and strength. We have derived soliton solutions for the examined equations and generated 3D contour and 2D visual representations of the resulting solution functions. Alongside obtaining the soliton solutions, we offer a graphical exploration of how the parameters in the considered equations influence the system. [ABSTRACT FROM AUTHOR]

The new types of exact solitons of the space–time fractional nonlinear (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) model are achieved by applying the unified technique and modified extended tanh-expansion function technique. A novel definition of the fractional derivative known as the truncated M-fractional derivative is also used. This model describes both the non-elastic and elastic interactions between internal waves. This model is used to represent intricate nonlinear phenomena like shallow-water waves, plasma physics, and others. The obtained results are in the form of kink, singular, bright, periodic, and dark solitons. The observed results are verified and represented by 2D and 3D graphs. The observed results are not present in the literature due to the use of fractional derivatives. The impact of the truncated M-fractional derivative on the observed results is also represented by graphs. Hence, our observed results are fruitful for the future study of these models. The applied techniques are simple, fruitful, and reliable in solving the other models in applied mathematics. [ABSTRACT FROM AUTHOR]