Dark Optical Solitons Research Articles

Sub-pico-second chirped optical solitons in birefringent fibers for space--time fractional Kaup-Newell equation

The present work is devoted to investigate the chirped bright and dark optical solitons of fractional Kaup-Newell equation (KNE) in birefringent fibers. The study is carried out analytically by the traveling wave hypothesis with the conformable derivative which reduces the governing model to an ordinary differential equation (ODE). The obtained equation is handled with the aid of an exotic integration scheme that utilizes the Jacobi elliptic equation in the form of a first-order nonlinear ODE with three-degree terms. Taking the modulus of Jacobi elliptic function to unity, distinct types of bright and dark optical solitons are derived with their corresponding chirping. The fractional order derivative is noted to have a significant influence on the pulse propagation. Additionally, the nonlinearity amount causes also marked variations in the amplitude and width of solitons. The modulation instability of the KNE is reported by implementing the linear stability analysis which confirms that all solutions are stable. The revealed results can be capitalized in improving the relevant physical and engineering applications in the field of birefringent fiber.

Bright and dark optical solitons in optical metamaterials using a variety of distinct schemes for a generalized Schrodinger equation

Purpose The purpose of this study is to investigate the nonlinear Schrödinger equation (NLS) incorporating spatiotemporal dispersion and other dispersive effects. The goal is to derive various soliton solutions, including bright, dark, singular, periodic and exponential solitons, to enhance the understanding of soliton propagation dynamics in nonlinear metamaterials (MMs) and contribute new findings to the field of nonlinear optics. Design/methodology/approach The research uses a range of powerful mathematical approaches to solve the NLS. The proposed methodologies are applied systematically to derive a variety of optical soliton solutions, each demonstrating unique optical behaviors and characteristics. The approach ensures that both the theoretical framework and practical implications of the solutions are thoroughly explored. Findings The study successfully derives several types of soliton solutions using the aforementioned mathematical approaches. Key findings include bright optical envelope solitons, dark optical envelope solitons, periodic solutions, singular solutions and exponential solutions. These results offer new insights into the behavior of ultrashort solitons in nonlinear MMs, potentially aiding further research and applications in nonlinear wave studies. Originality/value This study makes an original contribution to nonlinear optics by deriving new soliton solutions for the NLS with spatiotemporal dispersion. The diversity of solutions, including bright, dark, periodic, singular and exponential solitons, adds substantial value to the existing body of knowledge. The use of distinct and reliable methodologies to obtain these solutions underscores the novelty and potential applications of the research in advancing optical technologies. The originality lies in the novel approaches used to obtain these diverse soliton solutions and their potential impact on the study and application of nonlinear waves in MMs.

Dynamics of invariant solutions of the DNA model using Lie symmetry approach

The utilization of the Lie group method serves to encapsulate a diverse array of wave structures. This method, established as a robust and reliable mathematical technique, is instrumental in deriving precise solutions for nonlinear partial differential equations (NPDEs) across a spectrum of domains. Its applications span various scientific disciplines, including mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and several others. This research focuses specifically on the crucial molecule DNA and its interaction with an external microwave field. The Lie group method is employed to establish a five-dimensional symmetry algebra as the foundational element. Subsequently, similarity reductions are led by a system of one-dimensional subalgebras. Several invariant solutions as well as a spectrum of wave solutions is obtained by solving the resulting reduced ordinary differential equations (ODEs). These solutions govern the longitudinal displacement in DNA, shedding light on the characteristics of DNA as a significant real-world challenge. The interactions of DNA with an external microwave field manifest in various forms, including rational, exponential, trigonometric, hyperbolic, polynomial, and other functions. Mathematica simulations of these solutions confirm that longitudinal displacements in DNA can be expressed as periodic waves, optical dark solitons, singular solutions, exponential forms, and rational forms. This study is novel as it marks the first application of the Lie group method to explore the interaction of DNA molecules.

Highly Dispersive Optical Gap Solitons with Kundu-Eckhaus Equation Having Multiplicative White Noise

This paper explores the dynamics of highly dispersive gap solitons within the framework of the Kundu-Eckhaus equation, augmented by the inclusion of multiplicative white noise in the Itô sense, a novel addition to the model. Two integration schemes, the extended simplest equation approach and the generalized Riccati equation mapping scheme, are employed to analyze and integrate the model. Despite yielding bright, singular, and dark-singular straddled solitons, both methods independently fail to capture dark optical solitons. Additionally, our investigation highlights that the influence of white noise predominantly affects the phase aspect of the solitons, with negligible impact on their amplitude. Further details regarding the specific physical system or optical medium under study would provide readers with context, aiding in the comprehension of the significance of our findings.

Analyzing optical solitons in the generalized unstable NLSE in dispersive media

In this research, we utilize innovative methods to successfully address the challenges posed by the generalized unstable nonlinear Schrödinger equation (UNLSE) in dispersive media, yielding a rich array of solutions. Our investigation delves into a wide range of adapted exponential rational function (MERF) techniques, in addition to other unique and valuable approaches. By employing these methodologies in various scenarios, we unveil an extensive variety of wave solutions for the UNLSE. Furthermore, by satisfying specific parameter conditions, we identify novel optical solutions, encompassing both bright and dark solitons within the framework of this equation.

Spin dynamic soliton in ferromagnetic materials over the (2 + 1)-dimensional beta fractional HFSC model

Ferromagnetic materials have important exploitation practice in permanent information in electromagnets, electric motors, generators, recording in thin film etc. This research explores spin dynamic soliton in Ferromagnetic materials via the (2 + 1)-D Heisenberg ferro-magnetic Spin Chains (HFSC) model. The Extended (I,R) and Extended Simple schemes treatment are applied to integrate the model. Through the technique, various novel dynamical wave solutions afford as optical dark and bright soliton envelope, spin wave envelope Black soliton, optical bell-shock interaction, optical bell-anti-shock wave, optical Dark-bright bell soliton and optical bright periodic waves from modulus plots, besides these real and imaginary profiles exhibits bright periodic wave with dark bell wave, periodically increasing–decreasing carrier wave, periodic carrier wave increase its amplitude by shock, periodic carrier wave suddenly decrease its amplitude by shock waves. The effect of fractional derivative, neighboring interaction as well as uniaxial crystal field anisotropy parameters is explored on the envelope soliton and amplitude of carrier waves. The changing value of neighboring interaction parameters are exhibited as increases wave height with the increases of parametric values, but while increasing values of the uni-axial crystal arena anisotropy factor causes the reduction of wave heights.

The phase of darkness – measuring the phase of a dark pulse

Dark optical solitons are solutions to the nonlinear Schrödinger equation in normal dispersion media with positive Kerr nonlinearity, exhibiting a discrete π phase jump. These solitons are valuable to applications within telecommunication. Recent advancements have demonstrated the generation of two-colour bright-dark soliton pairs through cross-amplitude modulation in laser cavities, resulting in mode locking. In this study we present for the first time full field characterization of the electric field of a dark pulse. We achieved this by performing Blind Frequency Resolved Optical Gating measurements using the synchronous bright pulse as the gate pulse. The retrieved dark pulse verifies the existence of the expected π phase jump in the phase of the dark pulse, confirming theoretical predictions.

Optical dark, singular and bright soliton solutions with dual-mode fourth-order nonlinear Schrödinger equation involving different nonlinearities

This research article investigates the dynamic characteristics of solitary waves within a medium exhibiting weak nonlinearity in dual-mode. Firstly, the three models are converted to dual-mode. The primary focus is to investigate the behavior of solitary waves in a system described by a fourth-order nonlinear Schrödinger (NLS) equation, which incorporates Kerr nonlinearity, weak nonlocality, parabolic law nonlinearity and diffraction. By employing ansatz methods, the study successfully derives various localized waveforms, including bright, dark and singular solitons, based on the governing model. The examination extends to investigating the influence of nonlocality and power coefficients on dynamics of bright, dark and singular solitons. The findings presented in this article represent a crucial advancement in understanding the generation of solitary waves in nonlinear media. As the examined model finds application in various fields such as plasma physics, Bose-Einstein condensates and others, these findings not only enhance our theoretical understanding but also pave the way for potential practical applications and the advancement of innovative technologies.

Lie group analysis, solitons, self-adjointness and conservation laws of the nonlinear elastic structural element equation

This study is based on the Lie group method for the nonlinear elastic structural element equation (ESE Equation). We obtain a three-dimensional Lie algebra. By utilizing this Lie algebra a four-dimensional optimal system is constructed. The governing ESE Equation is converted to nonlinear ordinary differential equations (ODEs) via symmetry reduction. We use a modified auxiliary equation (MAE) procedure to deal with nonlinear ODEs. These ODEs reveal the dynamics of the periodic and soliton solutions. We obtain soliton solutions through rational, trigonometric, and hyperbolic functions. Wolfram Mathematica simulations vividly illustrate the wave characteristics of the derived solutions, affirming their properties as singular periodic solutions, a singular solution, an optical dark soliton solution, and a singular soliton solution. We also obtain the local conservation laws by a new conservation theorem introduced by Ibragimov.

Formation and manipulation of optical soliton pulse in a self- and cross-phase modulated four-level inverted Y-type system

We have proposed a theoretical model for controllable optical soliton pulse formation in a magnetic-field-assisted four-level inverted Y-type atomic system. The probability amplitudes of the pertinent levels are used to analyze the contrast between self- and cross-phase modulated probe absorption signals. The absorption profiles obtained from the probability amplitudes and the zeroth-order of the pulse propagation constant in the Taylor series are appropriately congruent. The nonlinear Schrödinger equation derived from the Maxwell–Bloch equation determines the respective bright and dark soliton conditions with positive and negative values of the pulse shape parameter. It is demonstrated that the coupling laser field strengths, laser frequency detunings, and the external magnetic field’s direction are the controlling parameters to produce both bright and dark optical soliton pulses. Alongside, the shape of the optical soliton pulses in self- and cross-phase modulation is influenced by the polarization angle of the probe and coupling fields. It is also found that the variation of coupling field strengths may be used to regulate the group velocity of the soliton probe pulse from superluminal to subluminal modes or vice-versa.

Variety of optical solitons for perturbed Fokas–Lenells equation through modified exponential rational function method and other distinct schemes

In this new work, new methods are proposed to solve the (2+1)-dimensional nonlinear perturbed Fokas–Lenells (pFL) equation and abundant solutions are reported. Novel techniques are employed to acquire exact solutions of the (pFL) equation. The current work addresses a variety of forms of a modified exponential rational function method and also other useful distinct schemes. Many variants of traveling wave solutions for the Fokas–Lenells (FL) equation are revealed upon using distinct cases of the proposed methods. Moreover, based on some certain conditions for the related parameters, some new optical modulated solutions, bright solitons and dark solitons are extracted for this equation.

Numerical Simulation of Highly Dispersive Dark Optical Solitons with Kerr Law of Nonlinear Refractive Index by Laplace–Adomian Decomposition Method

This work is a numerical perspective to highly dispersive dark optical solitons by using Laplace-Adomian decomposition method. The results are reported using this scheme with highly precise accuracy and the error measure is stunningly low. The surface plots, density plots and error plots are exhibited for different parameter choices. The simulations are almost an exact replica of such solitons that analytically arise from the governing system. The suggested iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions.

Temporal behavior of bright and dark spatial solitons in photorefractive crystals having both the linear and quadratic electro-optic effects based on low amplitude approximations

We investigate the formation or time evolution characteristics of bright and dark optical spatial solitons in novel photorefractive media with linear and quadratic electro-optic effects. This study was carried out at low amplitude. By the application of low amplitude approximation to the time-dependent wave (soliton) propagation equation, bright soliton solution (BSS) and dark soliton solution (DSS) were each obtained analytically, and both are exact solutions. The change in soliton width or the intensity full width at half maximum (FWHM) of two soliton types can also be calculated directly from the BSS and the DSS, providing a suitable description of the bright and dark solitons time evolution in novel photorefractive media. Only steady state solitons are formed in low amplitude, and the quasi-steady state soliton does not exist as reported at high saturation values. We compare analytical BSS with numerical calculation results of previous studies and find good agreement between analytics and numerics on the amplitude limit. In contrast to bright solitons, which can arise in the dominant linear or quadratic electro-optic effect, dark solitons in novel photorefractive media always exist under the dominant quadratic electro-optic effects only. We found that the time evolution of dark solitons from the DSS corresponds with time-independent optical spatial solitons studies in a similar media. Lastly, BSS and DSS found at low amplitude limits are fundamental soliton solutions or initial state soliton (at low peak intensity ratio regime) that can play an important role in further investigations of optical spatial solitons formation in novel photorefractive media.

Pure-cubic optical solitons to the Schrödinger equation with three forms of nonlinearities by Sardar subequation method

In this work, it is used three families of nonlinearities such as Kerr law, Power Law and Parabolic law over the High-order dispersive Nonlinear Schrödinger equation to inquire optical soliton solutions and diverse solutions by employing the Sardar Sub-equation method. Considering the constraint relation on some parameters of the NLSE, it is obtained bright and dark optical soliton solutions. However, under a certain condition on the method parameters, it is exposed singular soliton and trigonometric function solutions. More precisely, for ρ=a24b, a new form of the complex solutions are obtained. The obtained results are depicted in 2D and 3D to show out the peakon soliton in Figure 2 compared to Yıldırım et al. (2020); Zayed et al. (2020) and Rezazadeh et al. (2020). These results could probably give a new way for designing optical fibers devices to update the communication flux.

W-Shaped Bright Soliton of the (2 + 1)-Dimension Nonlinear Electrical Transmission Line

In this paper, we investigate solitary wave solutions of the nonlinear electrical transmission line by using the Jacobi elliptic function and the auxiliary equation methods. We obtain Jacobi elliptic function solutions as well as kink, bright, dark, and W-shaped solitons as a result. For specific values of the Jacobi elliptic modulus, we depict bright, dark, and W-shaped soliton solutions as suitable parameters of the structure. Using the auxiliary equation method gives the combined bright–bright and dark–dark optical solitons in optical fibers. One result emerges from this analysis: the potential parameters and free parameters of the method can be employed to degenerate W-shaped bright and dark solitons. The acquired results are general and can be used for many applications in nonlinear dynamic systems.

Optical Solitons and traveling wave solutions to Kudryashov’s equation

This paper employs a generalized transformation to investigate Kudryashov’s equation with four nonlinear terms in order to construct traveling wave solutions and optical solitons. The proposed unified ansätze framework is manipulated to derive analytical solutions, which are expressed as a combination of general solutions to simpler equations or systems of equations that are integrable or have unique known solutions. Using this technique, one can produce families of new exact solutions as well as solitary solutions. In addition to other model solutions, the scheme reveals the elliptic functions of Jacobi, Weierstrass, and Riccati. These are categorized as singular, bright, and dark optical solitons.

Bright and dark optical modulated soliton solutions for the fourth-order (2+1)-dimensional Schrödinger equation with higher-order odd and even terms

This work addresses a (2+1)-dimensional nonlinear Schrödinger (NLS) equation (NLSE) with fourth-order nonlinearity and dispersion. Some different solutions in the form of bright and dark optical modulated solitons (OMSs) to the current problem are derived using different approaches. Moreover, based on some certain conditions for the related parameters, some, bright, dark, periodic, exponential, and singular modulated solutions are obtained. The derived solutions can be helpful in understanding the bright and dark OMSs features of identical models in optical fields, fluid mechanics, and plasmas.

Bright and dark optical solitons for the concatenation model by the Laplace-Adomian decomposition scheme

Bright and dark optical solitons for the concatenation model by the Laplace-Adomian decomposition scheme

Dispersive Optical Solitons with Schrödinger–Hirota Equation by Laplace-Adomian Decomposition Approach

This paper studies dispersive bright and dark optical solitons, modeled by the Schrödinger–Hirota equation, numerically by the aid of the Adomian decomposition. The surface plots of the algorithm yielded an impressively small measure. The effects of soliton radiation are ignored.

Optical solitons to the fractional order nonlinear complex model for wave packet envelope

This research deals with the fractional order nonlinear complex model, which is a general form of nonlinear Schrodinger equation. This model demonstrates boson gas among 2 as well as 3 body collisions, nuclear hydro-dynamics allied with Skyrme energies, in addition the optical rhythm propagations within dielectric non-Kerr media. To analyze and generate complex dynamics, optical soliton solutions of the model are derived through improved generalized Riccati equation mapping scheme. As a result, bright, dark and combo bright-dark optical wave envelopes and oscillating pulse bright optical solitons are obtained. The effects of fractionality and parameters are also illustrated with few numerical graphics of the achieved results. The achieved optical (bright and dark) solitons solutions can be used in signal transmission through optical fiber communications system.