Higher-order Dispersion Effects Research Articles

Modulation instability, state transitions and dynamics of multi-peak rogue wave in a higher-order coupled nonlinear Schrödinger equation

The coupled nonlinear Schrödinger equation provides an effective description of the propagation of four optical fields in the fibers. The present work explores the dynamics of rogue waves that arise in a coupled nonlinear Schrödinger equation of higher-order that is acquired using the Ablowitz-Kaup-Newell-Segur method. The generalized (m,N−m)-fold Darboux transformation is used to determine the exact rogue wave solutions, which show multiple peaks and depressions. In addition, a detailed analysis is conducted of the modulation instability of three different kinds of plane-wave solutions. The results indicate that the coefficients of the second and third derivative terms, rather than just the coefficients of the third derivative term, have an impact on modulation instability. Lastly, the transition between rogue waves and solitons under the influence of higher-order dispersion effects is elucidated, with explicit soliton solutions provided within the modulation stability region.

Analytical and numerical solutions to the generalized Schrödinger equation with fourth-order dispersion and nonlinearity

This study aims to solve the generalized Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity (4-Order-GNLSE) using advanced analytical and numerical methods. The 4-Order-GNLSE is critical in understanding complex wave propagation phenomena in various physical contexts, including optical fibers and fluid dynamics, where higher-order dispersion and nonlinear effects are significant. We employ the Khater II (Khat II) and modified Rational (MRat) methods to construct analytical solutions. To validate these solutions, we utilize the trigonometric-quantic-B-spline (TQBS) method as a numerical scheme, demonstrating a close match between analytical and numerical results. Our findings indicate that the proposed methods effectively capture the intricate dynamics governed by the 4-Order-GNLSE. The results highlight the relevance of these solutions in practical applications, offering new insights into the wave propagation behaviors influenced by higher-order dispersion and nonlinearities. This research provides a significant contribution to the field of nonlinear wave equations, presenting novel solutions and validating methodologies that enhance our understanding and potential applications of these complex systems.

Cavity soliton formation in Kerr comb oscillators via input phase and amplitude modulation in the presence of high order dispersion

The generation of dissipative cavity solitons (CSs) in Kerr microresonators through pulsed driving is an effective method for generating coherent optical microcombs. Yet, the underlying physics has not been fully investigated. Our study investigates the effect of high-order dispersion on the generation of Kerr frequency combs in Si3N4 microresonator driven by phase- and amplitude-modulated inhomogeneities of the pump which had not been studied before. Here, we present a numerical and theoretical analysis of the effects of desynchronization and the pulsed driving chirp parameter on the train of solitons circulating in the cavity for various detunings. In this case, deterministic production of a K-band single soliton, as well single breather soliton has been investigated. Our results show that high-order dispersion can also affect CSs positioning, stability, and existence.

Modulation instability and localized wave excitations for a higher-order modified self-steepening nonlinear Schrödinger equation in nonlinear optics

This paper investigates a higher-order modified nonlinear Schrödinger equation with higher-order dispersion and self-steepening effects, which can be used to study the dynamics of asymmetric and steepened optical pulse transmission in optical fibres. The modulation instability of the plane-wave solution has been analysed, and the state transition of the rogue waves under the higher-order dispersion effect is presented. The findings demonstrate that the self-steepening effect may also produce an asymmetric unstable frequency band. Combined with the modulation instability, the rogue wave, rational soliton and mixed interaction of localized waves have a quantitative relationship with plane-wave parameters. Various exact solutions can be accurately located and obtained according to the generalized Darboux transformation. The asymptotic analysis of rational solutions demonstrates the state transition of higher-order rogue waves. This paper illustrates the significance of modulation instability for studying the integrable systems by the Darboux transformation. It is a new guidance to solve the difficulty of exact excitation of asymmetric localized waves in complex integrable systems. The results have prospective applications and references for the emergence, amplification and compression of asymmetric pulses in optical experiments.

Effective Control of Three Soliton Interactions for the High-Order Nonlinear Schrödinger Equation

We take the higher-order nonlinear Schrödinger equation as a mathematical model and employ the bilinear method to analytically study the evolution characteristics of femtosecond solitons in optical fibers under higher-order nonlinear effects and higher-order dispersion effects. The results show that the effects have a significant impact on the amplitude and interaction characteristics of optical solitons. The larger the higher-order nonlinear coefficient, the more intense the interaction between optical solitons, and the more unstable the transmission. At the same time, we discuss the influence of other free parameters on third-order soliton interactions. Effectively regulate the interaction of three optical solitons by controlling relevant parameters. These studies will lay a theoretical foundation for experiments and further practicality of optical soliton communications.

Wave-speed management of dipole, bright and W-shaped solitons in optical metamaterials

Wave-speed management of soliton pulses in a nonlinear metamaterial exhibiting a rich variety of physical effects that are important in a wide range of practical applications, is studied both theoretically and numerically. Ultrashort electromagnetic pulse transmission in such inhomogeneous system is described by a generalized nonlinear Schrödinger equation with space-modulated higher-order dispersive and nonlinear effects of different nature. We present the discovery of three types of periodic wave solutions that are composed by the product of Jacobi elliptic functions in the presence of all physical processes. Envelope solitons of the dipole, bright and W-shaped types are also identified, thus illustrating the potentially rich set of localized pulses in the system. We develop an effective similarity transformation method to investigate the soliton dynamics in the presence of the inhomogeneities of media. The application of developed method to control the wave speed of the presented solitons is discussed. The results show that the wave speed of dipole, bright and W-shaped solitons can be effectively controlled through spatial modulation of the metamaterial parameters. In particular, the soliton pulses can be decelerated and accelerated by suitable variations of the distributed dispersion parameters.

Novel optical soliton structures for a defocusing Lakshmanan–Porsezian–Daniel optical system

In this article, under investigation is a defocusing Lakshmanan–Porsezian–Daniel optical (DLPD) system holding higher-order dispersion and higher-order nonlinear effects. Via applying the Darboux transformation method, new analytical solutions are obtained for the system. Some novel intersection solitons and their characteristics are found out under the certain parameter setting. There are four basic soliton structures: (i) bright-dark soliton with reduced amplitude at the intersection point; (ii) bright-dark soliton with increased amplitude at the intersection point; (iii) doubly-bright soliton with bright peaks; (iv) doubly-dark soliton with dark troughs. Furthermore, as the order of the solutions increases, there are more abundant and more complex structures at the intersection point. This indicates that this system can carry out more optical information by exciting higher order solitons.

Peregrination, layers’ and multi-peaks’ generation induced by cubic-quintic-saturable nonlinearities and higher-order dispersive effects in a system of coupled nonlinear left-handed transmission lines

Peregrination, layers’ and multi-peaks’ generation induced by cubic-quintic-saturable nonlinearities and higher-order dispersive effects in a system of coupled nonlinear left-handed transmission lines

Nonlinear Landau-Zener tunneling under higher-order dispersion

We studied nonlinear Landau-Zener tunneling under the effect of higher-order dispersion, transformed the Gross-Pitaevskii (GP) equation with higher-order dispersion terms into a nonlinear two-energy level form using a two-mode approximation, and comprehensively analyzed the loop structure of the lowest-energy band and the nonlinear Landau-Zener tunneling. The results show that, when third-order dispersion coefficient is not zero, there is a spatial inversion symmetry breaking of the energy band structure and unbalanced Landau-Zener tunneling. By analyzing the fixed point's nature of the classical Hamiltonian, we obtain a law for the variation of the loop structure and adiabatic tunneling probability with the dispersion coefficient, consistent with our numerical analysis results. In addition, we analyzed the real-time evolution of solitons with transverse bias and observed the tunneling phenomenon. Consistent with our analysis, the adjustment of the dispersion term can effectively control optical tunneling, which provides an alternative idea for optical switching.

Exploration of new solitons in optical medium with higher-order dispersive and nonlinear effects via improved modified extended tanh function method

Throughout this article, the improved modified extended tanh function method is presented for deriving fresh travelling wave alternative solutions in different forms of nonlinear Schrödinger equation (NLSE) with higher-order dispersive and nonlinear effects that describes the propagation of ultra-short optical solitons in an optical fiber medium. For this model, we have a variety of new solutions, including solitons (bright, dark and singular), exponential, singular periodic and Jacobi elliptic solutions. Graphical representations of selected solutions are provided to demonstrate the physical properties of the raised solutions.

The modulation instability of shallow wake flows based on the higher-order generalized cubic-quintic complex Ginzburg–Landau equation

In the context of the parallel flow hypothesis, we derive a higher-order generalized cubic-quintic complex Ginzburg–Landau (GCQ-CGL) equation to describe the amplitude evolution of shallow wake flow from the dimensionless shallow water equations by using multi-scale analysis, perturbation expansion, and weak nonlinear theory. The evolution model includes not only the slowly changing envelope approximation but also the influence of higher-order dissipation, dispersion, and cubic and quintic nonlinear effects. We give the analytical solution of the higher-order GCQ-CGL equation based on the ansatz and coordinate transformation methods, and we discuss the influence of the higher-order dissipation coefficient on the amplitude and frequency of the wake flow by means of three-dimensional diagrams, contour maps, and plane graphs. The subsequent linear stability analysis gives a theoretical basis for the modulation instability (MI) of plane waves, and the linear theory predicts the instability of any amplitude of the main waves. Finally, we focus on the MI of shallow wake flows. Results show that the MI gain function is internally related to the background wave number, disturbance wave number, background amplitude, disturbance expansion parameter, and dissipation coefficient. The area of the MI decreases as the higher-order dissipation coefficient decreases.

SPICE-Compatible Equivalent Circuit Models for Accurate Time-Domain Simulations of Passive Photonic Integrated Circuits

Passive photonic integrated circuits (PICs) can be easily characterized in the frequency-domain, but their accurate time-domain performance evaluation is a hurdle for system-level designers, especially when dealing with resonant circuits having highly dispersive behavior, such as ring resonators. In this paper, a new equivalent circuit modeling and simulation approach is proposed, based on the Complex Vector Fitting algorithm, able to perform accurate and robust time-domain simulations of passive PICs directly in standard SPICE simulators. The proposed modeling technique starts from scattering parameters of passive PICs, and is able to capture linear and high order dispersion, backscattering, and wavelength dependent effects. Considering the different nature of optical and electronic signals, a novel concept of equivalent voltage and current for optical waveguides is proposed to simplify the optical to electronic ports conversion and to make it possible to connect and terminate the equivalent circuit models as needed in SPICE simulators, natively supporting bidirectional signal propagation in a waveguide. This work provides a precise and reliable solution to evaluate time-domain characteristics of passive PICs and to access any internal nodes within a circuit, such as the signals inside a ring resonator. Three examples of time-domain simulations of passive PICs in commercial SPICE simulators are presented to demonstrate the flexibility and advantages of the proposed technique.

Periodic and localized waves in parabolic-law media with third- and fourth-order dispersions.

Considering the higher-order nonlinearity is essential in a broad range of real physical media as it significantly influences the wave dynamics in these systems. We study the propagation of femtosecond light pulses inside an optical fiber medium exhibiting higher-order dispersion and cubic-quintic nonlinearities. Pulse evolution in such a system is governed by a higher-order nonlinear Schrödinger equationincorporating second-, third-, and fourth-order dispersions as well as cubic and quintic nonlinearities. The periodic and solitary wave solutions are identified using the equationmethod. Results presented indicated the potentially rich set of periodic waves in the system under the combined influence of higher-order dispersive effects and cubic-quintic nonlinearity. The velocity of these structures is uniquely dependent on all orders of dispersion. Conditions on the optical fiber parameters for the existence of these exact stable solutions are found by analytical stability analysis.

Linear pulse propagation with high-order dispersion

We present an approximate, but intuitively appealing theoretical study of the linear propagation of optical pulses in media with high-order dispersion. Our analysis, which is fully consistent with numerical simulations, is based on the pulses’ full-width at half maximum and shows that the effect of high-order dispersion differs significantly from that of the well-understood second order dispersion. For high dispersion orders m, the central part of the pulses, where the intensity is highest, evolve in the same way, independent of m, though at different rates, with a weak dependence on the initial pulse shape. We also find that all pulses, irrespective of initial pulse shape, eventually evolve to a sinc function. Our treatment allows us to find expressions for the characteristic dispersion lengths for high dispersion orders.

The effects of higher higher-order dispersion on pulse propagation

We give explicit formulas for the propagation of a pulse in a dispersive medium governed by a general dispersion relation. In particular, we consider the mean and standard deviation of a propagating pulse and relate them to the parameters of the initial pulse and the dispersion relation. If the dispersion relation is expanded in a Taylor series, higher-order dispersion is when there are terms that are higher than quadratic. We show the effects of the higher-order dispersion on the propagation of the pulse's mean, standard deviation, and contraction/expansion time. Explicit examples will be given.

Dark solitary pulses and moving fronts in an optical medium with the higher-order dispersive and nonlinear effects

We study the propagation dynamics of extremely short light pulses in an optical fiber medium exhibiting a plethora of higher-order dispersive and nonlinear effects of different nature. The wave propagation in such nonlinear media is described by an extended nonlinear Schrö dinger equation incorporating both even and odd higher-order terms. We introduce an appropriate nonlinear equation which enables us to achieve analytical shape-preserved and periodic wave solutions for the model. A new dark solitary wave is identified for the first time in the presence of all physical processes. In addition, we show that moving fronts or optical shock-type soliton solutions exist for the considered model. The periodic wave solutions are also found together with the parametric conditions for their existence. Results in this study may be useful for experimental realization of shape-preserved pulses in optical fibers and further understanding of their optical transmission properties.

Study of the effect of higher-order dispersions on photoionisation induced by ultrafast laser pulses applying a classical theoretical method

We investigated the effect of higher order dispersion on ultrafast photoionisation with Classical Trajectory Monte Carlo (CTMC) method for hydrogen and krypton atoms. In our calculations we used linearly polarised ultrashort 7 fs laser pulses, 6.5 times 10^{14} mathrm {W/cm^{2}} intensity, and a central wavelength of 800 nm. Our results show that electrons with the highest kinetic energies are obtained with transform limited (TL) pulses. The shaping of the pulses with negative second- third- or fourth- order dispersion results in higher ionisation yield and electron energies compared to pulses shaped with positive dispersion values. We have also investigated how the Carrier Envelope Phase (CEP) dependence of the ionisation is infuenced by dispersion. We calculated the left-right asymmetry as a function of energy and CEP for sodium atoms employing pulses of 4.5 fs, 800 nm central wavelength, and 4 times 10^{12}mathrm {W/cm^{2}} intensity. We found that the left-right asymmetry is more pronounced for pulses shaped with positive Group Delay Dispersion (GDD). It was also found that shaping a pulse with increasing amounts of GDD in absolute value blurs the CEP dependence, which is attributed to the increasing number of optical cycles.

M-shaped and other exotic solitons generated by cubic-quintic saturable nonlinearities in a nonlinear electrical transmission network with higher-order dispersion effects

This investigation presents the generalized cubic-quintic Guizburg-Landau (GCQGL) equation with higher-order dispersion effects in a nonlinear electrical transmission network. Based on the collective coordinates' theory with a conventional Gaussian Ansatz, the study reveals that the waves which propagate in this medium are exotic solitons such as M-shaped solitons. The investigation presents the stability of the dark soliton subjected to the quintic-phase modulation. Besides, the introduction of the cubic- and quintic-saturable nonlinearities induce the generation of twin narrow solitons, M-shaped soliton and Sasa-Satsuma soliton. Moreover, the introduction of the third-order dispersion (TOD) and the fourth-order dispersion (FOD) provokes the generation of twin large solitons associated to symmetric radiation lobes and twin narrow solitons associated to the damping effect. Some exact field solutions of the GCQGL equation are also illustrated. Moreover, some theoretical frequencies are found where significant results are exposed. In addition, physical conditions associated to the saturation lead to a particular internal perturbation. This internal excitation is measured in detail by the collective coordinates in order to build-up exotic solitons. This technique improves the comprehension of some exotic waves' mechanism of generation in nonlinear electrical transmission network.

Dust Acoustic Dressed Solitons in Jovian Magnetosphere

In this investigation, the evolution of dust acoustic solitons (DASs) and dust acoustic dressed solitons (DADSs) have been examined in the Jovian magnetosphere. A five components plasma in which positive dust grains interact with solar wind protons, superthermally distributed electrons, ions as well as solar wind electrons is considered to explore the formation of different kinds of nonlinear structures in the Jovian magnetosphere. The Korteweg-de Vries (KdV) and inhomogenous KdV-type equations encompassing the higher-order (HO) nonlinear and dispersion (N&D) effects are derived by employing the reductive perturbation technique (RPT). With the incorporation of HO effects, a new kind of DADSs are evolved. The influence of various plasma parameters viz., superthermality of ions as well as electrons, temperature, and concentration of different species on the salient features of DASs and DADSs has been encapsulated. The findings of this work may be useful to elucidate the formation of nonlinear dust acoustic coherent structures in the Jovian magnetosphere.

Influence of higher-order nonlinear effects on optical solitons of the complex Swift-Hohenberg model in the mode-locked fiber laser

The complex Swift-Hohenberg equation (CSHE) has been widely studied in recent years. It is a more real model in the mode-locked fiber laser with the saturable absorber due to the addition of the diffraction term compared with the complex Ginzburg-Landau equation. In this work, the dynamics process of the traditional soliton and M−type soliton pulses in the mode-locked fiber laser is demonstrated based on the CSHE model with higher-order nonlinear effects. The results show that with the increase of the small signal gain coefficient, the number of soliton molecules adds gradually. Under the same transmission conditions, the transmission of M−type solitons in the laser are more stable than that of single solitons. By adding the self-steepening effect, it can be found that the time-domain shift due to higher-order dispersion effects is compensated. The self-frequency shift effect caused by the Raman scattering can produce not only time domain shift, but also frequency domain shift. Moreover, the addition of higher-order diffraction term can describe the spectral response of multiple peaks, and makes the pulse spectrum show the asymmetric propagation in the transmission process. Finally, the increase of the length of the single-mode fiber on the right side of the gain fiber in the optical circuit will not only shift the center position of the output pulse backward, but also make the pulse energy show a ladder type downward trend.