Modulation Instability, Dark and Singular Soliton for Weakly Nonlocal Schrodinger Equation

This study explores the dynamics of highly dispersive optical solitons in nonlinear Schrödinger equations (NLSE) with non-local self-phase modulation (SPM) and polarization-mode dispersion (PMD). These nonlinear effects significantly influence soliton propagation and stability in advanced optical communication systems. Employing the Improved Modified Extended Tanh-Function Method (IMETFM), we derive exact soliton solutions, including bright, dark, singular, and combo solitons, under specific parametric conditions. The IMETFM effectively handles the complexity of the NLSE, incorporating higher-order dispersion terms (up to sixth-order) and non-local nonlinearities. Additionally, we perform a modulation instability (MI) analysis to examine the stability of steady-state solutions. This analysis uncovers the conditions under which instabilities emerge due to the interplay between dispersion and nonlinearity. The MI study offers critical insights into the growth of wave perturbations, thereby advancing the understanding of soliton stability dynamics. Graphical representations of the solutions illustrate their behavior, emphasizing the impact of non-local SPM, PMD, and MI on soliton dynamics. These findings offer valuable insights for optimizing high-capacity optical communication systems and fiber laser technologies, with broader implications for nonlinear wave phenomena in birefringent fibers and other nonlinear physical systems. This work advances the theoretical framework for soliton dynamics and lays a foundation for future experimental validations and practical applications in nonlinear optics.

Role of higher order dispersion on instability criterion of saturable fiber system with Non-Kerr Nonlinearities

Our study delves into the intricate interplay of various factors within metamaterials, with a focus on modulation instability. Through our research, we elucidate the intricate dynamics involving intermodal dispersion, self-steepening effect, higher order dispersion, and plane wave amplitude, showcasing their competition and influence on modulation instability phenomena. We aim to explore the impact of intermodal dispersion and higher-order effects by numerically solving the generalized nonlinear Schrödinger equation (NLSE), which models the propagation of a few-cycle pulse in a nonlinear metamaterial. Our modulation instability (MI) analysis captures the complex dynamics these factors introduce. We demonstrate the spatiotemporal evolution of MI under different parameter values, revealing how these variations influence the instability’s development and characteristics. This approach provides a detailed understanding of the system’s behavior across various conditions, highlighting the roles of intermodal dispersion and higher-order effects. We demonstrate that the behavior of modulation instability bands and their reliance on parameters such as self-steepening and wave amplitude is contingent upon the specific characteristics of the optical setup and medium dispersion properties

Study on the regulation of amplitude for the optical soliton through nonlinear effects in the optical communication system

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

We consider nonlinear effects associated with the propagation of a quasi‐monochromatic whistler wave packet along the inhomogeneous geomagnetic field. We show that under the specific conditions of the Siple experiment the whistler packet is subject to the modulational instability, resulting in longitudinal and transverse modulation of the envelope amplitude and the excitation of quasi‐neutral density perturbations akin to ion acoustic waves. In particular, we examine the behavior of the wave packet in the vicinity of the region where the group velocity of the packet is a maximum. In general, the wave packet is subject to the competing effects of dispersive broadening and the nonlinear tendency to steepen. In the vicinity of the region of interest the dispersive broadening is small, and the tendency for steepening prevails. This results in the creation of anomolously narrow solitary waves. We analyze the situation analytically using a modification of the nonlinear Schrodinger equation. We evaluate the conditions required for the onset of the modulational instability and find its typical growth rate. We then construct nonlinear traveling wave solutions. These take the form of solitons, which move toward the rear edge of the wave packet. Because the dispersive effects are small, these solitons are narrow structures with steep boundaries. We then analyze the possibility that these solitons act as drivers for triggered emission. The interaction of resonant particles with the wave at the soliton leads to particle bunching in velocity. The resulting current serves as a source term for monochromatic radiation, the frequency of which changes with time as the bunches move in the inhomogeneous geomagnetic field.

In this paper, we analytically solve the coupled equations of a \(\mathcal {PT}\)-Symmetric NLDC by considering high-order dispersion and nonlinear effects (Raman Scattering and self-steeping) simultaneously in normal dispersion regime. To the best of knowledge no works has been done in previous studies to decoupled these equations and obtain an exact analytical solution. The new exact bright solitary solutions are derived. In addition, to study the stability and instability of these propagated solitons in a \(\mathcal {PT}\)-Symmetric NLDC, perturbation theory is used. Numerical methods are applied to find perturbed eigenvalues and eigenfunctions. The Stability of obtained four perturbed eigenvalues and perturbed eigenfunctions for a \(\mathcal {PT}\)-Symmetric NLDC equations regard to high-order effects are examined. Using these results and simulating the propagation of perturbed temporal bright solitons through \(\mathcal {PT}\)-Symmetric NLDC show that perturbed solitons are mostly stable. This means that high-order dispersion and nonlinear effects canceled each other and do not affected the propagated solitons. Furthermore, the evolution of perturbed solitons energies match well the previous results and confirmed the stability of these solitons in a \(\mathcal {PT}\)-Symmetric NLDC. As seen the energies of pulses in bar and cross behave in two manner 1) the exchange of energy is happened in some periods, but the shape of each pulse in bar and cross is preserved. Therefore, the solitons under this eigenfunction perturbation are mostly stable. 2) the evolution of energy in the bar and cross, demonstrate that there is no changes in their energies and they remain constant. It is straightforward to show that in spite of considering high-order effects, the perturbed soliton conserve the shape and it remain stable. The deliverables of this article not only demonstrate a novel approach to ultrafast pulses, solitons and optical couplers, but more fundamentally, they could give insight for improving the new medical equipments technologies, enabling innovations in nonlinear optics and their usage in designing new communication systems and Photonic devices.

The impact of Raman scattering and modul ation instability is studied in Brillouin time-domain analysis systems. It turns out to be very detrimental for long-range sensing as a result of the extended interaction length combined to the high pump peak pulse power. The conditions under which these effects limit the sensing range are determined and the modeling is very well confirmed by experimental results. Keywords: Fibre optics, optical fibre sensor, stimulated Brillouin scattering, distributed fibre sensor, non-linear optics, stimulated Raman scattering, modulation instability. 1. INTRODUCTION The recent raising interest in Brillouin fiber sensors for application in civil engineering, in oil & gas industry, in perimeter security and in intrusion detection has stimulated the research efforts to extend the detection range. In time-domain distributed Brillouin sensors (BOTDA) pulses are used to interrogate locally the interaction along the fiber. In order to achieve better performances in terms of range and spatial resolution, pump pulse and signal powers must be raised. These powers cannot be made arbitrarily large and must be kept below the observation threshold of any other noise-fed nonlinear effects. In these systems the most critical nonlinear effects are those in which a signal wave is amplified through a forward interaction, in other words when the background noise is amplified while co-propagating with the pump pulse. In such a situation the interaction length may cover many tens of kilometers and even a weak nonlinear amplification much less efficient than stimulated Brillouin scattering (SBS) may result in a large amplified spontaneous signal that eventually depletes the pump. The id entified nonlinear effects are the modulation instability (MI), which occurs only in fibers presenting an anomalous dispersion at the pump wavelength, and the Raman scattering (RS). A spectral self-broadening of the pump due to MI was observed in sensors based on spontaneous Brillouin scattering

With the use of cubic, quintic, and septic nonlinearities, we demonstrate the influence of modified nonlinear saturation on modulational instability (MI) in a nonlinear complex parity–time (PT)-symmetric fiber Bragg grating (FBG) structure. Using a modified coupled nonlinear Schrodinger equation and linear stability analysis, we derive a dispersion relation for instability gain spectra in a complicated PT-symmetric system. Our main aim is to examine the MI in non-Kerr nonlinearities with nonlinear saturation in three PT-symmetric regimes: below threshold point, at threshold point (breaking point), and above threshold point. The occurrence of MI is known to be problematic at the PT-symmetry threshold point in a standard FBG structure (A.K. Sharma, 2014). At the same time, MI can exist in the normal group velocity dispersion domain when the modified nonlinear saturation effect is used. With the help of a modified form of saturable nonlinearity, we discovered that MI could exist in all three regimes in a complex PT-symmetric FBG structure. In anomalous group velocity dispersion alone, we found bistability behavior in a PT-symmetric FBG structure with higher-order saturable nonlinearity. In the presence of a modified nonlinear saturation effect and higher-order non-Kerr nonlinearities, we found a novel type of dynamics in the PT-symmetric FBG structure. All alterations in the photonic device bandgap directly result from changes in the refractive index of the medium caused by the interaction of PT-symmetric potential with the cubic–quintic–septic and modified form of nonlinear saturation. As a result, we provide approaches for generating and managing the MI in a complex PT-symmetric FBG structure under the influence of the modified nonlinear saturation effect.

In the present study, the nonlinear modulation of transverse waves propagating in a cubically nonlinear dispersive elastic medium is studied using a multi-scale expansion of wave solutions. It is found that the propagation of quasi-monochromatic transverse waves is described by a pair of coupled nonlinear Schrodinger (CNLS) equations. In the process of deriving the amplitude equations, it is observed that for a specific choice of material constants and wavenumber, the coefficient of nonlinear terms becomes zero, and the CNLS equations are no longer valid for describing the behaviour of transverse waves. In order to balance the nonlinear effects with the dispersive effects, by intensifying the nonlinearity, a new perturbation expansion is used near the critical wavenumber. It is found that the long time behaviour of the transverse waves about the critical wavenumber is given by a pair of coupled quintic nonlinear Schrodinger (CQNLS) equations. In the absence of one of the transverse waves, the CQNLS equations reduce to the single quintic nonlinear Schrodinger (QNLS) equation which has already been obtained in the context of water waves. By using a modified form of the so-called tanh method, some travelling wave solutions of the CQNLS equations are presented.

Modelling and simulation of surface water waves

This paper focuses on the study of a Nonlinear Electric Transmission Network (NETN). Addressing key engineering challenges such as modulation instability, insufficient numerical calculation accuracy, and limitations in system stability evaluation, it proposes three core innovations and clarifies their practical application value. First, by employing the second-kind Jacobi elliptic equation method, a large number of Jacobi elliptic voltage solutions with double-periodic characteristics are discovered in the modeling of complex high-order circuit transmission line differential equations. These solutions can accurately characterize the complex variation patterns of voltage physical quantities, providing a basis for predicting voltage fluctuations in the power supply systems of key facilities (e.g., communication base stations and core computer rooms). This effectively alleviates signal distortion caused by modulation instability, enhances the reliability of communication signal transmission, and reduces data error rates. Second, the equation is transformed into a unique dynamical structure, and a fourth-order NETN model is constructed to explore the three-dimensional dynamical system, phase portraits, and Lyapunov exponent. Breaking through the limitations of traditional second-order models, this model provides a more precise framework for system stability evaluation, enabling early identification of instability risks (e.g., equipment damage caused by harmonics) during power transmission and avoiding unexpected communication interruptions and economic losses. Third, the six-stage, fifth-order Kutta–Nyström method is innovatively applied to the numerical calculation of NETN, significantly improving calculation accuracy. This provides efficient support for circuit parameter optimization (e.g., adjusting the dispersive element Cs to balance dispersion and nonlinear effects), which not only reduces power loss caused by impure signal spectra but also helps lower carbon emissions of the power grid, aligning with the energy-efficient operation requirements of modern power systems. These innovations fill the research gaps related to NETN and provide key technical support for the engineering optimization and stable operation of nonlinear transmission networks.

Davydova-Lashkin-Fokas-Lenells equation (DLFLE) is a gauged equivalent form of Fokas-Lenells equation (FLE) that addresses both spatio-temporal dispersion (STD) and nonlinear dispersion (ND) effects. The balance between those effects results a soliton which has always been an interesting topic in research due to its potential applicability as signal carrier in information technology. We have induced a variation to the dispersion effects and apply Hirota bilinear method to realise soliton solution of the proposed DLFLE and explore how the soliton dynamic behaves in accordance to the variation of the dispersion effects. The proposed equation is applicable for number of systems like ultrashort optical pulse, ion-cyclotron plasma wave, Bose-Einstein condensate (BEC) matter-wave soliton under certain external fields, etc. The study on such systems under varying effects is very limited and we hope our work can benefit the researchers to understand soliton dynamics more and work on various other nonlinear fields under varying effects.

Chapter 7 - Fiber-Optic Communications

[1] Dissipation of ocean swell, inferred from published oceanographic data, is investigated to determine if laboratory results on the dissipative stabilization of narrow-banded wave trains are applicable to ocean swell. Three issues are addressed. (i) Dimensional decay rates of ocean swell are about a million times smaller than typical decay rates of laboratory waves. Nevertheless, when decay rates are nondimensionalized using scales of dispersive and nonlinear effects, the dimensionless decay rates of ocean swell are comparable to those of laboratory waves, indicating that dissipation and nonlinear effects can influence ocean swell on the same time scale. (ii) The stability of ocean swell to small perturbations is examined within the theoretical framework of nonlinear Schrödinger-type models that either do or do not include dissipation. As in laboratory experiments, for swell with small enough nonlinearity, dissipation can inhibit and eventually stop the growth of small perturbations before nonlinearity becomes important. And as in laboratory experiments, we document herein an example of ocean swell with stronger nonlinearity that exhibits frequency downshifting, which is not predicted by any nonlinear Schrödinger-type model, including higher-order models, with or without dissipation. (iii) Given that dissipation can influence the evolution of ocean swell, we compare the predicted decay rates of four (published) dissipative models with observed decay rates, both in the ocean and in a laboratory wave tank. The model that presupposes an inextensible film on the free surface agrees best with measured rates of dissipation of ocean swell.