Higher-order Dispersion Terms Research Articles

The quasi-Gramian solution of a non-commutative extension of the higher-order nonlinear Schrödinger equation

The nonlinear Schrödinger (NLS) equation, which incorporates higher-order dispersive terms, is widely employed in the theoretical analysis of various physical phenomena. In this study, we explore the non-commutative extension of the higher-order NLS equation. We treat real or complex-valued functions, such as g 1 = g 1(x, t) and g 2 = g 2(x, t) as non-commutative, and employ the Lax pair associated with the evolution equation, as in the commutation case. We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation. The soliton solutions are presented explicitly within the framework of quasideterminants. To visually understand the dynamics and solutions in the given example, we also provide simulations illustrating the associated profiles. Moreover, the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns within the context of modulational instability.

Study of Soliton Interaction in Optical Fibers with Third Order Dispersion and Higher Order Nonlinear Effects

In this paper, we present a numerical approach to solve the GNLSE and analyze soliton interaction phenomena using COMSOL environment. By leveraging the capabilities of COMSOL's PDE module, we can accurately capture the dynamics of solitons and investigate their interactions. We analyze the impact of different parameters such as soliton power, initial separation distance, and dispersion characteristics on the soliton dynamics. Furthermore, we examine the role of higher-order dispersion terms in shaping the soliton interactions. Our findings demonstrate the effectiveness of the proposed numerical approach in accurately simulating and analyzing soliton interaction phenomena. The COMSOL-based methodology provides a flexible and efficient framework for studying complex nonlinear optical systems, enabling researchers to gain insights into the behavior of solitons in different media and design optimized communication systems. This paper contributes to the understanding of soliton dynamics and provides a practical tool for investigating the behavior of solitons in nonlinear dispersive media. The presented numerical approach using COMSOL opens avenues for further research in nonlinear optics and fiber optic communication systems.

Modeling of Nonlinear Sea Wave Modulation in the Presence of Ice Coverage

A model accounting for the influence of ice coverage on the propagation of surface sea waves is suggested. The model includes higher-order linear and nonlinear terms in the equation of wave motion. The asymptotic solution is obtained to account for nonlinear modulated wave propagation and attenuation. Two kinds of attenuation are revealed. The influence of the higher-order nonlinear, dispersion, and dissipative terms on the shape and velocity of the modulated nonlinear wave is studied. Despite the presence of higher-order terms in the original equation, the modulated solitary wave solution contains free parameters, which is important for the possible generation of such waves.

New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion

The aim of the present paper is to obtain and analyze new exact travelling wave solutions and bifurcation behavior of modified Zakharov-Kuznetsov (mZK) equation with higher-order dispersion term. For this purpose, the first and second simplest methods are used to build soliton solutions of travelling wave solutions. Furthermore, the bifurcation behavior of traveling waves including new types of quasiperiodic and multi-periodic traveling wave motions have been examined depending on the physical parameters. Multistability for the nonlinear mZK equation has been investigated depending on fixed values of physical parameters with various initial conditions. The suggested methods for the analytical solutions are powerful and beneficial tools to obtain the exact travelling wave solutions of nonlinear evolution equations (NLEEs). Two and three-dimensional plots are also provided to illustrate the new solutions. Bifurcation and multistability behaviors of traveling wave solution of the nonlinear mZK equation with higher-order dispersion will add some value to the literature of mathematical and plasma physics.

On soliton solutions for higher-order nonlinear Schrödinger equation with cubic-quintic-septic law

The topic of dispersive optical solitons is one of the essential areas of telecommunications engineering. This paper presents the nonlinear Schrödinger equation (NLSE) with the cubic-quintic-septic (CQS) law which includes higher-order dispersion terms. This equation examines the dynamics of dispersive optical solitons propagation in nonlinear optical fibers. The presented model is significant in mathematics and theoretical physics sciences because long-distance optical fibers are used in optical communication utilized for transoceanic and intercontinental data transmission. First, the considered equation is transformed into a nonlinear ordinary differential equation (NLODE) using the traveling wave transformation. Then, two powerful techniques, Sinh-Gordon expansion approach and the auxiliary equation method (AEM), based on the homogeneous balance rule, are operated to produce analytical solutions for the NLSE with CQS law. Thus, bright, dark, singular, and periodic-singular soliton solutions are acquired. We demonstrate the three-dimensional (3D) and two-dimensional (2D) representations to interpret the physical behavior of some derived solutions. Besides, we investigate the effects of some parameters in the model with graphical illustrations. Therefore, it can be noted that the obtained soliton solutions have a significant role in soliton dynamics.

Modulated wave patterns brought by higher-order dispersion and cubic–quintic nonlinearity in monoatomic chains with anharmonic potential

This work investigates the modulation instability and wave patterns in the extended nonlinear Schrödinger equation with higher-order dispersion and cubic–quintic nonlinearity. We use the one-dimensional monoatomic chain with anharmonic potential to derive the discrete model. From the multiple scale method combined with a quasidiscreteness approximation, we derive the cubic–quintic nonlinear schrödinger equation, and thereafter an expression of the modulation instability gain is obtained by using a linearized expression. One notices that the modulation instability growth is sensitive to the higher-order dispersion and nonlinearities terms. A split-step Fourier method is used to assess the analytical predictions. A long evolution of the continuous wave is shown to lead to the formation of the bright soliton, and Akhemediev breathers also emerge to manifest the modulation instability. We have also demonstrated that the excitation wave number generates the train of waves to confirm the fact that the continuous wave can grow exponentially with any value of the latter. We mention equally that the model of the extended cubic–quintic nonlinearity with complex envelope has opened new features of the modulated wave patterns in monoatomic chains.

Dispersive Propagation of Terahertz Pulses in a Plasmonic Fiber

The dispersion properties of surface plasmon polaritons (SPPs) during propagation on metal wires with a dielectric coating in the terahertz frequency range were investigated theoretically. An analytical expression was obtained for a pulsed electric field using the solution of Maxwell equations taking into account high-order dispersion terms. The influence of the dielectric coating on the distortion of the pulse shape was investigated. Unlike uncoated wire, the propagation of surface plasmon pulses along a coated wire is highly dispersive. It was shown that the coating leads to the appearance of a long-chirped signal with a propagation of only a few millimeters, i.e., when a terahertz pulse propagates along a coated wire, it acquires a long oscillatory tail, the frequency of which depends on time.

Dynamical behaviors of solitons for NLSE with Kudryashov’s sextic power-law of nonlinear refractive index using improved modified extended tanh-function method

The higher order nonlinear Schrödinger equation (NLSE) describes ultrashort pulse propagation in optical fibres. In this paper, we discuss analytically the non-linear schrödinger equation with higher order dispersion terms up to sixth order and sextic power law of nonlinearity with the help of the improved modified extended tanh-function scheme. New families of solutions, such as dark solitons, singular solitons, rational solutions, periodic solutions and exponential type solutions, have been extracted. Extracted solutions will be useful in the development of the communication systems. Furthermore, some solutions are graphically depicted to demonstrate their physical nature by using distinct values of the parameters.

Impact of higher-order effects on the dynamics of soliton solutions in the (3+1)D cubic-quintic-septic complex Ginzburg–Landau equation with higher-order dispersion terms

We investigate the influence of the higher-order correction term of the nonlinear refractive index and the nonlinear-gain absorption on the dynamics of soliton solutions in a three-dimensional (3D) dissipative medium described by the higher-order (3+1)-dimensional cubic-quintic-septic complex Ginzburg–Landau [(3+1)D CQS-CGL] equation with the viscous (spectral-filtering) term, self steepening, Raman effect, dispersion terms up to six, diffraction and cubic-quintic-septic nonlinearities. By mean of direct numerical simulation of the dynamical model, we show that these two parameters can have a significant impact on the dynamics of solitons govern by the above mentioned equation, since their formation deeply depends of the value of each of the above cited parameters. Some interesting particular cases have been detected concerning the combined action of the two parameters or the isolated action of each of them. Among those solutions, we have bell-shaped, snail and creeping dissipative solitons. Also, we have found that the propagation distance z is another important parameter which has allowed us to appreciate the impact of each parameter. At a given distance z or for different values of z, the formation of soliton depends on the value of the higher-order correction term and the nonlinear-gain absorption involved. We have also shown that the above solutions can be self-trapped over a huge propagation distance even in the presence of random perturbations. The discovered nonlinear dependence of the higher-order correction term and the nonlinear-gain absorption on the soliton solution can also be used for the better understanding of the (3+1)D dissipative light bullets in a doped nonlinear kerr medium.

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.

Assorted exact explicit solutions for the generalized Atangana’s fractional BBM–Burgers equation with the dissipative term

In this study, the generalized Atangana’s fractional BBM–Burgers equation (GBBM-B) with the dissipative term is investigated by utilizing the modified sub-equation method and the new G'/(bG' + G + a)-expansion method; with the aid of symbolic computations, many types of new exact explicit solutions including solitary wave solutions, trigonometric function periodic solutions, and the rational function solutions are obtained. Some 3D and 2D plots of these solutions are simulated, which show the novelty and visibility of the propagation behavior and dynamical structure of the corresponding equation. Moreover, with the selection of different values on the parameters and orders, we can deduce many types of exact solutions in special cases. We also discussed the changes and characteristics of these solutions, which can help us further understand the inner structure of this equation. The obtained solutions indicate that the approach is easy and effective for nonlinear models with high-order dispersion terms.

Abundant exact solutions of higher-order dispersion variable coefficient KdV equation

Abstract In this article, various exact solutions of the fifth-order variable coefficient KdV equation with higher-order dispersion term are studied. Because of the complexity of the exact solution of the variable coefficient t, it has a certain influence on the tension waves at the fluid interface on the gravity surface. First, the bilinear KdV equation is derived by using the Hirota bilinear method, and four mixed solutions consisting of positive quartic function, quadratic function, exponential function, and hyperbolic function are constructed. Second, the linear superposition principle is used to obtain the resonance multisoliton solution, and two cases are taken as examples to illustrate the study of resonance multi soliton solution. In addition, 3D images and contour images are drawn by mathematical symbol calculation and appropriate parameters, and the process of tension fluctuation is vividly explained by physical phenomena. The results obtained greatly expand the exact solution of the KdV equation in the existing literature and enable us to understand nonlinear dynamical systems more deeply.

Dynamical characteristics of ultrashort terahertz pulse in metamaterials

We develop a time-dependent theoretical approach to study the interaction between the ultrashort terahertz pulse and the metamaterials beyond the steady-state approximation. By using photoconductive silicon, we show that the system dispersion is changed from abnormal to normal via the classical interference effect. As a result, the terahertz pulse propagates through the material with subluminal/superluminal group velocity. We show that it is not accurate to predict the propagation characteristics of the terahertz pulses by using the steady-state solution of coupled wave equations, and the high-order dispersion terms take a significant role which attributes to the pulse shape deformation and the position-dependent group velocity modification. The theoretical approach presented here is essential of describing the propagation properties of the terahertz pulse in metamaterials and is helpful to understand the nature of quantum interference in electromagnetically induced transparency metamaterial.

Multi-clad optical fiber design for ultra-wideband modulation instability

Developments of electro-optic (EO) modulators do not have a satisfactory pace and bandwidth restrictions are still limited to several hundred GHz, thus, obtaining methods that enable using broadband optical channels are key factors in high-speed optical communications. In this paper, Modulation Instability (MI) as an approach in order to improve the performance of EO modulators, is investigated in the anomalous and zero-dispersion regimes of proposed optical fiber. In present conventional optical fibers, MI is observed over a few hundred gigahertz bandwidths at low pump power. In order to obtain ultra-wideband (a few Terahertz) bandwidths and maximum gain in the MI process at low pump power, a new structure for optical fiber is proposed. The genetic algorithm is utilized for performance optimization in optical fiber design. It is shown that the proposed optical fiber is able to support MI for a few Terahertz bands at low pump power (100 mW). Furthermore, in this analysis, it is demonstrated that higher-order dispersion terms have a strong impact on modulation instability. Finally, it is concluded that the geometrical and optical parameters of optical fiber are key factors to control modulation instability parameters (gain and bandwidth). The proposed optical fiber can be used as an optical booster in order to compensate frequency response of EO modulators, which is a considerable step forward in high-speed optical communication.

Generation of dissipative solitons in a doped optical fiber modeled by the higher-order dispersive cubic-quintic-septic complex Ginzburg-Landau equation

The generation of dissipative optical solitons is explored in doped fibers with correction effects under the activation of modulational instability (MI). The model, described by the cubic-quintic-septic complex Ginzburg-Landau equation, includes higher-order dispersion and nonlinear gradient terms. The Lange-Newell's criterion for MI of Stokes wave, boundary domains of MI, and integrated gain of MI are obtained via the linear stability analysis. Particular attention is given to the physical effect on the critical frequency detuning, especially in the normal regime, when varying the values of odd dispersion coefficients. Numerical simulations are undertaken and confronted with analytical predictions. Beyond the agreement between the linear stability analysis and trains of soliton generation, the soliton map induced by MI, along with the subsequent physical effects, is debated via bifurcation diagrams. This allows accurate prediction of transitions between various types of localized modes and well-calibrated generation of a wide variety of solitons with different energies. It is argued that knowing the center of mass and the energy of the generated structures can better characterize the long-time evolution of MI and, eventually, its nonlinear manifestations.

Consequences of non-differentiable angular dispersion in optics: tilted pulse fronts versus space-time wave packets.

Conventional diffractive and dispersive devices introduce angular dispersion (AD) into pulsed optical fields, thus producing so-called 'tilted pulse fronts'. Naturally, it is always assumed that the functional form of the wavelength-dependent propagation angle[s] associated with AD is differentiable with respect to wavelength. Recent developments in the study of space-time wave packets - pulsed beams in which the spatial and temporal degrees of freedom are inextricably intertwined - have pointed to the existence of non-differentiable AD: field configurations in which the propagation angle does not possess a derivative at some wavelength. Here we investigate the consequences of introducing non-differentiable AD into a pulsed field and show that it is the crucial ingredient required to realize group velocities that deviate from c (the speed of light in vacuum) along the propagation axis in free space. In contrast, the on-axis group velocity for conventional pulsed fields in free space is always equal to c. Furthermore, we show that non-differentiable AD is needed for realizing anomalous or normal group-velocity dispersion along the propagation axis, while simultaneously suppressing all higher-order dispersion terms. We experimentally verify these and several other consequences of non-differentiable AD using a pulsed-beam shaper capable of introducing AD with arbitrary spectral profile. Non-differentiable AD is not an exotic phenomenon, but is rather an accessible, robust, and versatile resource for sculpting pulsed optical fields.

Stability and non-linear dynamic analysis of Kerr optical frequencycombs in dual-coupled microcavities with high-order dispersion

We investigate the stability and nonlinear dynamics of the Kerr optical frequency comb inside a dual coupled microcavity with high-order dispersion effect based on the theoretical model of coupled nonlinear Schrödinger equation. The effects of different high-order dispersion parameters on the evolution and spectral characteristics of the optical field in the cavity are also explored. Theoretical results indicate that the addition of the third-order dispersion enlarges the stability domain of the parametric space and transforms the periodically varying soliton breathers and chaos into stable bright soliton. In order to obtain an accurate Kerr optical frequency comb spectral envelope, higher-order dispersion should be considered. Moreover, high-order dispersion terms have a significant effect on the spectral characteristics of the optical frequency comb, such as the spectral envelope frequency shift and the dispersive wave spectral position. Specifically, the third-order dispersion and positive fourth-order dispersion can broaden the spectrum and enhance the dispersive waves; while the negative fourth-order dispersion can suppress the dispersive wave generation and obtain a symmetric soliton frequency comb; the fifth-order dispersion can regulate the drift direction and speed of the optical solitons. The theoretical results are of great value for dispersion regulation and design and also for stability studies in double-coupled microcavity experiments.

Observations of existence and instability dynamics of near-zero-dispersion temporal Kerr cavity solitons

Dissipative Kerr cavity solitons (CSs) are persisting pulses of light that manifest themselves in driven optical resonators and that have attracted significant attention over the last decade. Whilst the vast majority of studies have revolved around conditions where the resonator exhibits strong anomalous dispersion, recent studies have shown that solitons with unique characteristics and dynamics can arise under conditions of near-zero-dispersion driving. Here, we report on experimental studies of the existence and stability dynamics of Kerr CSs under such conditions. In particular, we experimentally probe the solitons' range of existence and examine how their breathing instabilities are modified when group-velocity dispersion is close to zero, such that higher-order dispersion terms play a significant role. On the one hand, our experiments directly confirm earlier theoretical works that predict (i) breathing near-zero-dispersion solitons to emit polychromatic dispersive radiation, and (ii) that higher-order dispersion can extend the range over which the solitons are stable. On the other hand, our experiments also reveal a novel cross-over scenario, whereby the influence of higher-order dispersion changes from stabilising to destabilising. Our comprehensive experiments sample soliton dynamics both in the normal and anomalous dispersion regimes, and our results are in good agreement with numerical simulations and theoretical predictions.

Design of Ultra-Compact On-Chip Discrete Phase Filters for Broadband Dispersion Management

This paper provides an in-depth theoretical and numerical analysis of a group-velocity dispersion (GVD) management scheme for broadband waveforms using ultra-compact on-chip discrete spectral phase filters based upon waveguide Bragg gratings (WBGs) in a silicon-on-insulator (SOI) platform. Through this technique, mm-long discrete phase filters can be designed to impart a target arbitrary GVD profile on a high-rate (typically, $\geq\!10$ GHz) periodic pulsed waveform, including fully customized and extremely large second and higher-order dispersion terms (e.g., equivalent to 10,000 km of a standard single-mode fiber), over a broad frequency bandwidth (up to $\sim\!3$ THz, demonstrated here). The capabilities and limitations of this technique to impart a target GVD profile over an arbitrary (generally, non-periodic) broadband signal are also studied. We show that customized GVD lines can be efficiently implemented offering a net group-delay excursion in the hundreds-of-ps range using mm-long integrated discrete phase filters. Additionally, we suggest and numerically demonstrate a simple and practical strategy to improve significantly the performance of the discrete phase filtering approach for application on non-periodic waveforms, by combining the discrete phase filter with a suitable periodic resonance (frequency-comb) amplitude filter. An extensive tolerance analysis is conducted, and we conclude that the proposed SOI design framework is well within practical fabrication requirements as well as robust to the expected variability in the main WBG device design parameters.

Pulse wave propagation in a deformable artery filled with blood: an analysis of the fifth-order mKdV equation with variable coefficients

In this paper, the propagation of pulse wave in a deformable elastic vessel filled with inviscid blood is studied. Starting from the stress–strain relationship of blood vessel wall, momentum conservation equation and the Naiver–Stokes equation, the basic equations describing the wall motion and blood flow are established. By utilizing reductive perturbation technique and long wave approximation theory, the basic equations are simplified into a classical third-order mKdV equation with variable coefficients. In order to describe the propagation characteristics of pulse wave more accurately, a fifth-order variable-coefficient mKdV equation is derived. Then, the tanh-function method is applied to find the localized traveling wave solutions of these equations. Based on these localized traveling wave solutions, we further investigate the effects of higher order terms and initial vessel deformation on the characteristics of pulse wave propagation, blood flow velocity and the volume of blood flow. The results show that the higher-order nonlinear and dispersion terms lead to the distortion of the wave, while the initial deformation of the tube wall will influence the wave amplitude and wave width.