Fourth-order Dispersion Terms Research Articles

Modulational instability, modulated wave, and optical solitons for a generalized highly dispersive cubic-quintic-septic-nonic medium with self-frequency shift and self-steepening nonlinear terms

In this research, we delve into a generalized highly dispersive (HD) nonlinear Schrödinger equation, enriched with cubic-quintic-septic-nonic (CQSN) nonlinearities. The core of our investigation revolves around the perturbation of plane waves, aiming to understand their stability characteristics in such a complex medium. We investigate the influence of various factors such as the amplitude of the plane wave, perturbed wave number, nonic nonlinear term, and fourth-order dispersion term. Our findings indicate that increasing the amplitude of the plane wave widens the modulation instability (MI) bands and amplifies the MI growth rate. In contrast, increasing the nonic nonlinear term has opposing effects, narrowing the MI bands and diminishing the amplitude of the MI growth rate. Increasing the fourth-order dispersion term does not affect the amplitude of the MI growth rate but narrows the MI bands. The observed pattern of increasing and then decreasing MI intensity with rising K can be attributed to the complex interplay among phase matching conditions, dispersion effects, and nonlinear saturation. Initially, higher K enhances phase matching and boosts MI growth. However, as K increases further, the combined influence of dispersion and nonlinear effects can diminish the effectiveness of phase matching, resulting in a reduction in MI intensity. A significant portion of our work is dedicated to identifying and analyzing modulated rational, polynomial Jacobi elliptic function solutions, and the emergence of optical solitons within this framework. These findings provide new insights into the nonlinear dynamics underpinning the generalized HDNLSE, enriched with CQSN nonlinearities, offering valuable contributions to the theoretical understanding of such phenomena.

Data-driven solutions and parameter discovery of the extended higher-order nonlinear Schrödinger equation in optical fibers

In this paper, we investigate an extended higher-order nonlinear Schrödinger equation that includes third-order and fourth-order dispersion terms. By considering higher-order effects, the equation can more accurately simulate and predict the propagation behavior of ultrashort pulses in optical fibers, e.g., pulse splitting, compression and stability. In the optical fiber sensors, the sensitivity and resolution of the sensor can be improved by utilizing nonlinear effects. The analysis of the extended higher-order nonlinear Schrödinger equation helps to understand how nonlinear effects affect the sensing signal, thereby designing more accurate sensors. Specific coefficients of each term in the equation are studied for different scenarios, such as optical fiber design and signal propagation in optical communications. Under certain conditions, the extended higher-order nonlinear Schrödinger equation can be degenerated into the Hirota equation or the Lakshmanan–Porsezian–Daniel equation. With the development of numerical algorithms, the utilization of deep neural networks for solving equations has become increasingly attractive. With given initial conditions and boundary conditions, the physical information is embedded into the neural network in the form of loss functions. We efficiently derive numerical solutions to the extended equation for one-soliton, two-soliton and rogue-wave cases by training weights and biases in physics-informed neural networks. Compared with traditional numerical methods, physics-informed neural networks yield lower prediction errors with less data volume, thus will reduce the cost and time consumptions for data sampling in practical engineering. Furthermore, we solve the inverse problem associated with the extended equation. This approach is critical for extracting fundamental parameters of optical systems from observable data, thereby deepening our understanding of complex optical behavior and helping to improve the performance of optical fiber communication technologies.

Optical soliton solutions of the perturbed fourth-order nonlinear Schrödinger-Hirota equation with parabolic law nonlinearity of self-phase modulation

This article introduces an examination of optical soliton solutions for the perturbed fourth-order nonlinear Schrödinger-Hirota equation, which plays a crucial role in optics. For the first time, it utilizes a novel approach by applying the extended auxiliary equation method. This equation models the propagation of optical pulses through nonlinear media, such as optical fibers, and has been the subject of many studies. Our goal extends beyond merely acquiring a significant number of soliton solutions using the method described in this article; we also aim to investigate the impact of the coefficients of group velocity dispersion, parabolic law, and fourth-order dispersion terms on soliton propagation in the problem examined. The 2D, 3D, and contour plots of the acquired dark and bright solitons, which represent the most fundamental soliton types, are presented. Additionally, all other calculations are performed using symbolic algebraic software. The results provide us with valuable insights, confirming that the introduced model can be analyzed from a physical perspective. It is demonstrated that the proposed technique is not only important but also efficient in analyzing various nonlinear scientific problems.

Effect of the Dispersion Orders on the Widths of the Coexistence Domain and Combs Spectra of Bright and Dark Solitons in Microresonators

Using the Lugiato–Lefever model, we base on the hysteresis approach to analyze the coexistence domain of bright and dark solitons in the zero, normal, and anomalous dispersion regimes. Our results also highlight that the fourth-order dispersion term affects the width of the frequency combs of both dark and bright solitons. It also allows the appearance of dispersive waves on the soliton spectra that disappear for high values of the fourth-order dispersion followed by the soliton destabilization into harmonic oscillations and oscillation packages.

Radial and non-radial multiple solutions to a general mixed dispersion NLS equation

We study the following nonlinear Schrödinger equation with a fourth-order dispersion term Δ2u−βΔu=g(u)in RN in the positive and zero mass regimes: in the former, and , where m > 0 depends on g; in the latter, and β > 0. In either regimes, we find an infinite sequence of solutions under rather generic assumptions about g; if N = 2 in the positive mass case, or N = 4 in the zero mass case, we need to strengthen such assumptions. Our approach is variational.

Dynamic behaviors of soliton solutions for a three-coupled Lakshmanan–Porsezian–Daniel model

In this paper, we use the Riemann–Hilbert (RH) approach to examine the integrable three-coupled Lakshmanan–Porsezian–Daniel (LPD) model, which describes the dynamics of alpha helical protein with the interspine coupling at the fourth-order dispersion term. Through the spectral analysis of Lax pair, we construct the higher-order matrix RH problem for the three-coupled LPD model, when the jump matrix of this particular RH problem is a $$4\times 4$$ unit matrix, the exact N-soliton solutions of the three-coupled LPD model can be exhibited. As special examples, we also investigate the nonlinear dynamical behaviors of the single-soliton, two-soliton, three-soliton and breather soliton solutions. Finally, an integrable generalized N-component LPD model with its linear spectral problem is discussed.

Variable coefficient higher-order nonlinear Schrödinger type equations and their solutions

In this communication we focus on certain higher-order partial differential equations of the nonlinear Schrödinger type with variable coefficients. By using a traveling wave reduction we obtain the general solution for the envelope in terms of the Jacobi elliptic sine function. The time dependence of the traveling wave coordinate and phase are explicitly determined as functions of the variable coefficients appearing in the equations. The equations investigated display cubic and fourth-order dispersion terms as well as higher nonlinearity and their constant coefficient versions have been the subject of a number of recent studies.

Parametric sideband generation in CMOS-compatible oscillators from visible to telecom wavelengths

We present an approach for generating widely separated first sidebands based solely on the four-wave-mixing process in optical parametric oscillators built on complementary metal–oxide–semiconductor-compatible photonic chips. Using higher-order transverse modes to perform dispersion engineering, we obtain zero-group-velocity dispersion near 796 nm. By pumping the chip in the normal dispersion region, at 795.6 nm, we generate a signal field in the visible band (at 546.2 nm) and the corresponding idler field in the telecom band (at 1465.3 nm), corresponding to a frequency span of approximately 346 THz. We show that the spectral position of signal and idler can be tailored by exploiting a delicate balance between second- and fourth-order dispersion terms. Furthermore, we explicitly demonstrate a change in the parametric oscillation dynamics when moving the pump field from the anomalous to normal dispersion, where the chip ceases producing multiple sidebands adjacent to the pump field and generates widely separated single sidebands. This provides a chip-scale platform for generating single-sideband fields separated by more than one octave, covering the visible and telecom spectral regions.

Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source

<p style='text-indent:20px;'>This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.</p>

Influence of fourth-order dispersion on the Anderson localization

Anderson localization, which consists in the absence of wave dispersion due to disorder, brought to the field of optics and matter waves has greatly improved our understanding of fundamental processes, such as transport and multiple dispersion of light. In this paper, we investigate the existence of Anderson localization in the interplay of a standard group velocity dispersion and a fourth-order dispersion term in the nonlinear Schrodinger (NLS) equation in the presence of a quasiperiodic linear coefficient. We employ a variational approach with a Gaussian ansatz to describe the center of the localized state, while the tails are studied by direct numerical simulations of the NLS equation. These two approaches are important to distinguish the region of the solution presenting exponential decay, which is the main signature of the Anderson localization. The existence of Anderson localization has been confirmed by numerical simulations even when the system presents a small defocusing nonlinearity. In this sense, we reported how the fourth-order dispersion effect changes the existence region of the localized state by changing the critical values for the transition between the localized and delocalized states.

Localization of optical pulses in guided wave structures with only fourth order dispersion

Inspired by the recent realization of pure-quartic solitons (Blanco-Redondo et al. (2016) [1]), in the present work we study the localization of optical pulses in a similar system, i.e., a silicon photonic crystal air-suspended structure with a hexagonal lattice. The propagation of ultrashort pulses in such a system is well described by a generalized nonlinear Schrödinger (NLS) equation, which in certain conditions works with near-zero group-velocity dispersion and third order dispersion. In this case, the NLS equation has only the fourth order dispersion term. In the present model, we introduce a quasiperiodic linear coefficient that is responsible to induce the localization. The existence of Anderson localization has been confirmed by numerical simulations even when the system presents a small defocusing nonlinearity.

Cubic-quartic optical soliton perturbation by semi-inverse variational principle

This paper secures bright optical solitons in presence of third and fourth order dispersion terms while group velocity dispersion term is absent. The perturbation terms that are included appear with full nonlinearity. The semi-inverse variational principle is implemented to secure such solutions.

Offner stretcher aberrations revisited to compensate material dispersion

We present simple analytical formulae for the calculation of the spectral phase and residual angular dispersion of an ultrashort pulse propagating through the Offner stretcher. Based on these formulae, we show that the radii of curvature of both convex and concave mirrors in the Offner triplet can be adapted to tune the fourth order dispersion term of the spectral phase of the pulse. As an example, a single-grating Offner stretcher design suitable for the suppression of material dispersion in the Ti:Sa PALS laser system is proposed. The results obtained by numerical raytracing well match those calculated from the analytical formulae.

New optical solitons of cubic-quartic nonlinear Schrödinger equation

The present paper acquires new optical solitons to the nonlinear Schrödinger equation (NLSE) which is investigated with third and fourth order dispersion terms. A modified form of Kudryashov approach is formally adopted to extract the exact analytical solutions of governing model. The constraint conditions for the existence of optical solitons in the case of Kerr law nonlinearity are also reported.

Cubic–quartic optical solitons in Kerr and power law media

In this paper, we present the exact bright and singular optical solitons of the nonlinear Schrödinger equation with third and fourth order dispersion terms. The method of undetermined coefficients is applied to obtain the reported solutions. The cases of Kerr law and power law nonlinearity are taken into account. We also find the conditions concerning the optical material parameters for the existence of these soliton structures. The results are useful in describing the propagation of optical solitons in highly dispersive media with Kerr and power law nonlinearity.

Dark solitons, Lax pair and infinitely-many conservation laws for a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in the inhomogeneous Heisenberg ferromagnetic spin chain

Under investigation in this paper is a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain. Lax pair and infinitely-many conservation laws are derived, indicating the existence of the multi-soliton solutions for such an equation. Via the Hirota method with an auxiliary function, bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interactions for the dark solitons are illustrated graphically: Velocity of the solitons is linearly related to the coefficients of the second- and fourth-order dispersion terms, while amplitude of the solitons does not depend on them. Interactions between the two solitons are shown to be elastic, while those among the three solitons are pairwise elastic.

New Exact Solutions for a Class of High-order Dispersive Cubic-quintic Nonlinear Schrodinger Equation

In the paper, the exact solutions to the cubic-quintic nonlinear Schrodinger equation with third and fourth-order dispersion terms is considered. The improved homogeneous balance method is used for constructing a series of new exact envelop wave solutions, including envelop solitary wave solutions, envelop periodic wave solutions and an envelop rational solution.

Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion

Modulational instability (MI) in negative refractive metamaterials with saturable nonlinearity, fourth-order dispersion (FOD), and second-order nonlinear dispersion (SOND) is investigated by using standard linear stability analysis and the Drude electromagnetic model. The expression for the MI gain spectrum is obtained, which clearly reveals the influence of the saturation of the nonlinearity, FOD, and SOND parameters on the temporal MI. The evolution of the MI in negative refractive metamaterials is numerically investigated. Special attention is paid to study the effects of the higher-order dispersion terms on the formation and evolution of the solitons induced by MI. It is shown that as the third-order dispersion term increases, the solitons travel toward the right. Moreover, the magnitude of the FOD term influences considerably the number of wave trains induced by MI.

Simple Method for Measuring the Zero-Dispersion Wavelength in Optical Fibers

We propose a very simple method for measuring the zero-dispersion wavelength of an optical fiber as well as the ratio between the third- and fourth-order dispersion terms. The method is based on the four-wave mixing process when pumping the fiber in the normal dispersion region, and only requires the measurement of two spectra. We provide an experimental demonstration of the method in a photonic crystal fiber and we show that the measured zero-dispersion wavelength is in good agreement with a low-coherence interferometry measurement.

Dispersion and nonlinear phase-shift compensation in high-peak-power short-pulse fiber laser sources using photonic-crystal fibers

Photonic-crystal fibers (PCFs) with a specifically designed dispersion profile and nonlinearity are shown to enable an accurate broadband compensation of the stretcher-compressor dispersion in fiber laser sources of high-peak-power ultrashort light pulses. We demonstrate that the nonlinear phase shift in such systems can partially compensate for the fourth-order dispersion, allowing the stretcher—compression group delay to be compensated up to the fourth-order dispersion terms by using a sequence of only two fibers—a standard optical fiber and a PCF.