Fourth-order Dispersion Research Articles

Numerical investigation of solitary-wave solutions for the nonlinear Schrödinger equation perturbed by third-order and negative fourth-order dispersion

Numerical investigation of solitary-wave solutions for the nonlinear Schrödinger equation perturbed by third-order and negative fourth-order dispersion

Investigating innovative optical solitons for a (3+1)- dimensional nonlinear Schrödinger’s equation under the influences of 4th-order dispersive and parabolic law of nonlinearities

This study tackles a nonlinear Schrödinger’s equation in three dimensions, using three dispersive components of fourth order, often resulting in pure-quartic solitons. Pure-quartic bullets are known to deviate from conventional solitons due to their balance of fourth-order dispersion and nonlinearity. As bright and dark optically modulated bullets, we derive many solutions. Experiments on bullet transmission using optical nanofibers can benefit from the solutions found. Our creative solutions are produced by implementing the well-known scheme which is the improved modified extended tanh-function scheme. We find novel kinds of solutions (dark, bright, singular solitons, exponential, singular periodic, Weierstrass elliptic doubly periodic solutions, and Jacobi elliptic functions) that make their originality for the problem at hand evident by using the previously described approach. Contour plots and 2D and 3D visualizations in Wolfram Mathematica software are used to show how the well-furnished results propagate for various values of the necessary free parameters. The results demonstrate the computational processes’ precise, well-informed, and efficient nature. Through their integration with representational computations, they may be applied to increasingly complex phenomena. This work constitutes a major advancement in our comprehension of the intricate and erratic behavior of this mathematical model.

Pure-quartic soliton self-frequency shift in a mode-locked fiber laser

The pure-quartic soliton (PQS) offers the advantage of a wide spectrum and approximately Gaussian temporal profile, which has the potential to obtain high-energy ultrafast laser pulses. Thus, exploring the role of stimulated Raman scattering in the formation and propagation of PQSs in fiber lasers is crucial, especially considering the greater propensity for highpower and short PQS pulses compared to traditional solitons. Here, we present the modeling of a self-frequency shift (SFS) fiber laser based on the PQS, emphasizing the impact of fourth-order dispersion and the Raman effect. The significant red-shift in wavelength is discovered as the pump energy increases, revealing that the emergence of SFS is a universal attractor in mode-locked PQS fiber laser systems. Our simulations demonstrate that such wavelength tunability is a direct byproduct of the gain bandwidth limit and the stimulated Raman scattering for PQSs inside the fiber cavity, while the evolution dynamics in the gain and passive fiber show obvious differences. However, we find that the effect of SFS in a PQS fiber laser is compromised by a trade-off with the degradation of pulse quality, such as reshaping the oscillating tail and destroying the symmetry of the emission pulse. This Raman-induced nonlinear process under high pump energy facilitates the discovery of the comprehensive complexity of science for PQSs suffering from higher-dimensional forms of nonlinear effects in fiber lasers.

M-shaped, W-shaped and dark soliton propagation in optical fiber for nonlocal fourth order dispersive nonlinear Schrödinger equation under distinct conditions

The formation of soliton in optical fiber governed by nonlocal nonlinear Schrödinger (NLS) equation with fourth order dispersion is studied. The model of nonlocal NLS equation with fourth order dispersion is solved using the new extended auxiliary method and the solutions are obtained. The solutions are in the form of hyperbolic and trigonometric functions which are based on Jacobi elliptic function m. Shape changing soliton in optical fiber for nonlocal fourth order dispersive NLS equation is discussed by suitably choosing the values of kerr and quintic nonlinearities and by varying fourth order dispersion term. The effect of fourth order dispersion on soliton in fiber for different conditions of kerr and quintic nonlinearity is also discussed. In addition, the phase portraits of the system have been investigated and the stability of wave in optical fiber for nonlocal NLS equation is discussed using fourth order Runge-Kutta algorithm. This paper addresses a significant gap in the current literature by examining the impact of fourth order dispersion on the nonlocal NLS equation in optical fiber.

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.

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.

Dynamic control of cascaded four-wave mixing via re-configurable dispersion.

We integrate a spatial light modulator-based dispersion controller into a cascaded four-wave mixing (CFWM) system. By tuning the group delay dispersion (GDD) and fourth-order dispersion (FOD) terms, we control the CFWM phase matching and demonstrate an output bandwidth tuning of over 3.3×. At the maximum bandwidth, our system covers the telecommunications S-, C-, and L-bands (1466-1641 nm) with an average output power of 300 mW, which is contained in 52 individual lines spaced 374 GHz apart. This method represents a reconfigurable alternative to photonic crystal fibers for dispersion engineering and allows for the use of step-index fiber and customizable power spectral density (PSD) profiles.

Coherence-controlled chaotic soliton bunch

Controlling the coherence of chaotic soliton bunch holds the promise to explore novel light-matter interactions and manipulate dynamic events such as rogue waves. However, the coherence control of chaotic soliton bunch remains challenging, as there is a lack of dynamic equilibrium mechanism for stochastic soliton interactions. Here, we develop a strategy to effectively control the coherence of chaotic soliton bunch in a laser. We show that by introducing a lumped fourth-order-dispersion (FOD), the soliton oscillating tails can be formed and generate the potential barriers among the chaotic solitons. The repulsive force between neighboring solitons enabled by the potential barriers gives rise to an alleviation of the soliton fusion/annihilation from stochastic interactions, endowing the capability to control the coherence in chaotic soliton bunch. We envision that this result provides a promising test-bed for a variety of dynamical complexity science and brings new insights into the nonlinear behavior of chaotic laser sources.

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.

Accurate optical dispersion identification for linear deformable mirror with spatial-spectral interferometry

In ultrashort laser pulse systems, optical dispersion can cause severe distortion of the spectral phase and thus significantly affect the pulse shape. To address this problem, this study places a linear deformable mirror (LDM) at the Fourier plane of a blazed grating for dispersion control, and reconstructs the spectral phase modulated by the LDM using single-shot spatial-spectral interferometry. An iterative algorithm and closed-loop optimization process are used to drive the LDM to generate orthogonal modes of Legendre polynomials with a signal-to-background ratio (SBR) of more than 10 dB. The identified LDM modes are then used as the basis for spectral phase modulation. It is shown that the group delay dispersion (GDD), third order dispersion (TOD), fourth order dispersion (FOD), and fifth order dispersion (5OD) can be accurately obtained by the linear combination of identified Legendre modes. In addition, the maximum phase error between the generated and target dispersions is less than 0.15 rad, while the error between the pulse widths produced by the generated dispersion and the target dispersion, respectively, is less than 3.5 %. Overall, the results confirm that the linear superposition of the generated Legendre modes enables the accurate dispersion modulation of ultrashort laser pulses.

Multi-soliton solutions of coupled Lakshmanan–Porsezian–Daniel equations with variable coefficients under nonzero boundary conditions

Abstract This paper aims to investigate the multi-soliton solutions of the coupled Lakshmanan–Porsezian–Daniel equations with variable coefficients under nonzero boundary conditions. These equations are utilized to model the phenomenon of nonlinear waves propagating simultaneously in non-uniform optical fibers. By analyzing the Lax pair and the Riemann–Hilbert problem, we aim to provide a comprehensive understanding of the dynamics and interactions of solitons of this system. Furthermore, we study the impacts of group velocity dispersion or the fourth-order dispersion on soliton behaviors. Through appropriate parameter selections, we observe various nonlinear phenomena, including the disappearance of solitons after interaction and their transformation into breather-like solitons, as well as the propagation of breathers with variable periodicity and interactions between solitons with variable periodicities.

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.

Dark solitons and their bound states in a nonlinear fiber with second- and fourth-order dispersion

We study the excitations of dark solitons in a nonlinear optical fiber with the second- and fourth-order dispersion, and find the emergence of striped dark solitons (SDSs) and some multi-dark-soliton bound states. The SDSs can exhibit time-domain oscillating structures on a plane wave, and they have two types: the ones with or without the total phase step, while the multi-dark-soliton bound states exhibit different numbers of amplitude humps. By the modified linear stability analysis, we regard the SDSs as the results of the competition between periodicity and localization, and analytically give their existence condition, oscillation frequency, and propagation stability, which show good agreements with numerical results. We also provide a possible interpretation of the formation of the existing striped bright solitons (SBSs), and find that SBS will become the pure-quartic soliton when its periodicity and localization carry equal weight. Our results provide the theoretical support for the experimental observation of striped solitons in nonlinear fibers, and our method can also guide the discovery of striped solitons in other physical systems.

Resonant Akhmediev breathers

Modulation instability is a phenomenon in which a minor disturbance within a carrier wave gradually amplifies over time, leading to the formation of a series of compressed waves with higher amplitudes. In terms of frequency analysis, this process results in the generation of new frequencies on both sides of the original carrier wave frequency. We study the impact of fourth-order dispersion on this modulation instability in the context of nonlinear optics that lead to the formation of a series of pulses in the form of Akhmediev breather. The Akhmediev breather, a solution to the nonlinear Schrödinger equation, precisely elucidates how modulation instability produces a sequence of periodic pulses. We observe that when weak fourth-order dispersion is present, significant resonant radiation occurs, characterized by two modulation frequencies originating from different spectral bands. As an Akhmediev breather evolves, these modulation frequencies interact, resulting in a resonant amplification of spectral sidebands on either side of the breather. When fourth-order dispersion is of intermediate strength, the spectral bandwidth of the Akhmediev breather diminishes due to less pronounced resonant interactions, while stronger dispersion causes the merging of the two modulation frequency bands into a single band. Throughout these interactions, we witness a complex energy exchange process among the phase-matched frequency components. Moreover, we provide a precise explanation for the disappearance of the Akhmediev breather under weak fourth-order dispersion and its resurgence with stronger values. Our study demonstrates that Akhmediev breathers, under the influence of fourth-order dispersion, possess the capability to generate infinitely many intricate yet coherent patterns in the temporal domain.

Bright, dark, and periodic soliton solutions for the (2+1)-dimensional nonlinear Schrödinger equation with fourth-order nonlinearity and dispersion

Bright, dark, and periodic soliton solutions for the (2+1)-dimensional nonlinear Schrödinger equation with fourth-order nonlinearity and dispersion

Pure-quartic soliton in a birefringence-managed fiber laser

Pure-quartic soliton (PQS) fiber lasers with a great variety of operating regimes have aroused considerable interest in high-energy ultrashort pulses and novel soliton patterns. Here, we numerically investigate the generation and dynamic properties of birefringence-managed pure-quartic soliton (BMPQS) in a positive fourth-order dispersion (FOD) fiber laser consisting of single-mode fibers and a section of polarization-maintaining fiber. The formation mechanism of the chirp-free BMPQS is dominated by the phase-matching effect, which arises from the co-actions of the birefringence, positive FOD, and nonlinear effects. It is observed that a higher FOD coefficient leads to energy transfer between the spectral sidebands inside the BMPQS. Moreover, the evolution of the BMPQS with the increase of the pump power confirms the phase-matching theory. Finally, the BMPQS molecule can be obtained with further increasing the pump power. This work offers a new avenue to create chirp-free pulses in positive FOD fiber lasers and enriches the sub-families of PQSs.

Extraction of Solitons in Optical Fibers for the (2+1)-Dimensional Perturbed Nonlinear Schrödinger Equation Via the Improved Modified Extended Tanh Function Technique

The presented research investigates the (2+1)-dimensional perturbed nonlinear Schrödinger model. This model takes into account various effects such as fourth order dispersion, intermodal dispersion, nonlinear dispersion, group velocity dispersion, Kerr nonlinearity and self steepening effects. This model simulates the estimation of optical solitons and a variety of broadcasting networks’ transmission in nonlinear fiber optics. The proposed model is studied using the improved modified extended tanh function technique. For the proposed model, several solitons and other solutions are generated. These solutions including {bright, dark and singular} solitons, singular periodic and Jacobi elliptic solutions. By introducing the two- and three-dimensional graphs, the physical behavior of the retrieved solutions is displayed.

Multiple solitons structures in optical fibers via PNLSE with a novel truncated M-derivative: modulated wave gain

This study introduces a novel truncated Mittage–Leffler (M)- proportional derivative (TMPD) and examines its impact on the perturbed nonlinear Schrödinger equation (PNLSE) that includes fourth-order dispersion and cubic-quintic nonlinearity. The TMPD-PNLSE is used to model light signals in nanofibers. In addition to dispersion and Kerr nonlinearity, which are characteristics of the NLSE, the PNLSE also exhibits self-steepening and self-phase modulation effects. The unified method is implemented to derive exact solutions for the model equation. These solutions provide a variety of phenomena; including breathers, geometric chaos, and complex solitons. The solutions also exhibit numerous structures, such as geometric chaos, where undulated M-shaped and M-shaped solitons are embedded. The modulation instability is analyzed, finding that it is triggered when the coefficient of the fourth-order dispersion surpasses a critical value.

The periodic soliton solutions for a nonlocal nonlinear Schrödinger equation with higher-order dispersion

A nonlocal nonlinear Schr¨odinger (NNLS) equation with fourth-order dispersion and cubic-quintic nonlinearities has been studied analytically and numeri- cally. Under the constraint conditions, auxiliary functions are introduced, and explicit one- and two-soliton solutions are obtained by the Hirota bilinear method. Accord- ing to the solutions, the propagation dynamics of soliton pulses are investigated. The influences of different parameters on the dynamics of one- and two-soliton solutions have been analyzed. The results show that the two-soliton solution exhibits diverse dy- namic characteristics under the suitable parameter selections. In addition, the stability of one- and two-soliton solutions against the constraint conditions deviations and under the initial perturbations are also studied numerically.

Impact of third-order dispersion on the dynamics of dissipative solitons in an ultrafast fiber laser.

Dissipative solitons (DSs), due to the complex interplay among dispersion, nonlinear, gain and loss, illustrate abundant nonlinear dynamics behaviors. Especially, dispersion plays an important role in the research of DS dynamics in ultrafast fiber lasers. Previous studies have mainly focused on the effect of even-order dispersion, i.e., group velocity dispersion (GVD) and fourth-order dispersion. In fact, odd-order dispersions, such as third-order dispersion (TOD), also significantly influences the dynamics of DSs. However, due to the lack of dispersion engineering tools, few experimental researches in this domain have been reported. In this work, by employing a pulse shaper in ultrafast fiber laser, an in-depth exploration of the DS dynamics influenced by TOD was conducted. With the increase of TOD value, the stable single DS undergoes a splitting into two solitons and then enters explosion state, and ultimately evolves into a chaotic state. The laser operation state is correlated to dispersion profile, which could be controlled by TOD. Here, the positive dispersion at long-wavelength side will be gradually shifted to negative dispersion by increasing the TOD, where soliton effect will drive the transitions. These findings offer valuable insights into the nonlinear dynamics of ultrafast lasers and may also foster applications involving higher-order dispersion.