Search Results (52)

This paper presents the construction of exact wave solutions for the generalized nonlinear Schrödinger equation (NLSE) with second-order spatiotemporal dispersion using the modified exponential rational function method (mERFM). The NLSE plays a vital role in various fields such as quantum mechanics, oceanography, transmission lines, and optical fiber communications, particularly in modeling pulse dynamics extending beyond the traditional slowly varying envelope estimation. By incorporating higher-order dispersion and nonlinear effects, including cubic–quintic nonlinearities, this generalized model provides a more accurate representation of ultrashort pulse propagation in optical fibers and oceanic environments. A wide range of soliton solutions is obtained, including bright and dark solitons, as well as trigonometric, hyperbolic, rational, exponential, and singular forms. These solutions offer valuable insights into nonlinear wave dynamics and multi-soliton interactions relevant to shallow- and deep-water wave propagation. Conservation laws associated with the model are also derived, reinforcing the physical consistency of the system. The stability of the obtained solutions is investigated through the analysis of modulation instability (MI), confirming their robustness and physical relevance. Graphical representations based on specific parameter selections further illustrate the complex dynamics governed by the model. Overall, the study demonstrates the effectiveness of mERFM in solving higher-order nonlinear evolution equations and highlights its applicability across various domains of physics and engineering. Full article

This study investigates the conformable nonlinear Schrödinger equation (NLSE) with self-phase modulation (SPM) and Kudryashov’s generalized refractive index, crucial for pulse propagation in optical fibers. By applying the modified simplest equation method, we derive several novel soliton solutions and investigate their dynamic behavior within the NLSE framework enhanced with a conformable derivative. The governing conformable NLSE also exhibits symmetry patterns that support the structure and stability of the constructed soliton solutions, linking this work directly with symmetry-based analysis in nonlinear wave models. Furthermore, various graphs are presented through 2D, 3D, and contour plots. These visualizations highlight different soliton profiles, including kink-type, wave, dark, and bell-shaped solitons, showcasing the diverse dynamics achievable under this model, influenced by SPM and Kudryashov’s generalized refractive index. The influence of the conformable parameter and temporal effects on these solitons is also explored. These findings advance the understanding of nonlinear wave propagation and have critical implications for optical fiber communications, where managing pulse distortion and maintaining signal integrity are vital. Full article

In this study, we investigate a nonlinear Schrödinger equation relevant to the evolution of optical beams in weakly nonlocal media. Utilizing the modified F-expansion method, we construct a variety of novel soliton solutions, including dark, bright, and wave solitons. These solutions are illustrated through comprehensive graphical simulations, including 2D contour plots and 3D surface profiles, to highlight their structural dynamics and propagation behavior. The effects of the temporal parameter on soliton formation and evolution are thoroughly analyzed, demonstrating its role in modulating soliton shape and stability. To further explore the system’s dynamics, chaos and sensitivity theories are employed, revealing the presence of complex chaotic behavior under perturbations. The outcomes underscore the versatility and richness of the present model in describing nonlinear wave phenomena. This work contributes to the theoretical understanding of soliton dynamics in weakly nonlocal nonlinear optical systems and supports advancements in photonic technologies. This study reports a novel soliton structure for the weak nonlocal cubic–quantic NLSE and also details the comprehensive chaotic and sensitivity analysis that represents the unexplored dynamical behavior of the model. This study further demonstrates how the underlying nonlinear structures, along with the novel solitons and chaotic dynamics, reflect key symmetry properties of the weakly nonlocal cubic–quintic Schrödinger model. These results enhanced the theoretical framework of the nonlocal nonlinear optics and offer potential implications in photonic waveguides, pulse shape, and optical communication systems. Full article

The method of deriving fractional-order differential equations using Riesz fractional-order calculus is a new breakthrough, and it is possible to use the inverse scattering transform (IST) to solve the analytical solutions of the derived equations. However, there are still relatively few examples of using discrete Riesz fractional (DRF) order calculations. This article focuses on using discrete Riesz fractional-order calculations to derive a variable-coefficient fractional integrable semi-discrete nonlinear Schrödinger (vcfISDNLS) equation. On the one hand, we derive the vcfISDNLS equation through dispersion relation (DR) and DRF order calculations. On the other hand, we obtain the explicit expressions of the one-soliton solution, two-soliton solution, and three-soliton solution of this equation without reflection potentials (RPs). By deriving and solving the equation and displaying the obtained soliton solutions, possible evidential support can be provided for control engineering, automotive engineering, image processing, and optical communication systems. Full article

The dissipative Kerr soliton (DKS) microcomb has been regarded as a highly promising multi-wavelength laser source for optical fiber communication, due to its excellent frequency and phase stability. However, in some specific application scenarios, such as direct modulation and direct detection (DM/DD), the relative intensity noise (RIN) performance of Kerr optical combs still fails to meet the requirements. Here, we systematically investigate the key factors that contribute to the power fluctuations in DKS combs. By exploiting the gain saturation effect of the semiconductor optical amplifier (SOA), the RIN of an on-chip DKS microcomb is effectively suppressed, achieving a maximum reduction of about 30 dB (@600 kHz offset frequency) for all comb lines. Moreover, such DKS comb RIN suppression technology based on an SOA chip can eliminate the need for additional complex feedback control circuits, showcasing the potential for further chip integration of the ultra-low-RIN DKS microcomb system. Full article

This paper examines the nonlinear behavior of the generalized stochastic intermediate dispersive velocity (SIdV) equation, which has been widely analyzed in a non-noise deterministic framework but has yet to be studied in any depth in the presence of varying forcing strength and noise types, in particular how it switches between periodic, quasi-periodic, and chaotic regimes. A stochastic wave transformation reduces the equation to simpler ordinary differential equations to make soliton overlap analysis feasible to analyze soliton robustness under deterministic and stochastic conditions. Lyapunov exponents, power spectra, recurrence quantification, correlation dimension, entropy measures, return maps, and basin stability are then used to measure the effect of white, Brownian, and colored noise on attractor formation, system stability, and spectral correlations. Order–chaos transitions as well as noise-induced complexity are more effectively described by bifurcation diagrams and by Lyapunov spectra. The results of this experiment improve the theoretical knowledge of stochastic nonlinear waves and offer information that will be useful in the fields of control engineering, energy harvesting, optical communications, and signal processing applications. Full article

The current study examines optical soliton solutions in a complicated system of metamaterial-based optical solutions coupled with extremely dispersive couplers. The conformable fractional derivative (CFD) influences the nonlinear refractive index, which is governed by a parabolic equation. Some soliton solutions are extracted, like bright, singular solitons, and singular periodic ones; also, Weierstrass elliptic doubly periodic, and several other exact solutions are systematically revealed by the study using the modified extended direct algebraic method. The findings shed important light on the many solitons in these intricate systems and the interactions between nonlinearity, dispersion, and metamaterial properties. The findings have significance beyond advancing our theoretical understanding of soliton behavior in metamaterial-based optical couplers; they might influence the advancement and development of optical communication technologies and systems. Complementary 2D and 3D representations show how stability parameters change throughout various dynamical regimes and confirm solution consistency. In order to comprehend the complex nonlinear phenomena of this system and its possible practical applications, this paper offers a comprehensive theoretical framework. Full article

This study investigates the dynamics of optical solitons for the variable-coefficient coupled higher-order nonlinear Schrödinger equation (VCHNLSE) enriched with $\beta$-derivatives. By employing an extended direct algebraic method (EDAM), we successfully derive explicit soliton solutions that illustrate the intricate interplay between nonlinearities and variable coefficients. Our approach facilitates the transformation of the complex NLS into a more manageable form, allowing for the systematic exploration of diverse solitonic structures, including bright, dark, and singular solitons, as well as exponential, polynomial, hyperbolic, rational, and Jacobi elliptic solutions. This diverse family of solutions substantially expands beyond the limited soliton interactions studied in conventional approaches, demonstrating the superior capability of our method in unraveling new wave phenomena. Furthermore, we rigorously demonstrate the robustness of these soliton solutions against various perturbations through comprehensive stability analysis and numerical simulations under parameter variations. The practical significance of this work lies in its potential applications in advanced optical communication systems. The derived soliton solutions and the analysis of their dynamics provide crucial insights for designing robust signal carriers in nonlinear optical media. Specifically, the management of variable coefficients and fractional-order effects can be leveraged to model and engineer sophisticated dispersion-managed optical fibers, tunable photonic devices, and ultrafast laser systems, where controlling pulse propagation and stability is paramount. The presence of $\beta$-fractional derivatives introduces additional complexity to the wave propagation behaviors, leading to novel dynamics that we analyze through numerical simulations and graphical representations. The findings highlight the potential of the proposed methodology to uncover rich patterns in soliton dynamics, offering insights into their robustness and stability under varying conditions. This work not only contributes to the theoretical foundation of nonlinear optics but also provides a framework for practical applications in optical fiber communications and other fields involving nonlinear wave phenomena. Full article

In this study, we explore the soliton solutions of the truncated M-fractional fifth-order Korteweg–de Vries (KdV) equation by applying the improved modified extended tanh function method (IMETM). Novel analytical solutions are obtained for the proposed system, such as brigh soliton, dark soliton, hyperbolic, exponential, Weierstrass, singular periodic, and Jacobi elliptic periodic solutions. To validate these results, we present detailed graphical representations of selected solutions, demonstrating both their mathematical structure and physical behavior. Furthermore, we conduct a comprehensive linear stability analysis to investigate the stability of these solutions. Our findings reveal that the fractional derivative significantly affects the amplitude, width, and velocity of the solitons, offering new insights into the control and manipulation of soliton dynamics in fractional systems. The novelty of this work lies in extending the IMETM approach to the truncated M-fractional fifth-order KdV equation for the first time, yielding a wide spectrum of exact analytical soliton solutions together with a rigorous stability analysis. This research contributes to the broader understanding of fractional differential equations and their applications in various scientific fields. Full article

This study examines the soliton dynamics in the time-fractional cubic–quintic nonlinear non-paraxial propagation model, applicable to optical signal processing, nonlinear optics, fiber-optic communication, and biomedical laser–tissue interactions. The fractional framework exhibits a wide range of nonlinear effects, such as self-phase modulation, wave mixing, and self-focusing, arising from the balance between cubic and quintic nonlinearities. By employing the Multivariate Generalized Exponential Rational Integral Function (MGERIF) method, we derive an extensive catalog of analytic solutions, multi-peakon structures, lump solitons, kinks, and bright and dark solitary waves, while periodic and singular solutions emerge as special cases. These outcomes are systematically constructed within a single framework and visualized through 2D, 3D, and contour plots under both anomalous and normal dispersion regimes. The analysis also addresses modulation instability (MI), interpreted as a sideband amplification of continuous-wave backgrounds that generates pulse trains and breather-type structures. Our results demonstrate that cubic–quintic contributions substantially affect MI gain spectrum, broadening instability bands and permitting MI beyond the anomalous-dispersion regime. These findings directly connect the obtained solution classes to experimentally observed routes for solitary wave shaping, pulse propagation, and instability and instability-driven waveform formation in optical communication devices, photonic platforms, and laser technologies. Full article

Quartic solitons in ultrafast fibre lasers (intra-cavity optical fibre modulation systems) have been theoretically and experimentally analysed in recent years. However, there are few reports about extra-cavity modulating quartic solitons. In this situation, the purpose of this work is to investigate a quartic soliton’s extra-cavity modulation. In this paper, we theoretically simulate an extra-cavity-modulating quartic soliton with negative fourth-order dispersion at 1550 nm. The simulation relies on a physical model of a single-mode optical fibre system. Through controlling soliton parameters in an extra-cavity modulation system, a quartic soliton’s orthogonal polarisation modes will show unique characteristics depending on which kind of parameter is changed (seven parameters are considered for variation). For example, through the variation in the projection angle, only a horizontally polarised quartic soliton pulse is generated. These results explore the dynamics of quartic solitons in single-mode optical fibre modulation systems and are applicable to optical soliton transmission techniques in the field of optical fibre communication. Full article

This study employs the improved modified extended tanh method (IMETM) to derive exact analytical solutions of a higher-order nonlinear Schrödinger (HNLS) model, incorporating $\beta$-fractional derivatives in both time and space. Unlike classical methods such as the inverse scattering transform or Hirota’s bilinear technique, which are typically limited to integrable systems and integer-order operators, the IMETM offers enhanced flexibility for handling fractional models and higher-order nonlinearities. It enables the systematic construction of diverse solution types—including Weierstrass elliptic, exponential, Jacobi elliptic, and bright solitons—within a unified algebraic framework. The inclusion of fractional derivatives introduces richer dynamical behavior, capturing nonlocal dispersion and temporal memory effects. Visual simulations illustrate how fractional parameters $\alpha$ (space) and $\beta$ (time) affect wave structures, revealing their impact on solution shape and stability. The proposed framework provides new insights into fractional NLS dynamics with potential applications in optical fiber communications, nonlinear optics, and related physical systems. Full article

This study focuses on the analysis of soliton solutions within the framework of the time-fractional cubic–quintic nonlinear Schrödinger equation (TFCQ-NLSE), a powerful model with broad applications in complex physical phenomena such as fiber optic communications, nonlinear optics, optical signal processing, and laser–tissue interactions in medical science. The nonlinear effects exhibited by the model—such as self-focusing, self-phase modulation, and wave mixing—are influenced by the combined impact of the cubic and quintic nonlinear terms. To explore the dynamics of this model, we apply a robust analytical technique known as the sub-ODE method, which reveals a diverse range of soliton structures and offers deep insight into laser pulse interactions. The investigation yields a rich set of explicit soliton solutions, including hyperbolic, rational, singular, bright, Jacobian elliptic, Weierstrass elliptic, and periodic solutions. These waveforms have significant real-world relevance: bright solitons are employed in fiber optic communications for distortion-free long-distance data transmission, while both bright and dark solitons are used in nonlinear optics to study light behavior in media with intensity-dependent refractive indices. Solitons also contribute to advancements in quantum technologies, precision measurement, and fiber laser systems, where hyperbolic and periodic solitons facilitate stable, high-intensity pulse generation. Additionally, in nonlinear acoustics, solitons describe wave propagation in media where amplitude influences wave speed. Overall, this work highlights the theoretical depth and practical utility of soliton dynamics in fractional nonlinear systems. Full article

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. Full article

The Airyprime beam, due to its adjustable focusing ability and controllable orbital angular momentum, has attracted significant attention in fields such as free-space optical communication and particle trapping. However, systematic studies on the propagation behavior of oscillating solitons in PT-symmetric optical lattices remain scarce, particularly regarding their formation mechanisms and self-accelerating characteristics. In this study, the propagation characteristics of Airyprime beams in PT symmetric optical lattices are numerically studied using the split-step Fourier method, and the generation mechanism and control factors of oscillating solitons are analyzed. The influence of lattice parameters (such as the modulation depth P, modulation frequency w, and gain/loss distribution coefficient W₀) and beam initial characteristics (such as the truncation coefficient a) on the dynamic behavior of the beam is revealed. The results show that the initial parameters determine the propagation characteristics of the beam and the stability of the soliton. This research provides theoretical support for beam shaping, optical path design, and nonlinear optical manipulation and has important application value. Full article