Search Results (55)

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 relationships among subjective well-being (SWB), active travel, and the built and social environment have been rarely studied, especially in southern cities of Chile. The goal of this research is to investigate the connections between SWB and active travel, along with the associated social, built environment, and individual aspects in Temuco. Furthermore, due to the high levels of socioeconomic segregation (SES) in the city’s various urban neighborhoods, these relationships were studied independently based on two categories of neighborhoods, namely low-SES (NLSES) and high-SES (NHSES), which represent the majority of the city’s areas and population. To ascertain the number of responders in each SES category, a power analysis and simple random sampling were used. Consequently, 481 and 301 respondents were identified for NLSES and NHSES, respectively. A quantitative method and hierarchical multiple regression analysis were used to investigate the goals. The findings indicate that SWB is generally higher in NHSES than in NLSES. It was also found that there was a correlation between subjective well-being and several factors, such as age, some job-related categories, social cohesion, role models, and accessibility to shops, parks, and bus stops. Less SWB is a result of a higher unemployment rate in NLSES as opposed to NHSES. Additionally, a certain lifestyle type in NHSES demonstrated a positive correlation with SWB. Furthermore, there was a positive association found between the NHSES’s SWB and access to the bus network. This study provides evidence from a highly segregated Latin American city that shows how SWB is shaped differently across low- and high-SES neighborhoods. Temuco’s urban policymakers could use these data to improve SWB according to the different types of neighborhoods within this city. Full article

Analysis of nonlinear plasma waves, formulated and applied for ionospheric HF heating experiments, indicates that Langmuir/upper hybrid waves excited by parametric instabilities can evolve into traveling solitary waves accompanied by self-induced cavitons. To explore these cavitons, a digisonde operating in fast mode was utilized. Significant results were observed in ionograms recorded two minutes after the activation of the O-mode heater. These ionograms displayed two distinct bumps in the virtual height spread, located slightly below both the HF reflection height and the upper hybrid resonance height. It is notable that these heights are also slightly below the excitation regions where Langmuir/upper hybrid Parametric Decay Instabilities (PDIs) are typically generated by an O-mode HF heater. These observations correlate with the theory and provide valuable insights into the dynamics of nonlinear plasma waves and their interaction with the ionosphere during HF heating experiments. Full article

To address the challenges of poor adaptability to spatial heterogeneity, easy breakage of amplitude–phase coupling relationships, and insufficient physical consistency in complex optical wavefield reconstruction, this paper proposes the DdONN-PINNs hybrid framework. Focused on preserving the intrinsic symmetries of wave physics, the framework achieves deep integration of optical neural networks and physics-informed information. Centered on an architecture of “SIREN shared encoding–domain-specific output”, it utilizes the periodic activation property of SIREN encoders to maintain the spatial symmetry of wavefield distribution, incorporates learnable Fourier diffraction layers to model physical propagation processes, and adopts native complex-domain modeling to avoid splitting the real and imaginary parts of complex amplitudes—effectively adapting to spatial heterogeneity while fully preserving amplitude-phase coupling in wavefields. Validated on rogue wavefields governed by the Nonlinear Schrödinger Equation (NLSE), experimental results demonstrate that DdONN-PINNs achieve an amplitude Mean Squared Error (MSE) of $2.94\times {10}^{-3}$ and a phase MSE of $5.86\times {10}^{-4}$, outperforming non-domain-decomposed models and ReLU-activated variants significantly. Robustness analysis shows stable reconstruction performance even at a noise level of $\sigma =0.1$. This framework provides a balanced solution for wavefield reconstruction that integrates precision, physical interpretability, and robustness, with potential applications in fiber-optic communication and ocean optics. 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 review summarizes the application of physics-informed neural networks (PINNs) for solving higher-order nonlinear partial differential equations belonging to the nonlinear Schrödinger equation (NLSE) hierarchy, including models with external potentials. We analyze recent studies in which PINNs have been employed to solve NLSE-type evolution equations up to the fifth order, demonstrating their ability to obtain one- and two-soliton solutions, as well as other solitary waves with high accuracy. To provide benchmark solutions for training PINNs, we employ analytical methods such as the nonisospectral generalization of the AKNS scheme of the inverse scattering transform and the auto-Bäcklund transformation. Finally, we discuss recent advancements in PINN methodology, including improvements in network architecture and optimization techniques. Full article

The central objective of this study is to develop some different wave solutions and perform a qualitative analysis on the nonlinear dynamics of the time-fractional chiral nonlinear Schrodinger’s equation (NLSE) in the conformable sense. Combined with the semi-inverse method (SIM) and traveling wave transformation, we establish the variational principle (VP). Based on this, the corresponding Hamiltonian is constructed. Adopting the Galilean transformation, the planar dynamical system is derived. Then, the phase portraits are plotted and the bifurcation analysis is presented to expound the existence conditions of the various wave solutions with the different shapes. Furthermore, the chaotic phenomenon is probed and sensitivity analysis is given in detail. Finally, two powerful tools, namely the variational method (VM) which stems from the VP and Ritz method, as well as the Hamiltonian-based method (HBM) that is based on the energy conservation theory, are adopted to find the abundant wave solutions, which are the bell-shape soliton (bright soliton), W-shape soliton (double-bright solitons or double bell-shaped soliton) and periodic wave solutions. The shapes of the attained new diverse wave solutions are simulated graphically, and the impact of the fractional order $\delta$ on the behaviors of the extracted wave solutions are also elaborated. To the authors’ knowledge, the findings of this research have not been reported elsewhere and can enable us to gain a profound understanding of the dynamics characteristics of the investigative equation. Full article

The well-known nonlinear Schrödinger equation (NLSE) plays a crucial role in describing the temporal evolution of disturbances in marginally stable or unstable media. However, when the media is a fractal form, it becomes ineffective. Thus, the fractal modification to the NLSE is presented based on the fractal derivative in this work for the first time. The semi-inverse method is employed to establish the fractal variational principle. The entire process of deriving the fractal variational principle is presented in detail. To our knowledge, the fractal variational principle mentioned in this article is the first exploration and report to date. The fractal variational principle established in this paper is expected to deepen our understanding of the essence of physical phenomena in the fractal space and offer new ideas for the application and exploration of the variational approaches. Full article

This study presents the synthesis of manganese molybdenum tetraoxide (MnMoO₄)-based nanoparticles and then their experimental demonstration as saturable absorbers (SAs) in erbium-doped fiber lasers (EDFLs). The MnMoO₄ nanoparticles were prepared and then embedded between the fiber ferrule to act as an SA to generate Q-switched pulsed operation in EDFLs. For the characterization, scanning electron microscopy (SEM) was employed to confirm the particle size of the prepared MnMoO₄ nanoparticles, and the SA optical properties were further investigated by measuring their modulation depth and saturation intensity. By implementing the prepared SA within the cavity, the measured results revealed that under pump power ranging from 28 to 312.5 mW, the laser exhibited Q-switched pulse durations varying from 15.22 to 2.35 µs and repetition rates spanning from 24.98 to 88.11 kHz. The proposed EDFL system delivered an average output power between 0.128 and 2.95 mW, pulse energies ranging from 5.12 to 33.49 nJ, and peak power from 0.281 to 6.26 mW. The laser stability was also confirmed by continuously noticing the pulse duration, emission wavelengths, and pulse repetition rates for 4 h. Finally, a numerical model based on a nonlinear Schrödinger equation (NLSE) was employed to validate both experimental and theoretical results of the passive Q-switched EDFL. These findings highlight the potential of EDFLs utilizing MnMoO₄-based SAs for potential applications in pulsed laser sources. Full article

Partially nonlocal (PNL) variable-coefficient nonlinear Schrödinger equations (NLSEs) represent a significant area of study in mathematical physics and quantum mechanics, particularly in scenarios where potential and coefficients vary spatially or temporally. The (3+1)-dimensional partially nonlocal (PNL) coupled nonlinear Schrödinger (NLS) model, enriched with different values of two transverse diffraction profiles and subjected to gain or loss phenomena, undergoes dimensional reduction to a (2+1)-dimensional counterpart model, facilitated by a conversion relation. This reduction unveils intriguing insights into the excited mechanisms underlying partially nonlocal waves, culminating in analytical solutions that describe high-dimensional extreme waves characterized by Hermite–Gaussian envelopes. This paper explores novel extreme wave solutions in (3+1)-dimensional PNL systems, employing Hirota’s bilinearization method to derive analytical solutions for ring-like bright–bright vector two-component one-soliton solutions. This study examines the dynamic evolution of these solutions under varying dispersion and nonlinearity conditions and investigates the impact of gain and loss on their behavior. Furthermore, the shape of the obtained solitons is determined by the parameters s and q, while the Hermite parameters p and n modulate the formation of additional layers along the z-axis, represented by $p+1$ and $n+1$, respectively. Our findings address existing gaps in understanding extreme waves in partially nonlocal media and offer insights into managing these phenomena in practical systems, such as optical fibers. The results contribute to the theoretical framework of high-dimensional wave phenomena and provide a foundation for future research in wave dynamics and energy management in complex media. Full article

C+L+S multiband (MB) optical transmission has the potential to increase the capacity of optical transport networks, and thus, it is a possible solution to cope with the traffic increase expected in the years to come. However, the introduction of MB optical technology needs to come together with the needed tools that support network planning and operation. In particular, quality of transmission (QoT) estimation is needed for provisioning optical MB connections. In this paper, we concentrate on modelling MB optical transmission for provide fast and accurate QoT estimation and propose machine learning (ML) approaches based on neural networks, which can be easily integrated into an optical layer digital twin (DT) solution. We start by considering approaches that can be used for accurate signal propagation modelling. Even though solutions such as the split-step Fourier method (SSFM) for solving the nonlinear Schrödinger equation (NLSE) have limited application for QoT estimation during provisioning because of their very high complexity and time consumption, they could be used to generate datasets for ML model creation. However, even that can be hard to carry out on a fully loaded MB system with hundreds of channels. In addition, in MB optical transmission, interchannel stimulated Raman scattering (ISRS) becomes a major effect, which adds more complexity. In view of that, the fourth-order Runge–Kutta in the interaction picture (RK4IP) method, complemented with an adaptive step size algorithm to further reduce the computation time, is evaluated as an alternative to reduce time complexity. We show that RK4IP provided an accuracy comparable to that of the SSFM with reduced computation time, which enables its application for MB optical transmission simulation. Once datasets were generated using the adaptive step size RK4IP method, two ML modelling approaches were considered to be integrated in the OCATA DT, where models predict optical signal propagation in the time domain. Being able to predict the optical signal in the time domain, as it will be received after propagation, opens opportunities for automating network operation, including connection provisioning and failure management. In this paper, we focus on comparing the proposed ML modelling approaches in terms of the models’ general and QoT estimation accuracy. Full article

Short time series are fundamental in the foreign exchange market due to their ability to provide real-time information, allowing traders to react quickly to market movements, thus optimizing profits and mitigating risks. Economic transactions show a strong connection to foreign currencies, making exchange rate prediction challenging. In this study, the exchange rate estimation between the US dollar (USD) and the Chilean peso (CLP) for a short period, from 2 August 2021 to 31 August 2022, is modeled using the nonlinear Schrödinger equation (NLSE) and calculated with the fourth-order Runge–Kutta method, respectively. Additionally, the daily fluctuations of the current exchange rate are characterized using the Hurst exponent, H, and later used to generate short synthetic fluctuations to predict the USD–CLP exchange rate. The results show that the USD–CLP exchange rate can be estimated with an error of less than $5\%$, while when using short synthetic fluctuations, the exchange rate shows an error of less than $10\%$. Full article

The measurement of deep water gravity wave elevations using in situ devices, such as wave gauges, typically yields spatially sparse data due to the deployment of a limited number of costly devices. This sparsity complicates the reconstruction of the spatio-temporal extent of surface elevation and presents an ill-posed data assimilation problem, which is challenging to solve with conventional numerical techniques. To address this issue, we propose the application of a physics-informed neural network (PINN) to reconstruct physically consistent wave fields between two elevation time series measured at distinct locations within a numerical wave tank. Our method ensures this physical consistency by integrating residuals of the hydrodynamic nonlinear Schrödinger equation (NLSE) into the PINN’s loss function. We first showcase a data assimilation task by employing constant NLSE coefficients predetermined from spectral wave properties. However, due to the relatively short duration of these measurements and their possible deviation from the narrow-band assumptions inherent in the NLSE, using constant coefficients occasionally leads to poor reconstructions. To enhance this reconstruction quality, we introduce the base variables of frequency and wavenumber, from which the NLSE coefficients are determined, as additional neural network parameters that are fine tuned during PINN training. Overall, the results demonstrate the potential for real-world applications of the PINN method and represent a step toward improving the initialization of deterministic wave prediction methods. Full article