Search Results (18)

In this work, we explore the nonlinear wave phenomena of the (3+1)-dimensional Boussinesq (II) equation, a significantly higher-dimensional model that describes dispersive wave propagation in fluid dynamics, plasma systems, and nonlinear optics. Using exact analytic and qualitative dynamic approaches, we study a wide range of solutions and stability characteristics of the model. Initially, we use the Hirota bilinear method to obtain a number of exact solutions, such as breather waves, two-wave interaction solutions, and other types of localized nonlinear waves. These solutions display remarkable physical properties, including periodic energy trapping, oscillatory modulations, and nonlinear wave interactions in higher dimensions. In addition, the $(m+{\displaystyle \frac{1}{G^{\prime }}})$-expansion method is used to derive new soliton solutions, such as bright solitary waves and W-shaped solitons, which are found to be stable and undergo pulse-shaping dynamics under certain conditions. Three-dimensional, two-dimensional, and contour plots are displayed for some of the solutions to demonstrate the physical significance of the results. The visualizations reveal the presence of localized waves, wave interactions, periodical breathing, and stable soliton profiles. Furthermore, we conduct modulation instability analysis to describe the conditions under which small perturbations of continuous wave backgrounds are unstable. The dispersion relation and the instability gain spectrum are obtained, which explain the formation of breathers, soliton trains, and other coherent structures. Furthermore, a Galilean transformation converts the governing equation into a planar nonlinear dynamical system, enabling its qualitative study. The Hamiltonian structure is revealed, and the fixed points are identified as centers, saddles, and cusps through bifurcation analysis. To investigate more complex dynamics, a periodic forcing term is introduced into the system, resulting in chaos in the forced system. The chaotic behavior is confirmed via phase portraits, three-dimensional attractors, time series, Poincaré sections, return maps, fractal dimension, and positive Lyapunov exponents. We also perform a sensitivity test to show the effect of initial condition variations on the system’s long-term dynamics. The findings greatly expand the exact solution set and dynamics of the (3+1)-dimensional Boussinesq equation (II). The analytical approach presented in this paper can also be applied to other multidimensional nonlinear evolution equations of mathematical physics. Full article

Ion-acoustic waves in dense quantum plasmas are strongly influenced by Fermi degeneracy, Landau quantization, and finite-temperature effects, and in many relevant environments, they also experience memory and nonlocal transport processes that cannot be captured within the planar integer Korteweg-de Vries (KdV) paradigm. In the present work, we revisit this problem by considering a two-fluid, partially degenerate electron-ion plasma in which electron trapping in the presence of a quantizing field and finite temperature is taken into account. Starting from the normalized fluid-Poisson system appropriate for such magnetized quantum plasmas, the reductive perturbation technique is used to derive the planar integer KdV equation for weakly nonlinear ion-acoustic disturbances. Within this integer-order KdV framework, we recast the evolution equation as a planar dynamical system, construct the associated Hamiltonian and effective Sagdeev-like potential, and demonstrate the existence of compressive solitary waves and nonlinear periodic modes via homoclinic and periodic phase-space orbits. Exact multi-soliton solutions and interaction states are then obtained by combining Hirota’s direct bilinear method with generalized Wronskian representations, allowing us to describe not only standard one-, two-, and three-soliton profiles but also positon-negaton interactions relevant to magnetized, partially degenerate plasmas. To incorporate hereditary and history-dependent effects that arise from anomalous transport and nonlocal temporal response in dense environments, we extend the model by introducing a Caputo time-fractional derivative, thereby obtaining a time-fractional KdV (FKdV) equation that continuously connects the classical KdV limit to fractional dynamics. The FKdV equation is analyzed using the Tantawy technique. This semi-analytical iterative scheme yields rapidly convergent series approximations for the fractional ion-acoustic soliton and provides explicit control of the approximation error. The fractional solutions show that varying the order of the Caputo derivative modifies the amplitude, width, and temporal relaxation of the solitary structures and can even split the pulse into two distinct lobes, in contrast with the nearly rigid propagation predicted by the integer-order KdV equation. Taken together, these results clarify how Landau quantization, finite electron temperature, and fractional-order memory jointly shape the morphology, robustness, and interaction properties of ion-acoustic structures in strongly magnetized quantum plasmas of astrophysical and high-energy-density laboratory interest. Full article

Monitoring urban subsidence over ultra-long periods using time-series Interferometric synthetic aperture radar (InSAR) technology is critically important. Conventional approaches, however, face two main limitations: significant atmospheric phase residuals in complex urban settings, and discontinuous temporal time-series with short temporal coverage due to single-platform data constraints. To address these limitations, this study presents a new method for estimating ultra-long-term subsidence time series in urban areas, which combines Interferometric Subset Stacking (ISS) with multi-platform data fusion (DF). The methodology firstly processes TerraSAR-X and Sentinel-1A datasets through differential interferometry and applies ISS for atmospheric phase suppression. Next, bilinear interpolation unifies the spatial resolution and aligns the spatial reference frames of the two datasets. Subsequently, joint modeling derives subsidence velocities. Finally, temporal integration via linear interpolation and moving averaging produces a unified spatio-temporal deformation sequence. Applied to the Beijing region, China, this approach generated a 12-year ultra-long-term subsidence time series result (2012–2024), revealing maximum cumulative subsidence of 1100 mm spatially correlated with groundwater extraction patterns. Validation against Global Navigation Satellite System (GNSS) data showed strong agreement (correlation coefficient: 0.94, Root Mean Square Error (RMSE): 6.3 mm). The method achieved substantial atmospheric reduction—67.7% for Sentinel-1A and 24.1% for TerraSAR-X—representing approximately 15–20% accuracy improvement over conventional Generic Atmospheric Correction Online Service (GACOS) for InSAR. By effectively utilizing multi-platform data, this approach makes fuller use of the available phase information and compensates for the temporal gaps inherent in single-satellite datasets. It thus offers a valuable framework for long-term urban deformation monitoring. Full article

This study presents a comparative analysis of wave buoy analogy models for sea state estimation. A nonparametric, response amplitude operator-based model is introduced as a physics-based approach, while a convolutional neural network is adopted as a machine learning approach. Using time-domain simulation data of wave-induced ship motions under various operating conditions, the accuracy and reliability of each model’s estimation are evaluated. The sensitivity of the physics-based model to operating conditions is examined, along with optimization strategies such as hyperparameter tuning. In particular, regularization techniques based on bilinear and B-spline surface fitting are applied to the nonparametric model, and the effects of interpolation techniques on model performance are assessed. For the machine learning model, a parametric study is conducted to determine input data types and formats, including time series and spectral representations, as well as the required length of the time window and dataset volume. Finally, the feasibility of the proposed neural network in estimating not only sea state parameters but also loading and navigational information, such as ship speed and GM, is discussed. Full article

Red-edge bands are the most essential spectral data for multispectral remote sensing images, with them playing a critical role in monitoring vegetation growth status at regional and global scales. However, the absence of red-edge bands limits the applicability of Landsat images, the most widely used remote sensing data, to vegetation monitoring. This study proposes an innovative method to reconstruct Landsat’s red-edge bands. The consistency in corresponding bands of Landsat OLI and Sentinel-2 MSI was first investigated using different resampling approaches and atmospheric correction algorithms. Three machine learning algorithms (ridge regression, gradient boosted regression tree (GBRT), and random forest regression) were then employed to build the red-edge reconstruction model for different vegetation types. With the optimal model, three red-edge bands of Landsat OLI were subsequently obtained in alignment with their derived vegetation indices. Our results showed that bilinear interpolation resampling, in combination with the LaSRC atmospheric correction algorithm, achieved high consistency between the matching bands of OLI and MSI (R² > 0.88). With the GBRT algorithm, three simulated OLI red-edge bands were highly consistent with those of MSI, with an R² > 0.96 and an RMSE < 0.0122. The derived Landsat red-edge indices coincide with those of Sentinel-2, with an R² of 0.78 to 0.95 and an rRMSE of 3.37% to 21.64%. This study illustrates that the proposed red-edge reconstruction method can extend the spectral domain of Landsat OLI and enhance its applicability in global vegetation remote sensing. Meanwhile, it provides potential insight into historical Landsat TM/ETM+ data enhancement for improving time-series vegetation monitoring. Full article

This paper introduces a novel numerical approach for solving fractional stochastic differential equations (FSDEs) using bilinear time-series models, driven by the Caputo–Katugampola (C-K) fractional derivative. The C-K operator generalizes classical fractional derivatives by incorporating an additional parameter, enabling the enhanced modeling of memory effects and hereditary properties in stochastic systems. The primary contribution of this work is the development of an efficient numerical framework that combines bilinear time-series discretization with the C-K derivative to approximate solutions for FSDEs, which are otherwise analytically intractable due to their nonlinear and memory-dependent nature. We rigorously analyze the impact of fractional-order dynamics on system behavior. The bilinear time-series framework provides a computationally efficient alternative to traditional methods, leveraging multiplicative interactions between past observations and stochastic innovations to model complex dependencies. A key advantage of our approach is its flexibility in handling both stochasticity and fractional-order effects, making it suitable for applications in a famous nuclear physics model. To validate the method, we conduct a comparative analysis between exact solutions and numerical approximations, evaluating convergence properties under varying fractional orders and discretization steps. Our results demonstrate robust convergence, with simulations highlighting the superior accuracy of the C-K operator over classical fractional derivatives in preserving system dynamics. Additionally, we provide theoretical insights into the stability and error bounds of the discretization scheme. Using the changes in the number of simulations and the operator parameters of Caputo–Katugampola, we can extract some properties of the stochastic fractional differential model, and also note the influence of Brownian motion and its formulation on the model, the main idea posed in our contribution based on constructing the fractional solution of a proposed fractional model using known bilinear time series illustrated by application in nuclear physics models. Full article

Interfacial adhesion is one of the key factors affecting the reliability of micro–nano systems. The adhesion contact mechanism is still unclear as the time-dependent viscoelasticity of soft materials. To clarify the adhesion interaction, the pull-off detachment between the rigid flat punch and viscoelastic substrate is explored considering the viscoelasticity of soft materials and rate-dependent adhesion. Taking the Lennard-Jones (L-J) potential characterizing interfacial adhesion and the Prony series defining the viscoelasticity of materials as references, the bilinear cohesion zone model (CZM) and standard Maxwell model are employed, and an adhesion analysis framework is established by combining finite element technology. The influence laws of the loading and unloading rates, material relaxation coefficients and size effect on adhesion pull-off behavior are revealed. The results show that the pull-off force is independent of the material relaxation effect and related to the unloading rate. When $\widehat{v}$ ≥ 50 or $\widehat{v}$ < 0.01, the pull-off force has nothing to do with the unloading rate, but when 0.01 < $\widehat{v}$ < 50, the pull-off force increases with the increasing unloading rate. Also, it is controlled by the size effect, and the changing trend conforms to the MD-n model proposed by Jiang. The energy required for interfacial separation (i.e., effective adhesion work) is a result of the comprehensive influence of unloading rates, material properties and the relaxation effect, which is consistent with Papangelo1’s research results. In addition, we derive the critical contact radius of the transition from the Kendall solution to the strength control solution. This work not only provides a detailed solution for the interfacial adhesion behavior but also provides guidance for the application of adhesion in Micro-Electro-Mechanical Systems (MEMSs). Full article

This paper introduces Tensor Visibility Graph-enhanced Attention Networks (TVGeAN), a novel graph autoencoder model specifically designed for MTS learning tasks. The underlying approach of TVGeAN is to combine the power of complex networks in representing time series as graphs with the strengths of Graph Neural Networks (GNNs) in learning from graph data. TVGeAN consists of two new main components: TVG which extend the capabilities of visibility graph algorithms in representing MTSs by converting them into weighted temporal graphs where both the nodes and the edges are tensors. Each node in the TVG represents the MTS observations at a particular time, while the weights of the edges are defined based on the visibility angle algorithm. The second main component of the proposed model is GeAN, a novel graph attention mechanism developed to seamlessly integrate the temporal interactions represented in the nodes and edges of the graphs into the core learning process. GeAN achieves this by using the outer product to quantify the pairwise interactions of nodes and edges at a fine-grained level and a bilinear model to effectively distil the knowledge interwoven in these representations. From an architectural point of view, TVGeAN builds on the autoencoder approach complemented by sparse and variational learning units. The sparse learning unit is used to promote inductive learning in TVGeAN, and the variational learning unit is used to endow TVGeAN with generative capabilities. The performance of the TVGeAN model is extensively evaluated against four widely cited MTS benchmarks for both supervised and unsupervised learning tasks. The results of these evaluations show the high performance of TVGeAN for various MTS learning tasks. In particular, TVGeAN can achieve an average root mean square error of 6.8 for the C-MPASS dataset (i.e., regression learning tasks) and a precision close to one for the SMD, MSL, and SMAP datasets (i.e., anomaly detection learning tasks), which are better results than most published works. Full article

Due to the prevalent challenges of inadequate accuracy, unstandardized parameters, and suboptimal efficiency with regard to icing prediction, this study introduces an innovative online method for icing prediction based on Robust STL–BTSF and IBWO–LSSVR. Firstly, this study adopts the Robust Seasonal Decomposition of Time Series and Bilinear Temporal–Spectral Fusion (Robust STL–BTSF) approach, which is demonstrably effective for short-term and limited sample data preprocessing. Subsequently, injecting a multi-faceted enhancement approach to the Beluga Whale Optimization algorithm (BWO), which integrates a nonlinear balancing factor, a population optimization strategy, a whale fall mechanism, and an ascendant elite learning scheme. Then, using the Improved BWO (IBWO) above to optimize the key hyperparameters of Least Squares Support Vector Regression (LSSVR), a superior offline predictive part is constructed based on this approach. In addition, an Incremental Online Learning algorithm (IOL) is imported. Integrating the two parts, the advanced online icing prediction model for transmission lines is built. Finally, simulations based on actual icing data unequivocally demonstrate that the proposed method markedly enhances both the accuracy and speed of predictions, thereby presenting a sophisticated solution for the icing prediction on the transmission lines. Full article

This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko–Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements. Full article

This paper focuses on the estimation and simulation of a specific subset of bilinear time series models characterized by dynamic exponential coefficients. Employing an exponential framework, we delve into the implications of the exponential function for our estimation process. Our primary aim is to estimate the coefficients of the proposed model using exponential coefficients derived from time-varying parameters. Through this investigation, our goal is to shed light on the asymptotic behaviors of the estimators and scrutinize their existence and probabilistic traits, drawing upon the foundational theorem established by Klimko and Nilsen. The least squares approach is pivotal in both estimating coefficients and analyzing estimator behavior. Moreover, we present a practical application to underscore the real-world implications of our research. By offering concrete examples of applications and simulations, we endeavor to provide readers with a comprehensive understanding of the implications of our work within the realm of time series analysis, specifically focusing on bilinear models and time-varying exponential coefficients. This multifaceted approach underscores the potential impact and practical relevance of our findings, contributing to the advancement of the field of time series analysis. To discern the symmetry characteristics of the model, we estimate it using coefficients that sum to zero and conduct a brief comparative analysis of two bilinear models. Full article

Predicting stock prices has long been the holy grail for providing guidance to investors. Extracting effective information from Limit Order Books (LOBs) is a key point in high-frequency trading based on stock-movement forecasting. LOBs offer many details, but at the same time, they are very noisy. This paper proposes a differential transformer neural network model, dubbed DTNN, to predict stock movement according to LOB data. The model utilizes a temporal attention-augmented bilinear layer (TABL) and a temporal convolutional network (TCN) to denoise the data. In addition, a prediction transformer module captures the dependency between time series. A differential layer is proposed and incorporated into the model to extract information from the messy and chaotic high-frequency LOB time series. This layer can identify the fine distinction between adjacent slices in the series. We evaluate the proposed model on several datasets. On the open LOB benchmark FI-2010, our model outperforms other comparative state-of-the-art methods in accuracy and F1 score. In the experiments using actual stock data, our model also shows great stock-movement forecasting capability and generalization performance. Full article

The increasing availability of Synthetic Aperture Radar (SAR) images facilitates the generation of rich Differential Interferometric SAR (DInSAR) data. Temporal analysis of DInSAR products, and in particular deformation Time Series (TS), enables advanced investigations for ground deformation identification. Machine Learning algorithms offer efficient tools for classifying large volumes of data. In this study, we train supervised Machine Learning models using 5000 reference samples of three datasets to classify DInSAR TS in five deformation trends: Stable, Linear, Quadratic, Bilinear, and Phase Unwrapping Error. General statistics and advanced features are also computed from TS to assess the classification performance. The proposed methods reported accuracy values greater than 0.90, whereas the customized features significantly increased the performance. Besides, the importance of customized features was analysed in order to identify the most effective features in TS classification. The proposed models were also tested on 15000 unlabelled data and compared to a model-based method to validate their reliability. Random Forest and Extreme Gradient Boosting could accurately classify reference samples and positively assign correct labels to random samples. This study indicates the efficiency of Machine Learning models in the classification and management of DInSAR TSs, along with shortcomings of the proposed models in classification of nonmoving targets (i.e., false alarm rate) and a decreasing accuracy for shorter TS. Full article

Time series prediction has been studied for decades due to its potential in a wide range of applications. As one of the most popular technical indicators, moving average summarizes the overall changing patterns over a past period and is frequently used to predict the future trend of time series. However, traditional moving average indicators are calculated by averaging the time series data with equal or predefined weights, and ignore the subtle difference in the importance of different time steps. Moreover, unchanged data weights will be applied across different time series, regardless of the differences in their inherent characteristics. In addition, the interaction between different dimensions of different indicators is ignored when using the moving averages of different scales to predict future trends. In this paper, we propose a learning-based moving average indicator, called the self-attentive moving average (SAMA). After encoding the input signals of time series based on recurrent neural networks, we introduce the self-attention mechanism to adaptively determine the data weights at different time steps for calculating the moving average. Furthermore, we use multiple self-attention heads to model the SAMA indicators of different scales, and finally combine them through a bilinear fusion network for time series prediction. Extensive experiments on two real-world datasets demonstrate the effectiveness of our approach. The data and codes of our work have been released. Full article

Recent literature reflects the substantial progress in combining spatial, temporal and spectral capacities for remote sensing applications. As a result, new issues are arising, such as the need for methodologies that can process simultaneously the different dimensions of satellite information. This paper presents PLS regression extended to three-way data in order to integrate multiwavelengths as variables measured at several dates (time-series) and locations with Sentinel-2 at a regional scale. Considering that the multi-collinearity problem is present in remote sensing time-series to estimate one response variable and that the dataset is multidimensional, a multiway partial least squares (N-PLS) regression approach may be relevant to relate image information to ground variables of interest. N-PLS is an extension of the ordinary PLS regression algorithm where the bilinear model of predictors is replaced by a multilinear model. This paper presents a case study within the context of agriculture, conducted on a time-series of Sentinel-2 images covering regional scale scenes of southern France impacted by the heat wave episode that occurred on 28 June 2019. The model has been developed based on available heat wave impact data for 107 vineyard blocks in the Languedoc-Roussillon region and multispectral time-series predictor data for the period May to August 2019. The results validated the effectiveness of the proposed N-PLS method in estimating yield loss from spectral and temporal attributes. The performance of the model was evaluated by the R² obtained on the prediction set (0.661), and the root mean square of error (RMSE), which was 10.7%. Limitations of the approach when dealing with time-series of large-scale images which represent a source of challenges are discussed; however, the N–PLS regression seems to be a suitable choice for analysing complex multispectral imagery data with different spectral domains and with a clear temporal evolution, such as an extreme weather event. Full article