Search Results (80)

Under climate change, the increasing typhoon intensity poses a severe threat to fishery harbor safety through storm surges and extreme waves. Traditional empirical management approaches fail to capture the complex wave-surge coupling inside harbors, leading to risk blind spots in berth allocation. This study enhances the fishery harbor disaster resilience by employing high-resolution coupled wave-storm surge modeling, taking Xinying Central Fishing Harbor (Hainan, China) during Super Typhoon Yagi (September 2024) as a case study. A Holland typhoon model integrated with ERA5 reanalysis data was used to reconstruct the wind field, which subsequently drove a one-way coupled MIKE 21 FM–SW model to simulate regional tides and deep-water waves. A Boussinesq wave model was then applied to resolve nearshore shallow-water wave transformations inside the harbor. Model validation showed strong agreement with observations: correlation coefficients of 0.97 for tides in Xinying station and 0.95, 0.97, 0.93 for significant wave heights in three buoys around Hainan island, with root-mean-square errors of 0.19 m and 0.67, 0.69, 0.31 m, respectively. The Boussinesq wave simulations revealed detailed spatial distributions of wave heights inside the harbor during the typhoon. Based on these simulations, a dynamic berth zoning strategy was developed, mapping safety zones for different vessel sizes according to wave-height tolerance (e.g., ≤0.6 m for medium-sized trawlers). This framework can provide potential support for decision-making regarding fishing vessel refuge during typhoons, maximizing safe capacity while minimizing capsizing risks. Overall, this study demonstrates a feasible pathway from advanced numerical modeling to practical engineering management, supporting a transition from experience-based to data- and model-driven disaster prevention for coastal fishery harbors. Full article

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

This paper examines the sixth-order generalized Boussinesq equation, which describes the dynamics of shallow-water waves, including the effects of surface tension. The study introduces Kudryashov’s R-function neural network approach for the first time, aiming to provide exact solutions to the nonlinear differential equation that represents the mathematical model of the sixth-order generalized Boussinesq equation. This technique incorporates the solutions of a nonlinear differential equation into neural networks, using them as an activation function within the hidden layer. In line with previous research on this topic, two categories of solutions are derived: solitary wave and shock wave solutions. Additionally, this paper includes 3D and 2D graphs to visually represent the solutions obtained. Full article

In this study, we consider an extended form of the Kadomtsev–Petviashvili–Boussinesq equation motivated by wave propagation phenomena in dissipative media. The primary aim of this work is to construct exact analytical solutions and clarify the types of nonlinear wave structure admitted by the considered model. For this purpose, the Riccati equation expansion method is applied for the first time within this framework. This method allows us to obtain several distinct families of solitary wave solutions whose qualitative behaviors and physical characteristics are illustrated through graphical representations. In addition, modulation instability analysis is carried out to assess the stability of continuous wave solutions and further elucidate the underlying nonlinear dynamics of the system. Full article

This study examines a nonlinear partial differential equation, namely the (3+1)-dimensional Boussinesq-type equation. To explore this model, three versatile analytical approaches are applied: the Exp-function method, the Kudryashov method, and the Riccati equation method. Using these techniques, a range of exact analytical solutions is derived, exhibiting diverse structural forms such as periodic, kink-type, rational, and trigonometric solutions. The analysis reveals the rich dynamical behavior of the equation and demonstrates its effectiveness in modeling a variety of nonlinear wave phenomena across different physical contexts. Several of the obtained solutions are illustrated through graphical representations for better interpretation. The results include hyperbolic, trigonometric, and rational function solutions, along with a sensitivity analysis. To highlight the physical relevance of the findings, suitable parameter values are selected, and the corresponding wave behaviors are visualized using three-dimensional and contour plots generated with Maple 2024. Overall, the study provides valuable insights into the mechanisms underlying the generation and propagation of complex nonlinear phenomena in fields such as fluid dynamics, optical fiber systems, plasma physics, and ocean wave transmission. Full article

The coupled nonlinear system of fractional Boussinesq–Burger equations that may be utilized to model the propagation of shallow water waves is solved in this study using a novel numerical approach. The fractional derivatives in Caputo–Fabrizio and Atangana–Baleanu manner are executed in the system under consideration. The exact solutions of the proposed nonlinear fractional system are shown in the classical scenario of fractional order at $\mathrm{ß}=1$, whereas the approximate solutions are derived using the natural decomposition method. The series solution is generated such that it is simple to compute. Our results are compared with the exact results which clearly show that the suggested approach solutions quickly converge to the known accurate results. We acquire some analysis of the absolute error by comparing the approximate values with their corresponding precise solutions throughout the provided computations. Numerical and graphical simulations are used to confirm the usefulness of the suggested approach, and the outcomes are compared with well-known methods like the fractional decomposition method (FDM) and Laplace residual power series method (LRPSM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of the proposed system. The obtained outcomes ensure that the suggested approach is more effective and examines the highly nonlinear problems arising in engineering and science. Full article

The (2+1)-dimensional Boussinesq equation is a fundamental model in nonlinear wave theory, governing shallow-water wave propagation, coastal dynamics in ocean engineering, and long waves in geophysical fluid systems such as atmospheric and oceanic currents. We present a novel neural network symbolic computation framework that seamlessly integrates neural architectures for powerful function approximation with symbolic manipulation for exact algebraic resolution, eliminating the need for bilinear transformations and thereby substantially reducing computational complexity. Applying this framework, we derive five previously unreported exact analytical solutions using carefully designed neural network configurations and probe functions. These solutions provide valuable tools for modeling ocean internal waves, coastal engineering simulations, and nonlinear optical pulse dynamics. In practice, the method delivers faster and more accurate simulations, improving engineering design and environmental prediction capabilities. By synergistically combining neural networks with symbolic computation, our approach surpasses traditional numerical methods and physics-informed neural networks in both accuracy and efficiency, opening new avenues for solving complex nonlinear partial differential equations. Full article

The dynamical processes in the atmosphere are coupled with the chemistry of the atmosphere. Internal gravity waves influence the distribution of chemical constituents in the atmosphere through their effects on the background wind or mean flow. We examine a coupled system of equations comprising a nonlinear transport equation of Fisher type for the distribution of the chemical species, along with nonlinear Boussinesq equations for internal gravity waves in a vertically stratified and vertically sheared fluid flow in a two-dimensional region. In our model, a horizontally localized gravity-wave packet is generated and propagates upward into a localized region where the chemical species is present. Numerical solutions show that the wave-induced mean flow resulting from nonlinear gravity-wave interactions in the vicinity of a critical level leads to modifications in the distribution of the chemical. An asymptotic analysis of a related qualitatively similar problem gives us information on the dominant behaviour of the chemical concentration perturbation. We conclude that nonlinearity and vertical shear play a vital role in the interplay between gravity-wave dynamics and chemical distributions in the atmosphere. Full article

This study investigates typhoon-induced oscillations within Youngan Harbor, southwestern Taiwan, which frequently compromise port operations and cause dock overtopping. Two representative wave-resolving models, the Boussinesq-type FUNWAVE-TVD and the non-hydrostatic XBeach-NH, were applied to simulate a typhoon event and evaluate their predictive performance against field observations. Both models underestimated significant wave height across all frequency bands. Spectral analysis revealed that FUNWAVE-TVD generated higher energy in the infragravity and very-low-frequency ranges, whereas XBeach-NH exhibited greater energy in the swell and wind-wave bands. Spatial resonance patterns further indicated that a berth, located in a nodal region, experienced reduced tranquility due to considerable horizontal currents. Conversely, wave overtopping at a dock was driven by amplified vertical water-level oscillations in an antinodal region. These contrasting responses highlight the sensitivity of the models to nonlinear wave interactions and underscore the critical role of simulating harbor currents, emphasizing the need for careful model selection in harbor tranquility assessment. Full article

In this study, we consider a (2+1)-dimensional integrable Boussinesq equation, where the Hirota method of positive logarithmic transformation is used to convert it into a bilinear form. We proceeded by employing different test functions, through which we obtained breather solutions, two-wave solutions, lump-periodic solutions, and new interaction solutions. The resulting soliton dynamics for the governing model are also derived using the enhanced modified extended tanh function method, where varieties of solutions, such as trigonometric, hyperbolic, and rational forms, were obtained. The derived solutions may hold significant potential for explaining real-world physical phenomena in fields like mathematical physics, plasma physics, and nonlinear optics. The accuracy and reliability of the solutions were tested by substituting them back into the original equation using Python, highlighting the method’s robustness, precision, and reliability. By choosing appropriate physical parameters, we showcased the rich diversity and dynamic behavior of the obtained soliton structures. In other words, the graphical representations in 3D, contour, and 2D were provided for some of the obtained results. The modulation instability analysis and gain spectrum of the model are also provided. The importance of the obtained results in the area of (2+1)-dimensional integrable equation application was also highlighted. Full article

We study here the Boussinesq-type Korteweg–de Vries (KdV) equation, a nonlinear partial differential equation, for describing the wave propagation of long, nonlinear, and dispersive waves in shallow water and other physical scenarios. In order to obtain novel families of wave solutions, we apply two efficient analytical techniques: the Modified Extended tanh (ME-tanh) method and the Modified Residual Power Series Method (mRPSM). These methods are used for the very first time in this equation to produce both exact and high-order approximate solutions with rich wave behaviors including soliton formation and energy localization. The ME-tanh method produces a rich class of closed-form soliton solutions via systematic simplification of the PDE into simple ordinary differential forms that are readily solved, while the mRPSM produces fast-convergent approximate solutions via a power series representation by iteration. The accuracy and validity of the results are validated using symbolic computation programs such as Maple and Mathematica. The study not only enriches the current solution set of the Boussinesq-type KdV equation but also demonstrates the efficiency of hybrid analytical techniques in uncovering sophisticated wave patterns in multimensional spaces. Our findings find application in coastal hydrodynamics, nonlinear optics, geophysics, and the theory of elasticity, where accurate modeling of wave evolution is significant. Full article

Rip currents at featureless beaches (i.e., beaches lacking sandbars or channels) are often hydrodynamically controlled, exhibiting intermittent and unpredictable behaviors that pose significant risks to recreational beach users. This study assessed occurrences of rip currents under a range of idealized morphology configurations and hydrodynamic wave forcing parameters using a wave-resolving Boussinesq-type model. Numerical experiments revealed that rip currents with durations on the time scale of 10 min are generated in the forms of vortex pairs, intensified eddies, mega-rips, and eddies shedding from longshore currents. In general, the key conditions that promote rip current formation at featureless beaches include shoreline curvature, headlands, moderately mild beach slopes (e.g., 0.02–0.03), normal or near-normal wave incidence, and large wave heights. Most importantly, this study highlights inherent uncertainties in rip current occurrences, particularly under conditions usually perceived as low risk: low wave heights, short wave periods, oblique wave incidence, and straight shorelines. These conditions can lead to transient rip currents and pose an unexpected hazard that coastal communities should be aware of. Full article

Accurately simulating wave breaking is crucial for modeling hydrodynamics over steep coral reef slopes, yet it remains a challenge for Boussinesq-type models like FUNWAVE-TVD. The model’s standard hybrid breaking mechanism, triggered by a fixed free surface elevation-to-depth ratio ($\eta /d>0.8$), often lacks physical sensitivity to local slope and wave conditions prevalent in reef environments and suffers from inaccuracies associated with using η as a direct proxy for wave height (H). This study introduces and validates a novel, enhanced hybrid breaking module within FUNWAVE-TVD, specifically designed to overcome these limitations on steep slopes. The core novelty lies in the synergistic implementation of two key components: (1) replacing the fixed threshold with a dynamic, physically-based criterion derived from the Modified Goda formula (MGO) by Rattanapitikon and Shibayama, which calculates the breaking wave height ($H_b$) based on local depth, slope, and deep-water wavelength; and (2) developing and applying a practical method, using the wave vertical asymmetry relationship proposed by Yu and Li, to dynamically convert the calculated $H_b$ into an equivalent breaking surface elevation threshold (${\eta }_b$). This derived dynamic threshold (${\eta }_b/d$) is then used to trigger the model’s existing switch from Boussinesq to Nonlinear Shallow Water Equations (NSWE), allowing for energy dissipation via shock-capturing while retaining the physical basis of the MGO criterion. The performance of this enhanced module was rigorously evaluated against five laboratory experiments of regular waves breaking on impermeable slopes ranging from mild (1:10) to extremely steep (1:1), contrasting results with the original FUNWAVE-TVD. The modified model demonstrates significantly improved accuracy (model skill increases ranging from 10.16% to 42.49%) compared to the original model for breaking location and wave height prediction on steeper slopes ($m\ge 1:6$). Conversely, tests on the 1:1 slope confirmed the inherent limitations of the MGO criterion itself under surging breaker conditions ($m\ge 1:2.3$), highlighting the applicability range. This work provides a validated methodology for incorporating slope-aware, dynamic breaking criteria effectively into hybrid Boussinesq models, offering a more robust tool for simulating wave processes on steep reef topographies. Full article

In this study, we propose a generalized framework based on the Simple Equations Method (SEsM) for finding exact solutions to systems of fractional nonlinear partial differential equations (FNPDEs). The key developments over the original SEsM in the proposed analytical framework include the following: (1) an extension of the original SEsM by constructing the solutions of the studied FNPDEs as complex composite functions which combine two single composite functions, comprising the power series of the solutions of two simple equations or two special functions with different independent variables (different wave coordinates); (2) an extension of the scope of fractional wave transformations used to reduce the studied FNPDEs to different types of ODEs, depending on the physical nature of the studied FNPDEs and the type of selected simple equations. One variant of the proposed generalized SEsM is applied to a mathematical generalization inspired by the classical Boussinesq model. The studied time-fractional Boussinesq-like system describes more intricate or multiphase environments, where classical assumptions (such as constant wave speed and energy conservation) are no longer applicable. Based on the applied SEsM variant, we assume that each system variable in the studied model supports multi-wave dynamics, which involves combined propagation of two distinct waves traveling at different wave speeds. As a result, numerous new multi-wave solutions including combinations of different hyperbolic, elliptic, and trigonometric functions are derived. To visualize the wave dynamics and validate the theoretical results, some of the obtained analytical solutions are numerically simulated. The new analytical solutions obtained in this study can contribute to the prediction and control of more specific physical processes, including diffusion in porous media, nanofluid dynamics, ocean current modeling, multiphase fluid dynamics, as well as several geophysical phenomena. Full article

A hybrid computational framework integrating the finite volume method (FVM) and finite difference method (FDM) is developed to solve two-dimensional, time-dependent nonlinear coupled Boussinesq-type equations (NCBTEs) based on Nwogu’s depth-integrated formulation. This approach models nonlinear dispersive wave forces acting on a stationary vessel and incorporates a frequency dispersion term to represent ship-wave generation due to a localized moving pressure disturbance. The computational domain is divided into two distinct regions: an inner domain surrounding the ship and an outer domain representing wave propagation. The inner domain is governed by the three-dimensional Laplace equation, accounting for the region beneath the ship and the confined space between the ship’s right side and a vertical quay wall. Conversely, the outer domain follows Nwogu’s 2D depth-integrated NCBTEs to describe water wave dynamics. Interface conditions are applied to ensure continuity by enforcing the conservation of volume flux and surface elevation matching between the two regions. The accuracy of this coupled numerical scheme is verified through convergence analysis, and its validity is established by comparing the simulation results with prior studies. Numerical experiments demonstrate the model’s capability to capture wave responses to simplified pressure disturbances and simulate wave propagation over intricate bathymetry. This computational framework offers an efficient and robust tool for analyzing nonlinear wave interactions with stationary ships or harbor structures. The methodology is specifically applied to examine the response of moored vessels to incident waves within Paradip Port, Odisha, India. Full article