Shallow Water Waves Research Articles

Orbital stability of smooth solitons for the modified Camassa-Holm equation

The modified Camassa-Holm equation with cubic nonlinearity is completely integrable and is considered a model for the unidirectional propagation of shallow-water waves. The localized smooth-wave solution exists uniquely, up to translation, within a certain range of the linear dispersive parameter. By constructing conserved H1 and L1 quantities in terms of the momentum variable m, this study demonstrates that the smooth soliton, when regarded as a solution of the initial-value problem for the modified Camassa-Holm equation, is orbitally stable to perturbations in the Sobolev space H3. Furthermore, the global well-posedness of the solution is established for certain initial data in Hs with s≥3.

The study of water wave scattering by series of submerged elastic scatterers

A submerged flexible breakwater can be used to control waves in shallow water as an advanced alternative to traditional rigid submerged structures. Submerged flexible breakwaters not only cost less to build than conventional submerged breakwaters but also allow ships and marine life to bypass them if they are deep enough. These marine structures decrease the intensity of the shock wave and prevent standing waves from forming. We present a solution for transmission/reflection using a series of submerged identical elastic scatterers. Scattering and transfer matrices will be used to compute the transmission and reflection coefficients for N-submerged elastic scatterers. The behavior of the transmission for different values of physical parameters such as submergence depth, length of the scatterer, the gap between two successive scatterers, stiffness of the scatterer, and the number of submerged elastic scatterers is given. In doing so, we can achieve the frequencies and the incident amplitudes at which these submerged elastic scatterers can be effectively used as breakwaters in water of finite depth. We have looked at the transmission and reflection behaviors for a range of physical parameter values, such as the submergence depth, the length of the scatterer, the spacing between successive scatterers, the stiffness of the elastic scatterer, and the number of scatterers over a wide frequency range. We have given a spectrum of frequencies for which, given certain values of physical parameters, we can achieve zero transmission. Furthermore, even at high frequencies, we have demonstrated an incidence amplitude range for which zero propagation of transmission is feasible.

Resonant Y-type soliton, interaction wave and other wave solutions to the (3+1)-dimensional shallow water wave equation

The current study aims to develop some novel wave solutions of the (3+1)-dimensional shallow water wave equation (SWWE). First, the Y-type soliton (YTS) solutions are developed via imposing the certain resonant condition to the N-solitons solutions (NSSs) that extracted by the Hirota bilinear approach. Then, the symbolic computation and ansatz function scheme are employed to explore the interaction wave solutions. Eventually, the sub-equation approach is utilized to find the diverse wave solutions which include the kink wave, singular wave and the singular periodic wave solutions. The dynamics of the obtained solutions are presented graphically for revealing the nonlinear physical characteristics. The investigative solutions in this article have not been probed elsewhere and can help us comprehend the nonlinear behaviors of the (3+1)-dimensional SWWE better.

Sufficient conditions of blowup to a shallow water wave equation

The blow-up features of a shallow water wave equation on the line R are investigated. The L2 conservation law is utilized to derive several estimates of solutions for the equation. Sufficient conditions for wave breaking and lifespan of the solutions are established. Our main results contain parts of the wave breaking conditions for the Fornberg–Whitham and Degasperis–Procesi equations in the previous literatures.

Optical soliton solutions of nonlinear differential Boussinesq water wave equation via two analytical techniques

In this article, we examine the optical soliton solutions of the 4th-order nonlinear Boussinesq water wave equation using the modified (G′/G2)− expansion and F-expansion methods. We utilize the similarity transformation to transform the partial differential form of the equation into an ordinary differential form. These techniques present a variety of solutions, including bright, dark, singular, dark-periodic, and singular-periodic soliton solutions. We plot the derived solutions in several profiles, such as 2D, 3D, and contour, to illustrate their physical appearance. The determined findings might be useful and have an enormous effect on the research of nonlinear phenomena in a variety of physical areas of science, like shallow water waves, acoustics, fluid dynamics, laser optics, communication systems, and heat transfer. The achieved results demonstrate the validity, applicability, and effectiveness of the presented method. We plot and validate the found solutions using the computational program MATHEMATICA.

Dynamics of geometric shape solutions for space-time fractional modified equal width equation with beta derivative

The modified equal width equation describes the propagation of shallow water waves in which nonlinear and dispersive effects are significant, including phenomena such as wave breaking, soliton interactions, ion-acoustic waves, energy transfer in plasma, and nonlinear stress waves. The aim of this article is to establish some novel and generic solutions to the space-time fractional modified equal width (MEW) equation using the two variable (θ′/θ, 1/θ)-expansion technique, a modification of the (θ′/θ)-expansion method. A wide range of geometric shapes and inclusive soliton solutions have been constructed, comprising rational, trigonometric, and hyperbolic solutions, along with their integration to the equation under consideration. The two- and three-dimensional graphs of the solitons, including periodic, V-shaped, bell-shaped, singular periodic, flat kink-shaped, plane-shaped, peakon, and parabolic, illustrate the physical aspects of the solitary wave solution and the effect of the fractional parameter. The results demonstrate the efficiency, appropriateness, and reliability of the adopted technique for investigating fractional-order nonlinear evolution equations in science, technology, and engineering.

For the Shallow Water Waves: Bilinear-Form and Similarity-Reduction Studies on a Boussinesq-Burgers System

For the Shallow Water Waves: Bilinear-Form and Similarity-Reduction Studies on a Boussinesq-Burgers System

Rogue wave, lump, kink, periodic and breather-like solutions of the (2+1)-dimensional KdV equation

In this paper, the ([Formula: see text])-dimensional KdV equation is investigated by using the bilinear neural network method (BNNM). We construct six neural network models, extending beyond single hidden layer models to create deeper and broader network structures (e.g. [3-3-1], [3-4-1], [3-1-3-1], [3-4-1-1], [3-2-2-1] and [3-2-3-1-1] models). Introducing specific activation functions into the neural network model enables the generation of various test functions, resulting in novel solutions for equations that include rogue wave solutions, lump-kink solutions, periodic soliton solution, breather-like solutions and lump solutions. The physical properties of these novel solutions are vividly depicted through three-dimensional plots, density plots, and curve plots. The findings contribute to a better understanding of the propagation behavior of shallow water waves.

Novel insights into the propagation of the generalized Boussinesq equation: Dynamics of bidirectional symmetric solitary waves

In this work, we explore the propagation of bidirectional solitary wave solutions in the generalized (2+1)-dimensional Boussinesq equation. The proposed model was recently introduced by Wazwaz and Kaur as an application to describe wave propagation in various multidimensional physical systems, including shallow water waves and similar phenomena. Via implementing a collection of effective approaches, we extract a diversity physical structures of the proposed model manifesting their propagation in a form of moving simultaneous symmetric bidirectional waves. The proposed methodologies will include the use of sine/cosine-function method, sinh/cosh-function method, tanh–coth expansion approach, and the rational form trial solutions including trigonometric and hyperbolic functions. By conducting graphical analyses, we illustrate the various physical structures and dynamic behaviors of this class of nonlinear partial differential equations. The results of this work are expected to facilitate the study of wave interactions, propagation dynamics, and soliton formation in multidimensional settings.

Short-term extreme response estimation methodology for marine umbilical cables with two configuration types

This study investigates the critical engineering problem of the short-term extreme responses of marine umbilical cables, which are essential components in offshore oil and gas operations. Umbilical cables play a pivotal role in connecting subsea systems to surface platforms, necessitating a safe design to withstand external loads. This study focused on two primary umbilical cable configuration types: deep-water catenary and shallow-water lazy wave. For each configuration type, 100 3-h dynamic numerical simulations were performed, with maximum values fitted by using the Gumbel distribution, and the most probable maximum (MPM) values were set as benchmark values. The average conditional exceedance rate (ACER) method and uncertainty-incorporated ACER method were applied to study the short-term extreme values of umbilical cable responses based on limited sample data. The predicted results have been compared with the MPM values, and the accuracy and uncertainty of different efficient extreme-value analysis methods are discussed. This study provides a comprehensive methodological framework for the reliability analysis and safety design of umbilical cables in marine engineering.

Integration of the Kaup–Boussinesq system via inverse scattering method

In this paper, we investigate the Cauchy problem for the KB system in the class of rapidly decreasing functions and give an algorithm of the constructing a solution to the Cauchy problem of the KB system in the class of rapidly decreasing functions via IST method when the scattering coefficient a(k) of the problem (6) has multiple zeros. In this case, the class of initial functions will be expanded by removing the assumption about simplicity of the scattering coefficient a(k) of the problem (6). We also present an efficient method to obtain the time evolution of scattering data. The advantage of this method is its reliability and the possibility of using other soliton equations to obtain the time evolution of scattering data. The resulting equalities completely determine the scattering data at any t, which allows applying the IST method to solve the Cauchy problem for the KB system. The results obtained will be useful in carrying out further analysis in the context of shallow water waves that arises in the context of oceanography.

Research on the design and optimal control of the power take-off (PTO) system for underwater eel-type power generators

Wave energy and ocean current energy are considered stable, reliable, and highly predictable renewable energy sources. The development of ocean current energy conversion technology is crucial in addressing power shortages. However, the low velocity of most ocean currents worldwide, typically less than 1.5 m/s, poses a challenge for traditional ocean current energy capture devices. Current estimations of power generation from underwater devices often significantly differ from actual results. This study presents an eel-type power generation device designed for underwater use, investigates the design and optimization of a hydraulic power take-off (PTO) system suitable for such devices, and examines key components of the hydraulic PTO system like variable hydraulic motors and generators. Analytical research was conducted to understand its operational characteristics, with a focus on components such as the accumulator, speed control valve, and energy storage flywheel. The article also delves into the hydrodynamic and energy conversion characteristics of the device under shallow water wave and current conditions, enhancing the hydraulic PTO system model and establishing an integrated calculation model for real-time data transmission during the calculation process. Through evaluations of pitch angle and power generation under varying sea conditions, the study explores the impact of hydraulic motor displacement and damping coefficient on the hydrodynamic and power generation characteristics of the device, further validated through sea trials. The study findings indicate a high level of parameter adaptation among the components of the hydraulic system. The addition of an accumulator to the system results in smoother output response curves, suggesting that the accumulator absorbs impacts and stabilizes the hydraulic power transmission system. The proposed comprehensive calculation method enables a more precise prediction of the performance of the eel power generation device in real marine environments, capturing its movement behavior and power generation characteristics. Adjusting the motor displacement or damping coefficient under specific conditions can optimize the total damping of the hydraulic PTO system for maximum power output. The optimal power point of the hydraulic power generation system varies for different sea conditions, with a potential power generation efficiency of up to 80.3% under certain circumstances.

Analytical Scheme for Time Fractional Kawahara and Modified Kawahara Problems in Shallow Water Waves

The Kawahara equation exhibits signal dispersion across lines of transmission and the production of unstable waves from the water in the broad wavelength area. This article explores the computational analysis for the approximate series of time fractional Kawahara (TFK) and modified Kawahara (TFMK) problems. We utilize the Shehu homotopy transform method (SHTM), which combines the Shehu transform (ST) with the homotopy perturbation method (HPM). He’s polynomials using HPM effectively handle the nonlinear terms. The derivatives of fractional order are examined in the Caputo sense. The suggested methodology remains unaffected by any assumptions, restrictions, or hypotheses on variables that could potentially pervert the fractional problem. We present numerical findings via visual representations to indicate the usability and performance of fractional order derivatives for depicting water waves in long-wavelength regions. The significance of our proposed scheme is demonstrated by the consistency of analytical results that align with the exact solutions. These derived results demonstrate that SHTM is an effective and powerful scheme for examining the results in the representation of series for time-fractional problems.

Kink, Dark, Bright, and Singular Optical Solitons to the Space–Time Nonlinear Fractional (41)-Dimensional Davey–Stewartson–Kadomtsev–Petviashvili Model+

The new types of exact solitons of the space–time fractional nonlinear (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) model are achieved by applying the unified technique and modified extended tanh-expansion function technique. A novel definition of the fractional derivative known as the truncated M-fractional derivative is also used. This model describes both the non-elastic and elastic interactions between internal waves. This model is used to represent intricate nonlinear phenomena like shallow-water waves, plasma physics, and others. The obtained results are in the form of kink, singular, bright, periodic, and dark solitons. The observed results are verified and represented by 2D and 3D graphs. The observed results are not present in the literature due to the use of fractional derivatives. The impact of the truncated M-fractional derivative on the observed results is also represented by graphs. Hence, our observed results are fruitful for the future study of these models. The applied techniques are simple, fruitful, and reliable in solving the other models in applied mathematics.

Exploring multi-soliton patterns, bifurcation analysis, and chaos in (2+1) dimensions: A study on nonlinear dynamics

This study explores the intrinsic characteristics of the (2+1)-dimensional Schwarz-Korteweg-de Vries equation used to describe shallow water waves. The multiple solitons are successfully constructed using a technique involving multiple exponential functions. Graphical representations of the results are provided in 3D, 2D, and density plots to assess the compatibility of the solutions. Also, the dynamic nature of studied equation is examined based on bifurcation and chaos theory for nonlinear systems. Bifurcation signifies how our dynamical system is affected by physical parameters in planar dynamical system. After that, we apply the external force on planar dynamical system to show the chaotic like behavior of the studied model. Such behavior is confirmed by utilizing different chaos detecting tools. The obtained results serve to explain the effectiveness and applicability of the utilized methodologies in comprehend the exact solution and qualitative behavior of nonlinear physical models.

Exploring the dynamics of shallow water waves and nonlinear wave propagation in hyperelastic rods: Analytical insights into the Camassa–Holm equation

The objective of this paper is to examine the analytical properties of the nonlinear [Formula: see text]-dimensional Camassa–Holm equation ([Formula: see text]), a fundamental model within the domain of nonlinear evolution equations. The aforementioned equation serves as a valuable tool in elucidating the unidirectional propagation of shallow water waves over a level terrain, as well as in representing certain nonlinear wave phenomena seen in cylindrical hyperelastic rods. We use the Khater III ([Formula: see text]hat.III) and improved Kudryashov ([Formula: see text]ud) technique to provide accurate solutions, drawing inspiration from the intricate mathematical framework of the [Formula: see text] issue. He’s variational iteration ([Formula: see text]) technique is used as a numerical methodology to assess the correctness of the generated answers. This strategy reveals a notable concurrence between the analytical and numerical outcomes. This alignment guarantees the suitability of the acquired solutions within the framework of the studied model. The importance of this investigation lies in its ability to improve our comprehension of the intricate dynamics regulated by the [Formula: see text] equation and its connections with other nonlinear evolution equations that describe shallow water wave behaviors and nonlinear wave propagation in cylindrical hyperelastic rods. The results of the study demonstrate novel analytical approaches, expanding the range of potential solutions and offering valuable insights into the physical characteristics of the interconnected wave phenomena. This research offers valuable insights and methodologies for addressing intricate mathematical models in shallow water wave theory and studying nonlinear waves in hyperelastic materials, therefore making substantial advances to the subject of nonlinear partial differential equations.

Fully screened two-dimensional magnetoplasmons and rotational gravity shallow water waves in a rectangle

Fully screened two-dimensional magnetoplasmons and rotational gravity shallow water waves in a rectangle

Dimensionless dynamics: Multipeak and envelope solitons in perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

In this work, the dimensionless form of the improved perturbed nonlinear Schrödinger equation with Kerr law of fiber nonlinearity is solved for distinct exact soliton solutions. We examined the multi-wave solitons and rational solitons of the governing equation using the logarithmic transformation and symbolic computation using an ansatz functions approach. Multi-wave solitons in fluid dynamics describe the situation in which a fluid flow shows several different regions (or peaks) of high concentration or intensity of a particular variable (e.g., velocity, pressure, or vorticity). Multi-wave solitons in turbulent flows might indicate the existence of several coherent structures, like eddies or vortices. These formations are areas of concentrated energy or vorticity in the turbulent flow. Understanding how these peaks interact and change is essential to comprehending the energy cascade and dissipation in turbulent systems. Furthermore, a sub-ordinary differential equation approach is used to create solutions for the Weierstrass elliptic function, periodic function, hyperbolic function, Chirped free, dark-bright (envelope solitons), and rational solitons, as well as the Jacobian elliptic function, periodic function, and rational solitons. Also, as the Jacobian elliptic function's' modulus m approaches values of 1 and 0, we find trigonometric function solutions, solitons-like solutions, and computed chirp free-solitons. Envelope solitons can arise in stratified fluids and spread over the interface between layers, such as layers in the ocean with varying densities. Their research aids in the management and prediction of wave events in artificial and natural fluid settings. In fluids, periodic solitons are persistent, confined wave structures that repeat on a regular basis, retaining their form and velocity over extended distances. These structures occur in a variety of settings, including internal waves in stratified fluids, shallow water waves, and even plasma physics.

Shallow-water waves through two new generalized multi-dimensional variable coefficient equations

The objective of this study is to propose and investigate two new forms of generalized variable coefficients within multi-dimensional equations describing shallow-water waves. We employ the Mathematica program to rigorously establish Painlevé's integrability for these two nonlinear equations. Subsequently, we constructed their bilinear forms and utilized Hirota's bilinear method to examine the dispersion relations and phase shifts of these two models that enable the derivative of multi-soliton solutions. Furthermore, diverse forms of lump-wave solutions are also considered. To illustrate the physical characteristics of these two models, we establish several graphical representations of the discovered solutions. These visualizations offer insights into the behavior, shape, and dynamics of both the multi-soliton, Peregrine soliton, lump wave, and rogue wave, enhancing our understanding of their physical significance. The two soliton solutions effectively replicate the shallow water waves, encompassing the T-, X-, and Y-types, along with other intricate interactions. Additionally, the lump and rogue wave structures are displayed to visually represent their spatial structures. These graphical representations offer a comprehensive view of the diverse wave phenomena observed in shallow water systems, aiding in the understanding of their spatial characteristics and interactions. Therefore, our findings indicate that the introduction of the two newly proposed integrable nonlinear evolution equations enhances the repertoire of integrable system models and aids in comprehending the distinctive characteristics of nonlinear dynamics in real-world applications.

Advancing computational models for wave dynamics applications with quartic trigonometric tension b-spline techniques

This paper presents a computational model using Quartic Trigonometric Tension B-spline (QT-TB) function with collocation approach for nonlinear dispersive wave equation. Shallow water wave modeling plays a crucial role in various fields, including coastal engineering, oceanography, and natural hazard assessment. The QT-TB function is employed for spatial derivatives, integrating the nonlinear term through the linearization technique. While maintaining most characteristics of traditional polynomial B-splines, this method improves numerical solutions, enhancing accuracy in modeling. The performance of the method is rigorously assessed across three benchmark problems, with results compared to those of prior studies employing identical parameters. Detailed numerical illustrations are presented. Graphical representations are utilized to illustrate the single solitary wave motion, dynamics of coupled solitary waves and dynamics of triplet solitary waves in this study. Numerical experiments validate the method's accuracy and efficiency in capturing RLW-driven wave dynamics, establishing it as a reliable tool for various applications. The presented work includes an analysis of the stability of the proposed scheme, employing the Fourier method and shows that its unconditional stable.