Propagation Of Shallow Water Waves Research Articles

Numerical study of nonlinear time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising in propagation of waves

In this manuscript, the numerical approximation of the solution for time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation is described involving the Caputo fractional derivative with the help of two novel techniques namely, Sumudu Residual Power Series (SRPS) method and Homotopy Perturbation Sumudu Transform (HPST) method. CDGSK equation describe the propagation of capillary waves and shallow water waves. The results obtained through numerical approximation shows that both SRPS and HPST under consideration are reliable and efficient. The present framework precisely reflects the behavior of the obtained result for various fractional orders. Additionally, we conduct a comparison between the exact solution and an approximate series solution to affirm the efficacy of the proposed techniques. This analysis highlights the efficiency and precision of both techniques in solving nonlinear fractional differential equations. All the numerical simulations are demonstrated graphically using Maple and Matlab.

The dynamical perspective of soliton solutions, bifurcation, chaotic and sensitivity analysis to the (3+1)-dimensional Boussinesq model

In this study, we examine multiple perspectives on soliton solutions to the (3+1)-dimensional Boussinesq model by applying the unified Riccati equation expansion (UREE) approach. The Boussinesq model examines wave propagation in shallow water, which is derived from the fluid dynamics of a dynamical system. The UREE approach allows us to derive a range of distinct solutions, such as single, periodic, dark, and rational wave solutions. Furthermore, we present the bifurcation, chaotic, and sensitivity analysis of the proposed model. We use planar dynamical system theory to analyze the structure and characteristics of the system’s phase portraits. The current study depends on a dynamic structure that has novel and unexplored results for this model. In addition, we display the behaviors of associated physical models in 3-dimensional, density, and 2-dimensional graphical structures. Our findings demonstrate that the UREE technique is a valuable mathematical tool in engineering and applied mathematics for studying wave propagation in nonlinear evolution equations.

New Traveling Wave Solutions to the Simplified Modified Camassa–Holm Equation and the Landau-Ginsburg-Higgs Equation

Researchers are interested in the (1+1)-dimensional Camassa-Holm and Landau-Ginzburg-Higgs equations as they allow for the study of unidirectional wave propagation in shallow waters with a flat seabed, as well as nonlinear media exhibiting dispersion systems and superconductivity. This work has effectively developed exact wave solutions to the stated models, which may have significant consequences for characterising the nonlinear dynamical behaviour related to the phenomena. The extended -expansion technique is employed to procure a diverse array of progressive wave solutions characterized by hyperbolic, trigonometric, and rational functions. The solutions are shown as 3D profiles with a variety of shapes, including kink, singular kink, periodic, singular periodic, etc. The physical significance of the solutions is discussed by these plots, and the approach used in this study is considered efficient and capable of finding analytical solutions for the nonlinear models. Dhaka Univ. J. Sci. 72(1): 7-13, 2024 (January)

Dynamical behavior of water wave phenomena for the 3D fractional WBBM equations using rational sine-Gordon expansion method

To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

Experimental Investigation of Pore Pressure on Sandy Seabed around Submarine Pipeline under Irregular Wave Loading.

The propagation of shallow-water waves may cause liquefaction of the seabed, thereby reducing its support capacity for pipelines and potentially leading to pipeline settlement or deformation. To ensure the stability of buried pipelines, it is crucial to consider the excess pore pressure induced by irregular waves thoroughly. This paper presents the findings of an experimental study on excess pore pressure caused by irregular waves on a sandy seabed. A series of two-dimensional wave flume experiments investigated the excess pore pressure generated by irregular waves. Based on the experimental results, this study examined the influences of irregular wave characteristics and pipeline proximity on excess pore pressure. Using test data, the signal analysis method was employed to categorize different modes of excess pore-water pressure growth into two types and explore the mechanism underlying pore pressure development under the influence of irregular waves.

Some Novel Fusion and Fission Phenomena for an Extended (2+1)-Dimensional Shallow Water Wave Equation

An extended (2+1)-dimensional shallow water wave (SWW) model which can describe the evolution of nonlinear shallow water wave propagation in two spatial and temporal coordinates, is systematically studied. The multi-linear variable separation approach is addressed to the extended (2+1)-dimensional SWW equation. The variable separation solution consisting of two arbitrary functions is obtained, by assumption, from a specific ansatz. By selecting these two arbitrary functions as the exponential and trigonometric forms, resonant dromion, lump, and solitoff solutions are derived. Meanwhile, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions, and periodic waves are studied with graphical and analytical methods. The results can be used to enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions

PurposeThe purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in shallow water, as a function of the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of three types of initial conditions.Design/methodology/approachAn implicit, first-order accurate in time, finite difference method valid for semipositive relaxation times has been used to solve the equation in a truncated domain for three different initial conditions, a first-order time derivative initially equal to zero and several constant wave speeds.FindingsThe numerical experiments show a very rapid transient from the initial conditions to the formation of a leading propagating wave, whose duration depends strongly on the shape, amplitude and width of the initial data as well as on the coefficients of the bidirectional equation. The blowup times for the triangular conditions have been found to be larger than those for the Gaussian ones, and the latter are larger than those for rectangular conditions, thus indicating that the blowup time decreases as the smoothness of the initial conditions decreases. The blowup time has also been found to decrease as the relaxation time, degree of nonlinearity, linear drift coefficient and amplitude of the initial conditions are increased, and as the width of the initial condition is decreased, but it increases as the viscosity coefficient is increased. No blowup has been observed for relaxation times smaller than one-hundredth, viscosity coefficients larger than ten-thousandths, quadratic and cubic nonlinearities, and initial Gaussian, triangular and rectangular conditions of unity amplitude.Originality/valueThe blowup of a one-dimensional, bidirectional equation that is a model for the propagation of waves in shallow water, longitudinal displacement in homogeneous viscoelastic bars, nerve conduction, nonlinear acoustics and heat transfer in very small devices and/or at very high transfer rates has been determined numerically as a function of the linear and nonlinear drift coefficients, power of the nonlinear drift, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of the initial conditions for nonzero relaxation times.

Bosonization, symmetry reductions, mapping and deformation method for B-extension of Sawada–Kotera equation

An extended (2+1)-dimensional shallow water wave (SWW) model governs the evolution of nonlinear shallow water wave propagation in two spatial and a temporal coordinate. The multi-linear variable separation approach is applied to the SWW equation. The variable separation solution consisting of two arbitrary functions is given. By choosing the arbitrary functions as the exponential and trigonometric forms, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions and periodic waves are graphically and analytically studied. The results enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

Analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation in shallow water wave propagation

In this study, the analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation, a mathematical model used to describe the propagation of shallow water waves characterized by weak dispersion and nonlinearities, is determined. The (G′/G,1/G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({G}^{\\prime}/G,1/G)$$\\end{document}-expansion method, a powerful and reliable technique, is exploited to formulate the soliton solutions for the equation. The process yields a wide range of solutions, including trigonometric, rational, and hyperbolic functions with free parameters. Various original soliton solutions are generated for different parameter values, including bell-shaped, anti-bell-shaped, periodic, compacton, singular bell-shaped, singular periodic and flat kink solitons. To visually comprehend the physical characteristics of the obtained solutions, two- and three-dimensional graphs, as well as contour plots, are plotted. Thorough comparisons with previous results are conducted to ensure the originality of the derived solutions. The insights gained from understanding soliton behavior in shallow water can be helpful in improving tsunami warnings, coastal protection systems, underwater data transmission, submarine cables, and marine sensing networks.

Line-soliton, lump and interaction solutions to the (2+1)-dimensional Hirota-Satsuma-Ito equation with time-dependent via Hirota bilinear forms

In this paper, we consider the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation with time-dependent, which has applications in describing the propagation of shallow water waves. Based on the bilinear formalism and with the aid of symbolic computation, we obtain line-soliton, lump, one-lump-one-stripe and one-lump-one-soliton using various ansatze’s function. To realize dynamics, we use diverse plots to analyze these solutions. Obtained solutions are reliable in the mathematical physics and engineering.

2D Modeling of Wave Propagation in Shallow Water by the Method of Characteristics

2D Modeling of Wave Propagation in Shallow Water by the Method of Characteristics

A new paradigm of reservoir computing exploiting hydrodynamics

Nonlinear waves have played a significant role in the development of the science of complexity throughout history. More recently, they have also contributed to the emergence of a novel paradigm in reservoir computing called neuromorphic computing by waves. In this paradigm, information is transmitted through wave dynamics, which process the data to perform complex tasks with minimal energy consumption. Leveraging nonlinear hydrodynamic waves for computational purposes, we have created the Aqua-Photonic-Advantaged Computing Machine by Artificial Neural Networks (Aqua-PACMANNs). This system utilizes wave propagation in shallow water as the primary physical phenomenon, with minimal electronic components required, limited to a camera for detection purposes. As a proof of concept, we have successfully implemented an exclusive-not-or logic gate using the Aqua-PACMANN architecture. This accomplishment demonstrates the potential of fluid dynamic neuromorphic computing and opens up possibilities for a new class of computing systems.

Directing Shallow-Water Waves Using Fixed Varying Bathymetry Designed by Recurrent Neural Networks

Directing shallow-water waves and their energy is highly desired in many ocean engineering applications. Coastal infrastructures can be protected by reflecting shallow-water waves to deep water. Wave energy harvesting efficiency can be improved by focusing shallow-water waves on wave energy converters. Changing water depth can effectively affect wave celerity and therefore the propagation of shallow-water waves. However, determining spatially varying bathymetry that can direct shallow-water waves to a designed location is not trivial. In this paper, we propose a novel machine learning method to design and optimize spatially varying bathymetry for directing shallow-water waves, in which the bathymetry is assumed fixed in time without considering morphodynamics. Shallow-water wave theory was applied to establish the mapping between water wave mechanics and recurrent neural networks (RNNs). Two wave-equivalent RNNs were developed to model shallow-water waves over fixed varying bathymetry. The resulting RNNs were trained to optimize bathymetry for wave energy focusing. We demonstrate that the bathymetry optimized by the wave-equivalent RNNs can effectively reflect and refract wave energy to various designed locations. We also foresee the potential that new engineering tools can be similarly developed based on the mathematical equivalence between wave mechanics and recurrent neural networks.

On H 2-solutions for a Camassa-Holm type equation

Abstract Camassa-Holm type equations arise as models for the unidirectional propagation of shallow water waves over a flat bottom. They also describe finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods. Under appropriate assumption on the initial data, on the time T T , and on the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem.

On dynamical behavior for approximate solutions sustained by nonlinear fractional damped Burger and Sharma–Tasso–Olver equation

In this paper, we study a nonlinear fractional Damped Burger and Sharma–Tasso–Olver equation using a new novel technique, called homotopy perturbation transform method (FHPTM). There are three examples used to demonstrate and validate the proposed algorithm’s efficiency. This nonlinear model depicts nonlinear wave processes in fluid dynamics, ecology, solid-state physics, shallow-water wave propagation, optical fibers, fluid mechanics, plasma physics, and other applied science, engineering, and mathematical physics disciplines, as well as other phenomena. Numerous algebraic properties of the fractional derivative Caputo–Fabrizio operator are illustrated concerning the Laplace transformation to demonstrate their utility. Different graphs and tables compare the results obtained by R. Nawaz et al. [Alex. Eng. J. 60, 3205 (2021)] and M. S. Rawashdeh [Appl. Math. Inform. Sci. 9, 1239 (2015)]. The proposed scheme accelerates the convergence of the series solution and guarantees the convergence associated with the homotopy parameter. Furthermore, the physical nature of various fractional orders has been captured in plots. The obtained results demonstrate that the employed solution procedure is dependable and methodical in investigating the behaviors of nonlinear models of both integer and fractional orders.

Tsunami scenario triggered by a submarine landslide offshore of northern Sumatra Island and its hazard assessment

Abstract. Near the northern border of Sumatra, the right-lateral strike–slip Sumatran fault zone splits into two branches and extends into the offshore, as revealed by seismic sounding surveys. However, due to its strike–slip faulting characteristics, the Sumatran fault zone's activity is rarely believed to cause tsunami hazards in this region. According to two reprocessed reflection seismic profiles, the extended Sumatran fault zone is strongly associated with chaotic facies, indicating that large submarine landslides have been triggered. Coastal steep slopes and new subsurface characteristics of submarine landslide deposits were mapped using recently acquired high-resolution shallow bathymetry data. Slope stability analysis revealed some targets with steep morphology to be close to failure. In an extreme hypothetical case, an earthquake of Mw 7 or more occurred, and the strong ground shaking triggered a submarine landslide off the northern shore of Sumatra. Based on a simulation of tsunami wave propagation in shallow water, the results of this study indicate that a potential tsunami hazard from several submarine landslide sources triggered by the strike–slip fault system can generate a tsunami as high as 4–8 m at several locations along the northern coast of Aceh. The landslide tsunami hazard assessment and early warning systems in this study area can be improved on the basis of this proposed scenario.

Bilinear method and semi-inverse variational principle approach to the generalized (2+1)-dimensional shallow water wave equation

In this article, we investigate the generalized (2+1)-dimensional shallow water wave equation which enables an unidirectional propagation of shallow-water waves. By noticing that the system is integrable, we could get the diverse forms of the solitary wave solutions by using the rogue wave and semi-inverse variational principle (SIVP) schemes. In particular, we investigate four solutions including rogue wave, soliton, bright soliton, dark soliton, and lump solutions. To achieve this, an illustrative example of the Helmholtz equation is provided to demonstrate the feasibility and reliability of the used procedure in this study. The effect the free parameters on the behavior of acquired figures to a few obtained solutions containing two nonlinear rational exact cases was also analyzed due to the nature of nonlinearities. The dynamic properties the obtained results are shown and analyzed by some density, two and three-dimensional images and also are presented the physical nature of obtained solutions.

Diverse solitary wave solutions of fractional order Hirota-Satsuma coupled KdV system using two expansion methods

The generalized Hirota-Satsuma coupled KdV system with fractional-order derivative plays a significant role to simulate the interaction of nearby identical-weight particles in a crystal lattice structure, two long waves interaction with different dispersion relationships, the description of shallow-water wave propagation, ion-acoustic waves, plasma physics and some other fields. In this study, diverse analytical wave solutions in general and standard structures are established of the stated equation by introducing two viable techniques, namely, the generalized Kudryashov scheme and the two variables G'/G,1/G-expansion approach. The solutions are established in terms of elementary functions, specifically trigonometric functions, rational functions, exponential functions, and hyperbolic functions. For definite values of free parameters, the obtained analytical wave solutions transform into solitary wave solutions. The graphical patterns of the wave solutions with there-, two-, and contour plots are depicted magnificently to elucidate the internal structure of the phenomenon. The methods contribute as powerful mathematical tools and appear to be further effective, computerized, and user-friendly to investigate nonlinear fractional equations as well as comprehensive analytical solutions of nonlinear evolution equations in engineering, technology, and mathematical physics.

Solitary, kink and periodic wave solutions of the (3+1)-dimensional Hirota–Satsuma–Ito-like equation

In this article, we consider new solutions to the (3+1)-dimensional Hirota–Satsuma–Ito-like (HSIl) equation by using the exp-function and its expansion methods. This model describes the unidirectional propagation of shallow water waves. The is rewritten as an ordinary differential equation by the traveling wave transformation. Some exact traveling wave solutions of this equation are obtained based on the homogeneous balance method, which includes solitons, kinks, periodic wave solutions, etc. The derived results are innovative and have important applications in the current field of mathematical physics research. To further investigate the properties of these solutions, the dynamical behaviors of these solutions are analyzed. The results show that the scheme is completely powerful and effective.