Dynamics Of Shallow Water Waves Research Articles

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.

Dark Localized Waves in Shallow Waters: Analysis within an Extended Boussinesq System

We study dark localized waves within a nonlinear system based on the Boussinesq approximation, describing the dynamics of shallow water waves. Employing symbolic calculus, we apply the Hirota bilinear method to transform an extended Boussinesq system into a bilinear form, and then use the multiple rogue wave method to obtain its dark rational solutions. Exploring the first- and second-order dark solutions, we examine the conditions under which these localized solutions exist and their spatiotemporal distributions. Through the selection of various parameters and by utilizing different visualization techniques (intensity distributions and contour plots), we explore the dynamical properties of dark solutions found: in particular, the first- and second-order dark rogue waves. We also explore the methods of their control. The findings presented here not only deepen the understanding of physical phenomena described by the (1+1)-dimensional Boussinesq equation, but also expand avenues for further research. Our method can be extended to other nonlinear systems, to conceivably obtain higher-order dark rogue waves.

Bifurcation analysis, chaotic analysis and diverse optical soliton solutions of time-fractional (2+1) dimensional generalized Camassa-Holm Kadomtsev-Petviashvili equation arising in shallow water waves

This paper presents a study on (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation, which is used to describe the behavior of shallow water waves in nonlinear media. The considered equation provides a more accurate description of wave behavior compared to linear wave equations and can account for wave breaking and other nonlinear effects. This model can be used to describe and study the behavior of nonlinear waves such as rogue waves in complex fluid dynamics scenarios. This includes the behavior of waves in stratified fluids, nonlinear dispersive media and wave interactions in fluid flows with varying velocities and densities. The bifurcation analysis of the governing equation has been performed using the planar dynamical system method. The chaotic behavior of the dynamical system has been examined by utilizing various techniques such as time series analysis and the construction of 2D and 3D phase space trajectories. Furthermore, the introduction of a perturbed term has resulted in the observation of chaotic and quasi-periodic behaviors across a range of parameter values. The considered equation has been reduced to ordinary differential equation by performing symmetry reduction. The Kudryashov method has been used to obtain the exact solution of reduced equation. The single soliton solution of governed equation has been obtained by using Hirota method and impact of fractional parameter on the obtained solution has been studied using graphical representation. The extended sinh-Gordon equation expansion method and modified generalized exponential rational function method have been exploited to obtain dark, bright and singular soliton solutions of considered equation. The motivation for this study arises from the need to understand and analyze the complex dynamics of shallow water waves in nonlinear media with a particular focus on the (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation. By performing symmetry reduction and applying various analytical methods, we aim to unravel the intricate behavior and soliton solutions of considered equation, contributing to the broader understanding of nonlinear wave phenomena.

Homoclinic breathers and soliton propagations for the nonlinear [formula omitted]-dimensional Geng dynamical equation

We investigate the rational solitons of the nonlinear (3+1)-dimensional Geng (NG)-equation via symbolic computation with ansatz functions technique and the logarithmic transformation. The (3+1) NG-equation uses in shallow water wave dynamics. Multiwave solitons are reported by using three-waves technique. Homocentric breathers (HBs) approach is also applied to built new forms of exact solutions. Furthermore periodic cross rational solitons (PCRS) and kink cross rational solitons (KCRS) are examined through suitable transformations. We construct M-shape solitons and their dynamics are shown via the appropriate values of parameters. Additionally, two types of interactions for rational soliton with kink wave are investigated.

Hybrid solitary wave solutions of the Camassa–Holm equation

Abstract The Camassa–Holm equation governs the dynamics of shallow water waves or in its reduced form models nonlinear dispersive waves in hyperelastic rods. By using the straightforward Bogning-Djeumen Tchaho-Kofané method, explicit expressions of many solitary wave solutions with different profiles not previously derived in the literature are constructed and classified. Geometric characterizations of the solutions in terms of three new mappings are presented. Intensive numerical simulations carried confirm the stability of the solutions even with relatively high critical velocities and reveal that solitary waves with large widths are more stable than the ones with small widths.

Modeling nearshore dynamics of extreme storms in complex environments of Connecticut

Flood hazard planning requires the accurate estimation of total water elevation due to predicted tide, surge, and wave runup to design flood protection structures and improve coastal risk planning for severe storms. The beach geomorphology and nearshore hydrodynamic conditions impact the conclusive flood inundation mapping in complex environments. The conventional approaches of flood modeling are limited due to either i) simple static estimates, ii) the application of a coupled circulation and phase-averaged wave models in coarse resolution, iii) failing to calibrate and validate with in-situ data, or iv) not considering sea-level rise projections in mapping the flood extent. We used a fully nonlinear Boussinesq wave model (FUNWAVE-TVD) on the nearshore area with a high-resolution grid to determine total water elevation on the shores. The model adopted the boundary conditions from coupled circulation and wave model of the Long Island Sound (Liu et al., 2020).We applied the model to Branford, Norwalk, and New Haven, CT to represent the complex topo-bathymetric features and the structural interference in shallow water wave dynamics. We compared the model results with the FEMA base flood elevation, the North Atlantic Coast Comprehensive Study (NACCS), and (Liu et al., 2020)’s Long Island Sound FVCOM-SWAVE model. The FUNWAVE-TVD model is found to model wave processes more accurately in shallow water regions compared to the empirical equation application of FEMA and coupled circulation-phase averaged model application of NACCS and FVCOM-SWAVE. We also examined local sea-level rise predictions of storms with 1% and 10% annual exceedance probability by the year 2050 in Connecticut and found that the flood extent of these two storms showed little to no difference. We suggest the planning approaches should consider the increase in the frequency of the storms in the predicted inundation zones due to sea-level rise.

Dynamical analysis of lump, lump-triangular periodic, predictable rogue and breather wave solutions to the (3 + 1)-dimensional gKP–Boussinesq equation

This paper studies the dynamics of shallow water waves with the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq (gKP–Boussinesq) equation arising in fluid mechanics. By the virtue of the Hirota bilinear method, a series of lump, single stripe, lump-triangular periodic, predictable rogue, and breather wave solutions are constructed to the (3 + 1)-D gKP–Boussinesq equation. For generating the breather wave solutions, two homoclinic test approaches are executed to the mentioned equation. It is seen that the breather wave propagates periodically in a certain direction with a constant period. Furthermore, for lump solutions, the shapes and amplitudes of the lump wave are found to be unchanged during its propagation. Besides, the lump-stripe soliton solutions illustrate the fission and fusion behaviors that one-stripe wave splits into one lump and one-stripe waves, and one lump and one-stripe waves merge into one-stripe wave, respectively. To interpret the underlying propagation characteristics, some attained solutions are displayed by making their 3D, density, and contour plots. All the graphs are presented to show their proper wave profiles via the obtained solutions to the studied equation. Moreover, it may be concluded that the attained solutions and their physical features might be helpful to comprehend the localized wave propagation in shallow water waves.

Equatorial Magnetohydrodynamic Shallow Water Waves in the Solar Tachocline

Abstract The influence of a toroidal magnetic field on the dynamics of shallow water waves in the solar tachocline is studied. A sub-adiabatic temperature gradient in the upper overshoot layer of the tachocline causes significant reduction of surface gravity speed, which leads to trapping of the waves near the equator and to an increase of the Rossby wave period up to the timescale of solar cycles. Dispersion relations of all equatorial magnetohydrodynamic (MHD) shallow water waves are obtained in the upper tachocline conditions and solved analytically and numerically. It is found that the toroidal magnetic field splits equatorial Rossby and Rossby-gravity waves into fast and slow modes. For a reasonable value of reduced gravity, global equatorial fast magneto-Rossby waves (with the spatial scale of equatorial extent) have a periodicity of 11 years, matching the timescale of activity cycles. The solutions are confined around the equator between latitudes ±20°–40°, coinciding with sunspot activity belts. Equatorial slow magneto-Rossby waves have a periodicity of 90–100 yr, resembling the observed long-term modulation of cycle strength, i.e., the Gleissberg cycle. Equatorial magneto-Kelvin and slow magneto-Rossby-gravity waves have the periodicity of 1–2 years and may correspond to observed annual and quasi-biennial oscillations. Equatorial fast magneto-Rossby-gravity and magneto-inertia-gravity waves have periods of hundreds of days and might be responsible for observed Rieger-type periodicity. Consequently, the equatorial MHD shallow water waves in the upper overshoot tachocline may capture all timescales of observed variations in solar activity, but detailed analytical and numerical studies are necessary to make a firm conclusion toward the connection of the waves to the solar dynamo.

Dynamics of Shallow Water Waves with Various Boussinesq Equations

Dynamics of Shallow Water Waves with Various Boussinesq Equations

Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics

This paper studies a few nonlinear evolution equations that appear with fractional temporal evolution and fractional spatial derivatives. These are Benjamin-Bona-Mahoney equation, dispersive long wave equation and Nizhnik-Novikov-Veselov equation. The extended Jacobi’s elliptic function expansion method is implemented to obtain soliton and other periodic singular solutions to these equations. In the limiting case, when the modulus of ellipticity approaches zero or unity, these doubly periodic functions approach solitary waves or shock waves or periodic singular solutions emerge.

Dynamics of Shallow Water Waves with Kawahara Equation

Dynamics of Shallow Water Waves with Kawahara Equation

Random focusing of tsunami waves

Fluctuations in the profile of the ocean floor can lead to large variations in tsunami wave height. A theory linking this behaviour to the branched flow characteristics of electron waves through semiconductors may provide a framework for prediction. Tsunamis exhibit surprisingly strong height fluctuations. An in-depth understanding of the mechanisms that lead to these variations in wave height is a prerequisite for reliable tsunami forecasting. It is known, for example, that the presence of large underwater islands1 or the shape of the tsunami source2 can affect the wave heights. Here we show that the consecutive effect of even tiny fluctuations in the profile of the ocean floor (the bathymetry) can cause unexpectedly strong fluctuations in the wave height of tsunamis, with maxima several times higher than the average wave height. A novel approach combining stochastic caustic theory and shallow water wave dynamics allows us to determine the typical propagation distance at which the strongly focused waves appear. We demonstrate that owing to this mechanism the small errors present in bathymetry measurements can lead to drastic variations in predicted tsunami heights. Our results show that a precise knowledge of the ocean’s bathymetry is absolutely indispensable for reliable tsunami forecasts.

Dynamics of shallow water waves with Gardner-Kadomtsev-Petviashvili equation

This paper obtains soliton and other solutions to the Gardner-Kadomtsev-Petviashvili equation that models shallow water wave equation in (1+2)-dimensions. There are three types of integration architectures that will be employed in order to obtain several forms of solution to this model. These are traveling wave hypothesis, improved $G^{\prime}/G$-expansion method and finally the tanh-coth hypothesis. The constraint conditions that are needed, for these solutions to exist, are also reported.

Solitons, Shock Waves and Conservation Laws of Rosenau-KdV-RLW Equation with Power Law Nonlinearity

This paper obtains solitary waves, shock waves and singular solitons alon with conservation laws of the Rosenau Korteweg- de Vries regularized long wave (R-KdV-RLW) equation with power law nonlinearity that models the dynamics of shallow water waves. The ansatz approach and the semi-inverse variational principle are used to obtain these solutions. The constraint conditions for the existence of solitons are also listed.

Two types of bounded traveling-wave solutions of a two-component Camassa–Holm equation

The Camassa–Holm equation (CH) is a well known integrable equation, with is used to describe the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of peakons from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in dynamics. In this paper, the bifurcation method of dynamical systems and numerical approach of differential equations are employed to study CH2 equation. Two types of bounded traveling waves are found. One of them is called compacton, another is called generalized kink wave. The planar graphs of compactons and generalized kink awaves are simulated by using software Maple. Exact explicit parameter expressions of the compactons and implicit expressions of the generalized kink waves solutions are given.

Dynamics of shallow water waves with Boussinesq equation

This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The tanh method is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact 1-soliton solution is obtained. The results are also supported by numerical simulations.

Solitons and other nonlinear waves of the Boussinesq equation

This paper studies the dynamics of shallow water waves that are governed by the Boussinesq equations. A few perturbation terms are taken into account. The ansatz method is used to carry out the perturbed Boussinesq equation. Later on, the mapping method is used to extract a few more analytical solutions. Additionally, the Weierstrass elliptic function method is also used to obtain solitary waves and singular soliton solutions. Finally, the Lie symmetry approach is used to extract a few more additional solutions.

Singular solutions of a modified two-component Camassa-Holm equation

The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow a dependence on the average density as well as the pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in the velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. Numerical results for the MCH2 system are given and compared with the pure CH2 case. These numerics show that the modification in the MCH2 system to introduce the average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for the MCH2 system shows a different asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, the MCH2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes.

A cost-function approach to the assimilation of asynoptic data

This paper describes a method for reconstructing a synoptic state by fitting dynamics to asynoptic data. The best fit is defined by the minimum of a quadratic cost function and dynamics are enforced through the use of a penalty term. When the coefficient of the penalty term is identified as the inverse of the variance of model error, the method yields the same results as Kalman filtering, and in the limit of infinitely large coefficient, the same as strong-constraint formalisms. The self-adjoint nature of the equations for the best fit motivated the use of a relaxation method for their solution. The method is illustrated within the context of one-dimensional, linear, shallow-water wave dynamics, where computational examples indicate that a synoptic state is properly determined only if the asynoptic data are equivalent to complete initial conditions.