Propagation Of Long Waves Research Articles

Galilei-invariant and energy-preserving extensions of Benjamin–Bona–Mahony-type equations

The classical Benjamin–Bona–Mahony equation (BBM equation) models unidirectional propagation of long gravity surface waves of small amplitude. Unlike many other water wave models, it lacks the Galilean invariance, which is an essential property of physical systems. It is shown that by an addition of a higher asymptotic order nonlinear term, this deficiency can be corrected, giving rise to a new Galilei invariant Benjamin–Bona–Mahony equation (iBBM equation). Moreover, further additional higher-order terms can be chosen in a way that the augmented model preserves the energy conservation property along with Hamiltonian and Lagrangian structures. The resulting equation is referred to as energy-preserving Benjamin–Bona–Mahony equation (eBBM).It is shown that both the classical BBM equation and the energy-preserving eBBM equations belong to a one-parameter (α) family that shares essentially the same local and nonlocal symmetries, conservation laws, Hamiltonian, and Lagrangian structures, with the BBM and eBBM equations corresponding to parameter values α=0 and α=1, respectively. Symmetry and conservation law classifications reveal a special case α=1/3, which is shown to correspond to a rescaled version of the celebrated integrable Camassa–Holm (CH) equation. Local symmetries and conservation laws are computed, and numerical solution behaviour is compared for the three BBM-type modes and the CH-equivalent eBBM1/3 model.

On the numerical approximation of Boussinesq/Boussinesq systems for internal waves

AbstractThe present paper is concerned with the numerical approximation of a three‐parameter family of Boussinesq systems. The systems have been proposed as models of the propagation of long internal waves along the interface of a two‐layer system of fluids with rigid‐lid condition for the upper layer and under a Boussinesq regime for the flow in both layers. We first present some theoretical properties of the systems on well‐posedness, conservation laws, Hamiltonian structure, and solitary‐wave solutions, using the results for analogous models for surface wave propagation. Then the corresponding periodic initial‐value problem is discretized in space by the spectral Fourier Galerkin method and for each system, error estimates for the semidiscrete approximation are proved. The spectral semidiscretizations are numerically integrated in time by a fourth‐order Runge–Kutta‐composition method based on the implicit midpoint rule. Numerical experiments illustrate the accuracy of the fully discrete scheme, in particular its ability to simulate accurately solitary‐wave solutions of the systems.

Abundant explicit solutions for the M-fractional coupled nonlinear Schrödinger–KdV equations

In this study, the famous fractional generalized coupled cubic nonlinear Schrödinger–KdV equations arising in many domains of physics and engineering such as depicting the propagation of long waves in dispersive media and the dynamics of short dispersive waves for narrow-bandwidth packet have been investigated. We propose two significant methods named the modified (Gʹ/G, 1/G)-expansion method and the Gʹ/(bGʹ + G + a)-expansion method. After utilizing these two efficient techniques, many types of explicit soliton pulse solutions including the well-known bell-shape bright soliton pulse, the kink and anti-kink dark solitons pulse, the mixed bright-dark soliton pulse, the W-shaped soliton pulse, the periodic waves pulse, and the blow-up soliton pattern solutions are obtained. If we select different values of the frequency, coefficients and orders, the dynamic properties and physical structures of these solutions are discussed, these important results can help us to further understand the inner characteristic of the model.

Particle trajectories in a weakly nonlinear long-wave model on a shear flow

This work centers on the numerical examination of particle trajectories associated with the propagation of long water waves of small but finite amplitude on a background shear flow over a flat bottom. Taking into consideration the assumption that the nonlinear and dispersive effects are small and of the same magnitude, the Boussinesq-type equations for two-dimensional water waves on a background flow with constant vorticity are derived. Restricting attention to waves propagating in a single direction, we find a new version of the Benjamin-Bona-Mahony (BBM) equation which takes into account the effect of vorticity. In order to investigate the particle trajectories of the flow, an approximate velocity field associated with the derivation of the BBM equation over a shear flow is obtained. Several cases of particle paths under surface waves profiles such as solitary waves and periodic traveling waves are examined.

Petrov–Galerkin approximation of time-fractional coupled Korteweg–de Vries equation for propagation of long wave in shallow water

The time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) describe various interesting real-world phenomena including wave propagation and the description of shallow water waves on a viscous fluid. This paper presents an accurate and robust numerical technique to solve the TFCKdVE. The cubic B-spline is introduced as a basis function and a quadratic B-spline is used as a test function in a finite element method (FEM) is known as Petrov–Galerkin method. The temporal fractional part is simplified via L1 formula, while the B-spline is employed for the space approximation. The Lax–Richtmyer stability criterion is applied to analyze the stability of the proposed scheme. Four test problems are solved to check performance and validation of the scheme. The accuracy and efficiency of the proposed method are checked via various error norms. The obtained results show good agreement with the exact solutions and earlier work available in the literature.

Initial value problem of nonlinear KdV-Schrödinger system

The Korteweg-de Vries (KdV) equation is a mathematical model that describes the propagation of long waves in dispersive media. It takes into account both nonlinearity and dispersion, and is particularly useful for modeling phenomena like solitons. The Nonlinear Schrödinger (NLS) equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. It is a useful model for describing many physical systems, including Bose-Einstein condensates, optical fibers, and water waves. A system that couples the KdV and NLS equations can model the interaction of long and short waves. This system combines the strengths of both models. The long waves described by the KdV equation can affect the behavior of the short waves described by the NLS equation, while the short waves can in turn affect the behavior of the long waves. Such a coupled system has been studied extensively over the last few decades, and has led to important insights into many physical systems. This paper considers the existence of local solutions to the Cauchy problem of KdV-Schrödinger nonlinear system on the basis of literature (Bernard D, Nghiem V N, Benjamin L S <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1088/1751-8113/49/41/415501">2016 <i>J. Phys. A: Math. Theor.</i> <b>49</b> 415501</ext-link>), and also gives the existence space of the local solutions.

Fully nonlinear investigation on energy transfer between long waves and short-wave groups over a reef

Long waves are amplified as short-wave groups shoal and break over reefs, therefore, having significant impacts on coastal inundation, structure stability, and sediment transport. This study investigated the cross-reef variation of long-wave energy exchange with short-wave group over a reef using fully nonlinear analysis of simulation results by the non-hydrostatic model SWASH. The objective was to elucidate the mechanisms of long-wave transformation under nonlinear short-wave group forcing over a reef, and to assess the consequences of simplifications in linear and weakly nonlinear analyses in this problem. The energy transfer between short and long waves is the work done by wave radiation stress on long waves, which is the product of radiation-stress gradient and long-wave velocity. Unlike conventional linear and weakly nonlinear analysis, the Stokes transport and long-wave modulation of local water depth are included in the fully nonlinear analysis. It was found that only the long-wave energy flux gradient predicted by the fully nonlinear analysis was balanced by the work done by wave radiation stress over a shallow reef. The fully nonlinear analysis showed that strict mass conservation has to be used to extract long wave velocity properly. In contrast, in linear and weakly nonlinear analysis, the long-wave velocity is extracted from single-point velocity measurements. The fully nonlinear analysis demonstrated that the generation and growth of incoming breakpoint-forced long waves overcame the dissipation of bound long waves in the surf zone, leading to amplification of incoming long-wave energy flux. This phenomenon occurred even when short waves mainly broke over the horizontal reef flat with large submergence, indicating that long-wave evolution is not locally controlled but dependent on wave spatial evolution history. Outgoing breakpoint-forced long waves were dissipated considerably during de-shoaling over the forereef due to substantial energy transfer to incoming short waves, though both of them are free waves. The phase coupling between outgoing long waves and incoming short-wave groups occurred at all frequencies at the breakpoint, which was found to be the main driving mechanism for the energy transfer. According to the fully nonlinear analysis, the reef-flat submergence may affect the long wave in a complex fashion, i.e. , reducing the submergence may enhance the energy transfer from short waves to long waves or suppress long-wave growth by increasing its frictional dissipation at the same time. • Long-wave energy balance is achieved with fully nonlinear analysis, but not linear and weakly nonlinear ones. • Strict mass conservation has to be used to extract long-wave velocity properly. • Long wave amplitude is dependent on short-wave spatial evolution history. • Radiation stress does positive work on incoming long waves in both shoaling and surf zones. • Substantial energy is transferred from outgoing free long waves to incident short waves during de-shoaling.

From a magnetoacoustic system to a J-T black hole: A little trip down memory lane

<abstract><p>We assign a Riemannian metric to a system of nonlinear equations that describe the one-dimensional propagation of long magnetoacoustic waves (also called magnetosonic waves) in a cold plasma under the inference of a transverse magnetic field. The metric, which in general is expressed in terms of the density of the plasma and its speed across the magnetic field, when specialized to a particular solution of the nonlinear system (the Gurevich-Krylov (G-K) solution) is mapped explicitly to a Jackiw-Teitelboim (J-T) black hole metric, which is the main result. Dilaton fields, constructed from data involved in the G-K solution, are presented - which with the plasma metric provide for elliptic function solutions of the J-T equations of motion in 2d dilaton gravity. A correspondence between solutions of the nonlinear plasma system (whose Galilean invariance is also established) and certain solutions of a resonant nonlinear Schrödinger equation is set up, along with some other general background material to render an expository tone in the presentation.</p></abstract>

Bifurcation of Exact Solutions for the Space-Fractional Stochastic Modified Benjamin–Bona–Mahony Equation

This paper studies the influence of space-fractional and multiplicative noise on the exact solutions of the space-fractional stochastic dispersive modified Benjamin–Bona–Mahony equation, driven in Ito’s sense by a multiplicative Wiener process. The bifurcation of the exact solutions is investigated, and novel fractional stochastic solutions are presented. The dependence of the solutions on the initial conditions is discussed. Due to the significance of the fractional stochastic modified Benjamin–Bona–Mahony equation in describing the propagation of surface long waves in nonlinear dispersive media, the derived solutions are significantly more helpful for and influential in comprehending diverse, crucial, and challenging physical phenomena. The effect of the Wiener process and the fractional order on the exact solutions are studied.

Predictability and empirical dynamics of fisheries time series in the North Pacific

Previous studies have documented a strong relationship between marine ecosystems and large-scale modes of sea surface height (SSH) and sea surface temperature (SST) variability in the North Pacific such as the Pacific Decadal Oscillation and the North Pacific Gyre Oscillation. In the central and western North Pacific along the Kuroshio-Oyashio Extension (KOE), the expression of these modes in SSH and SST is linked to the propagation of long oceanic Rossby waves, which extend the predictability of the climate system to ~3 years. Using a multivariate physical-biological linear inverse model (LIM) we explore the extent to which this physical predictability leads to multi-year prediction of dominant fishery indicators inferred from three datasets (i.e., estimated biomasses, landings, and catches). We find that despite the strong autocorrelation in the fish indicators, the LIM adds dynamical forecast skill beyond persistence up to 5-6 years. By performing a sensitivity analysis of the LIM forecast model, we find that two main factors are essential for extending the dynamical predictability of the fishery indicators beyond persistence. The first is the interaction of the fishery indicators with the SST/SSH of the North and tropical Pacific. The second is the empirical relationship among the fisheries time series. This latter component reflects stock-stock interactions as well as common technological and human socioeconomic factors that may influence multiple fisheries and are captured in the training of the LIM. These results suggest that empirical dynamical models and machine learning algorithms, such as the LIM, provide an alternative and promising approach for forecasting key ecological indicators beyond the skill of persistence.

Mixing and nonlinear internal waves in a shallow flow of a three-layer stratified fluid

We propose a mathematical model describing the mixing and propagation of nonlinear long waves in a shear three-layer flow of a stratified fluid under a lid. The shallow water equations describe the fluid flow in the outer almost potential homogeneous layers. In the intermediate mixing layer, the fluid is inhomogeneous and its flow is turbulent. Kinematic boundary conditions at the interfaces ensure the interaction between the layers. In the Boussinesq approximation, we reduced the governing equations to an evolutionary system of balance laws that is hyperbolic for a small difference in velocities in the outer layers. Using the characteristic velocities of the proposed model, we define the concept of a supercritical (subcritical) three-layer flow. We study classes of stationary flows and construct examples of continuous and discontinuous oscillating solutions that describe the spatial evolution of the mixing layer in a stratified flow with upstream shear. The problem of transcritical flow over an obstacle is considered. Depending on the relative velocity, the obtained solutions describe qualitatively different flow regimes on the leeward side of the obstacle. The proposed one-dimensional model was compared against experimental data and numerical results based on the Navier–Stokes equations. The model quite accurately describes the region of intense mixing and characteristic features of the flow.

Some closed-form solutions, conservation laws, and various solitary waves to the (2 + 1)-D potential B-K equation via Lie symmetry approach

The main objective of our work is to obtain some exact closed-form solutions for the (2 + 1)-D potential Bogoyavlensky–Konopelchenko (B-K) equation, which describes the interaction between the Riemann wave propagating and the long-wave propagation along the y-axis and x-axis, by utilizing two effective methods namely: Lie symmetry method and generalized Kudryashov method (GKM). First, we obtained infinitesimals and vector fields of the B-K equation. After that, we used linear combinations of vectors to obtain various reductions of the governing equation. Four ordinary differential equations are obtained; two of them are solved using a series form, while the other two are solved using GKM. Many solitary wave solutions are obtained. The behavior of solutions includes solitary wave solutions, kink wave solutions, anti-kink wave solutions and singular wave solutions. By taking advantage of symbolic computation, the physical phenomena of the demonstrated results are visualized graphically by means of 3D and 2D plots. Using the new conservation theorem and Noether operators, conservation laws and nonlinear self-adjointness for the potential B-K equation are also constructed. It is clearly evident that the obtained results are very useful in studying interactions and processes in optical fibers, mathematical physics, fluid dynamics, engineering and many other areas of science.

New impressive representations for the soliton behaviors arising from the (2+1)-Boussinesq equation

In the present work, we will detect unpredicted conducts “which weren't realized before” to the soliton solutions of the (2+1)-Boussinesq equation by using three distinct algorithms. The proposed equation is the famous nonlinear one which distinguishes the waves of coastal and ocean engineering that involve the nonlinearity and dispersion terms. Moreover, it is a developed form of the standard Boussinesq equation which describes the solution interaction mechanism of shallow-water waves that involve many waves and shallow water influences refraction, diffraction, shoaling and weak nonlinearity properties arising in fluid dynamics. It also plays principal role in many physics branches, such as propagation of long waves in shallow water, vibrations in a nonlinear string and ion sound waves in plasma and one-dimensional nonlinear lattice waves. The three algorithms that will select for this purpose are the (G′G) -expansion method, the extended direct algebraic method (EDAM) and the extended simple equation method (ESEM). These three distinct manners are applied in the same time and parallel. We will show comparison between our new soliton solution behaviours with that constructed before.

A novel approach to study generalized coupled cubic Schrödinger–Korteweg-de Vries equations

The Kortewegde Vries (KdV) equation represents the propagation of long waves in dispersive media, whereas the cubic nonlinear Schrödinger (CNLS) equation depicts the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A model that couples these two equations seems intriguing for simulating the interaction of long and short waves, which is important in many domains of applied sciences and engineering, and such a system has been investigated in recent decades. This work uses a modified Sardar sub-equation procedure to secure the soliton-type solutions of the generalized cubic nonlinear Schrödinger–Korteweg-de Vries system of equations. For various selections of arbitrary parameters in these solutions, the dynamic properties of some acquired solutions are represented graphically and analyzed. In particular, the dynamics of the bright solitons, dark solitons, mixed bright-dark solitons, W-shaped solitons, M-shaped solitons, periodic waves, and other soliton-type solutions. Our results demonstrated that the proposed technique is highly efficient and effective for the aforementioned problems, as well as other nonlinear problems that may arise in the fields of mathematical physics and engineering.

DYNAMICAL BEHAVIOR OF THE LONG WAVES IN THE NONLINEAR DISPERSIVE MEDIA THROUGH ANALYTICAL AND NUMERICAL INVESTIGATION

This paper studies the well-known mathematical model’s analytical wave solutions (modified Benjamin–Bona–Mahony (BBM) equation), which demonstrates the propagation of long waves in the nonlinear dispersive media in a visual illusion. Six recent analytical and semi-analytical schemes (extended simplest equation (ESE) method, modified Kudryashov (MKud) method, sech–tanh expansion method, Adomian decomposition (ADD) method, El Kalla (EK) expansion method, variational iteration (VI) method) are applied to the considered model for constructing abundant analytical and semi-analytical novel solutions. This variety of solutions aims to investigate the analytical techniques’ accuracy by calculating the absolute error between analytical and semi-analytical solutions that shows the matching between them. The analytical results are sketched through two-dimensional (2D), three-dimensional (3D), contour plot, spherical plot and polar plot. The stability characterization of the analytical solutions is investigated through the Hamiltonian system’s features. The originality and novelty of this paper are discussed, along with previously published papers.

An efficient computational technique for class of generalized Boussinesq shallow-water wave equations

In this study, a modification of the homotopy perturbation method called the homotopy perturbation transform method (shortly, HPTM) is applied to the generalized Boussinesq and generalized perturbed Boussinesq equations. These equations depict the propagation of long waves in shallow water under gravity, one-dimensional nonlinear lattices, and nonlinear strings. They are also significantly important in coastal and ocean engineering. The numerical simulations have been implemented to verify that the proposed scheme is very reliable and accurate. Significant results and their graphical depictions revealed that HPTM is more appropriate for nonlinear problems, and it is immensely active for nonlinear models of classical and fractional order type. The obtained results illuminate that the proposed method is simple to implement and very effective to analyze the complex models arising in science, technology, and engineering.

On Lie symmetries and invariant solutions of Broer–Kaup–Kupershmidt equation in shallow water of uniform depth

The dynamics of atmosphere and ocean can be examined under different circumstances of shallow water waves like shallow water gravity waves, Kelvin waves, Rossby waves and inertio-gravity waves. The influences of these waves describe the climate change adaptation on marine environment and planet. Therefore, the present work aims to derive symmetry reductions of Broer–Kaup–Kupershmidt equation in shallow water of uniform depth and then a variety of exact solutions are constructed. It represents the propagation of nonlinear and dispersive long gravity waves in two horizontal directions in shallow water. The invariance of test equations under one parameter transformation leads to reduction of independent variable. Therefore, twice implementations of symmetry method result into equivalent system of ordinary differential equations. Eventually, the exact solutions of these ODEs are computed under parametric constraints. The derive results entail several arbitrary constants and functions, which make the findings more admirable. Based on the appropriate choice of existing parameters, these solutions are supplemented numerically and show parabolic nature, intensive and non-intensive behavior of solitons.

Diversity of wave structures to the conformable fractional dynamical model

This manuscript examines the recently developed conformable three-dimensional Wazwaz–Benjamin–Bona–Mahony (3D-WBBM) equation’s dynamical behavior in terms of its spatial and temporal variables. The governing equation is stretch for the Korteweg-de-Vries equation that represents the unidirectional propagation of small amplitude long waves on the surface of hydro magnetic and acoustic waves in a channel, especially for shallow water. Solitary wave solutions of various types, such as kink and shock, as well as singleton, combined solitons, and complex solitons, are all retrieved. Additionally, solutions to hyperbolic, exponential, and trigonometric functions are obtained through the use of recently developed methods, namely the Kudryashov method (KM), the modified Kudryashov method (MKM), and the new extended direct algebraic method (NEDAM). The study conducts a comparison of our findings to well-known findings, and concludes that the solutions reached here are novel. Additionally, the earned results are sketched in different shapes to demonstrate their dynamics as a function of parameter selection. We can assert from the obtained results that the applied techniques are simple, vibrant, and quite well, and will be helpful tool for addressing more highly nonlinear issues in various of fields, especially in ocean and coastal engineering. Furthermore, our findings are first step toward understanding the structure and physical behavior of complicated structures. We anticipate that our results will be highly valuable in better understanding the waves that occur in the ocean. We feel that this work is timely and will be of interest to a wide spectrum of experts working on ocean engineering models.

Integrable systems of the intermediate long wave type in [formula omitted] dimensions

We classify 2+1 dimensional integrable systems with nonlocality of the intermediate long wave type. Links to the 2+1 dimensional waterbag system are established. Dimensional reductions of integrable systems constructed in this paper provide dispersive regularisations of hydrodynamic equations governing propagation of long nonlinear waves in a shear flow with piecewise linear velocity profile (for special values of vorticities).

Travelling waves in the Boussinesq type systems

Considered herein are a number of variants of the Boussinesq type systems modelling surface water waves. Such equations were derived by different authors to describe the two-way propagation of long gravity waves. A question of existence of special solutions, the so called solitary waves, is of a particular interest. There are a number of studies relying on a variational approach and a concentration-compactness argument. These proofs are technically very demanding and may vary significantly from one system to another. Our approach is based on the implicit function theorem, which makes the treatment easier and more unified.