Shallow Water Wave Equation Research Articles

An enormous diversity of soliton solutions to the (2+1)-dimensional extended shallow water wave equation using three analytical methods

In this paper, we obtain a variety of analytical wave solutions of a ([Formula: see text])-dimensional extended shallow water wave equation. The applications of the governing equation are enormous in ocean modeling and investigation of moist-convection properties in atmospheric dynamics. Two powerful mathematical approaches, the [Formula: see text]-expansion method (EEM) and the generalized projective Riccati equations method (GPREM), were used to find closed-form traveling wave solutions. In addition, the breather wave solutions were described using the Cole-Hopf transform, derived with the help of Hirota bilinear method (HBM). Eventually, we obtain 46 closed-form solutions in explicit form. The general form of obtained solutions include trigonometric, rational, exponential and hyperbolic function solutions. To illustrate the physical significance of some novel results, we provided contour, two- and three-dimensional graphics under the suitable free choices of unknown parameters. The dynamical shapes of these solutions are one soliton, multiple solitons, kink, anti-kink, lump and periodic solutions. We believe that the applied methods of this work are well-organized, genuine and powerful mathematical tools for solving the nonlinear evolution equations (NLEEs) occurring in the study of ocean science and engineering.

Lumps and interactions, fission and fusion phenomena in multi solitons of extended Shallow Water Wave Equation of (2+1)-dimensions

In this manuscript, we study novel exact solutions of the extended shallow water wave equation of (2+1)-dimensions. To acheive the exact solutions the Hirota bi-linear approach is utilized. The first, second, third and fourth order solitons are presented. Further, the fission or fusion of the first to fourth order solitons are also presented which are obtained with the help of consideration of different dispersion coefficients as zeros. Similarly, lumps and other their interactions with some new combinations of trigonometric and hyperbolic functions. The solutions are graphically presented with help of MATLAB R2020a.

Modified Hirota bilinear method to (3+1)-D variable coefficients generalized shallow water wave equation

Modified Hirota bilinear method to (3+1)-D variable coefficients generalized shallow water wave equation

The Soliton Solutions of the Stochastic Shallow Water Wave Equations in the Sense of Beta-Derivative

The stochastic shallow water wave equation (SSWWE) in the sense of the beta-derivative is considered in this study. The solutions of the SSWWE are obtained using the F-expansion technique with the Riccati equation and He’s semi-inverse method. Since the shallow water equation has many uses in ocean engineering, including river irrigation flows, tidal waves, tsunami prediction, and weather simulations, the solutions discovered can be utilized to represent a wide variety of exciting physical events. We create many 2D and 3D graphs to demonstrate how the beta-derivative and Brownian motion affect the analytical solutions of the SSWWE.

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

In this paper, we propose a high-order nonlinear algorithm based on a finite difference method modification to the regularized long wave equation and the Benjamin–Bona–Mahony–Burgers equation subject to the homogeneous boundary. The consequence system of nonlinear equations typically trades with high computation burden. This dilemma can be overcome by establishing a fast numerical algorithm procedure without a reduction of numerical accuracy. The proposed algorithm forms a linear system with constant coefficient matrix at each time step and produces numerical solutions, which remarkably gains many computational advantages. In terms of analysis, a priori estimation for the numerical solution is derived to obtain the convergence and stability analysis. Additionally, the algorithm is globally mass preserving to avoid nonphysical behavior. Two benchmarks, including a single solitary wave to both equations, are given to validate the applicability and accuracy of the proposed method. Numerical results are obtained and compared to other approaches available in the literature. From the comparisons it is clear that the proposed approach produces accurate and precise results at low computational cost. Besides, the proposed scheme is applied to study the effect of the viscous term on a single solitary wave. It is shown that the viscous term results in the amplitude and width of the initial condition but not in its velocities in the case of a single solitary wave. As a consequence, theoretical and numerical findings provide a new area to investigate and expand the high-order algorithm for the family of wave equations.

Green-Naghdi Theory, Part B: Green-Naghdi Equations for Deep Water Waves

“Green — Naghdi Theory, Part A: Green — Naghdi (GN) Equations for Shallow Water Waves” have investigated the linear dispersion relations of high-level GN equations in shallow water. In this study, the GN equations for deep water waves are investigated. In the traditional GN equations for deep water waves, the velocity distribution assumption involves only one representative wave number. Herein, a new velocity distribution shape function with multiple representative wave numbers is employed. Further, we have derived the three-dimensional GN equations and analyzed the linear dispersion relations of the GN-3 and GN-5 equations. In this study, the finite difference method is used to simulate focus waves in the time domain. Additionally, the GN-5 equations are used to validate the wave profile and horizontal velocity distribution along water depth for different focused waves.

The energy method for high-order invariants in shallow water wave equations

Third-order dispersive evolution equations are widely adopted to model one-dimensional long waves and have extensive applications in fluid mechanics, plasma physics and nonlinear optics. The typical representatives are the KdV equation, the Camassa–Holm equation and the Degasperis–Procesi equation. They share many common features such as complete integrability, Lax pairs and bi-Hamiltonian structure. In this paper we revisit high-order invariants for these three types of shallow water wave equations by the energy method in combination of a skew-adjoint operator (1−∂xx)−1. Several applications to seek high-order invariants of the Benjamin–Bona–Mahony equation, the regularized long-wave equation and the Rosenau equation are also presented.

An investigation of a closed-form solution for non-linear variable-order fractional evolution equations via the fractional Caputo derivative

Determining the non-linear traveling or soliton wave solutions for variable-order fractional evolution equations (VO-FEEs) is very challenging and important tasks in recent research fields. This study aims to discuss the non-linear space–time variable-order fractional shallow water wave equation that represents non-linear dispersive waves in the shallow water channel by using the Khater method in the Caputo fractional derivative (CFD) sense. The transformation equation can be used to get the non-linear integer-order ordinary differential equation (ODE) from the proposed equation. Also, new exact solutions as kink- and periodic-type solutions for non-linear space–time variable-order fractional shallow water wave equations were constructed. This confirms that the non-linear fractional variable-order evolution equations are natural and very attractive in mathematical physics.

Simulasi Terbentuknya Gelombang Permukaan Akibat Adanya Longsoran Bawah Laut (Metode Lax- Friedrich)

The problem studied in this paper is the formation of surface waves due to underwater landslides. The shallow water wave equation is employed to study the phenomenon. The equation is solved numerically by using the Lax-Friedrich method. The numerical result show that this method is able to simulate the generation of surface wave due to underwater landslide.

Approximate solutions to shallow water wave equations by the homotopy perturbation method coupled with Mohand transform

In this paper, the Mohand transform-based homotopy perturbation method is proposed to solve two-dimensional linear and non-linear shallow water wave equations. This approach has been proved suitable for a broad variety of non-linear differential equations in science and engineering. The variation trend of the water surface elevation at different time levels and depths are given by some graphs. Moreover, the obtained solutions are compared with the existing results, which show higher efficiency and fewer computations than other approaches studied in the literature.

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.

Linear barycentric rational collocation method for solving a class of generalized Boussinesq equations

<abstract><p>This paper is concerned with solving a class of generalized Boussinesq shallow-water wave (GBSWW) equations by the linear barycentric rational collocation method (LBRCM), which are nonlinear partial differential equations (PDEs). By using the method of direct linearization, those nonlinear PDEs are transformed into linear PDEs which can be easily solved, and the corresponding differentiation matrix equations of their discretization linear GBSWW equations are also given by a Kronecker product. Based on the error estimate of a barycentric interpolation, the rates of convergence for numerical solutions of GBSWW equations are obtained. Finally, three examples are presented to show theoretical results.</p></abstract>

The entropy weak solution to a nonlinear shallow water wave equation including the Degasperis-Procesi model

<abstract><p>A nonlinear model, which characterizes motions of shallow water waves and includes the famous Degasperis-Procesi equation, is considered. The essential step is the derivation of the $ L^2(\mathbb{R}) $ uniform bound of solutions for the nonlinear model if its initial value belongs to space $ L^2(\mathbb{R}) $. Utilizing the bounded property leads to several estimates about its solutions. The viscous approximation technique is employed to establish the well-posedness of entropy weak solutions.</p></abstract>

Propagation of lump-type waves in nonlinear shallow water wave.

In this work, a new extended shallow water wave equation in (3+1) dimensions was studied, which represents abundant physical meaning in a nonlinear shallow water wave. We discussed the interaction between a lump wave and a single solitary wave, which is an inelastic collision. Further, the interaction between a lump wave and two solitary waves and the interaction between a lump wave and a periodic wave was also studied using the Hirota bilinear method. Finally, the interaction among lump, periodic and one solitary wave was investigated. The dynamic properties of the obtained results are shown and analyzed by some three-dimensional images.

Quasi-Linear Model of Tsunami Run-Up on a Beach with a Seafloor Described by the Piecewise Continuous Function

The purpose of this paper is to propose the quasi-linear theory of tsunami run-up and run-down on a beach with complex bottom topography. We begin with the one-dimensional nonlinear shallow-water wave equations, which we consider over a beach of complex geometry that can be modeled by a piecewise continuous function, along with several natural initial and boundary conditions. The primary obstacle in solving this problem is the moving boundary associated with the shoreline motion. To avoid this difficulty, we replace the moving boundary with a stationary boundary by applying a transformation to the spatial variable of the computational domain. A characteristic feature of any tsunami problem is the smallness of the parameter ε=η0/h0, where η0 is the characteristic amplitude of the wave, and h0 is the characteristic depth of the ocean. The presence of this small parameter enables us to effectively linearize the problem by using the method of perturbations, which leads to an analytical solution via an integral transformation. This analytical solution assumes that there is no wave breaking. In light of this assumption, we introduce the wave no-breaking criterion and determine bounds for the applicability of our theory. The proposed model can be readily used to investigate the tsunami run-up and draw-down for different sea bottom profiles. The novel particular solution, when the seafloor is described by the piecewise linear function, is obtained, and the effects of the different beach profiles and initial wave locations are considered.

Wave breaking to a shallow water wave equation involving the Fornberg-Whitham model

A nonlinear shallow water wave model involving the Fornberg-Whitham equation is investigated. From the equation itself, we derive several estimates, which are helpful to study the dynamical properties of its solutions. Finally, we give conditions on the initial value that lead to wave breaking.

Symmetric waves are traveling waves of some shallow water scalar equations

Following a straightforward proof for symmetric solutions to be traveling waves by Pei (Exponential decay and symmetry of solitary waves to Degasperis‐Procesi equation. Journal of Differential Equations. 2020;269(10):7730‐7749), we prove that classical symmetric solutions of the highly nonlinear shallow water equation recently derived by Quirchmayr (A new highly nonlinear shallow water wave equation. Journal of Evolution Equations. 2016;16(3):539‐556) are indeed traveling waves, with further information on their steady structures. We also provide a simple proof that symmetric waves are traveling waves to the free surface evolution equation of moderate amplitude waves in shallow water.

Fractal soliton solutions for the fractal-fractional shallow water wave equation arising in ocean engineering

The generalized shallow water wave equation is an important mathematical model that is used to elaborate ocean engineering, weather simulations, tsunami prediction and tidal currents. In this work, the generalized (3 + 1)-dimensional fractal-fractional shallow water wave equation (FFSWWE) is investigated where fractal-fractional derivative is taken in the conformable derivative sense. Some new fractal soliton solutions of FFSWWE are successfully derived by the fractal-fractional variational wave method (FFVWM), which is a new mathematical technology. This new method has the advantages of being simple, efficient, and direct. The 3D graphics that describe these new fractal soliton solutions that were obtained are tremendously important for improving our understanding of physical oceanography.

Some new soliton solutions to the higher dimensional Burger–Huxley and Shallow water waves equation with couple of integration architectonic

In this paper, we retrieve some traveling wave, periodic solutions, bell shaped, rational, kink and anti-kink type and Jacobi elliptic functions of Burger’s equation and Shallow water wave equation with the aid of various integration schemes like improved F-expansion scheme and Jacobi elliptic function method respectively. We also present our solutions graphically in various dimensions.

Exact solutions of the generalized [formula omitted]-dimensional shallow water wave equation

In this paper, we construct abundant exact solutions of generalized (2+1)-dimensional shallow water wave equation via the Hirota bilinear method and test functions. We obtain exact interaction solutions, such as solitons, lump solutions and lump-periodic solutions for the generalized (2+1)-dimensional shallow water wave equation degenerated by the generalized (3+1)-dimensional shallow water wave equation which introduced by Sakkaravarthi et al.