Long Waves In Shallow Water Research Articles

Symmetry Reductions of the (1 + 1)-Dimensional Broer–Kaup System Using the Generalized Double Reduction Method

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs.

Nonlinear long waves in shallow water for normalized Boussinesq equations

In this work, we investigate a partial differential equation (PDE) derived from a Boussinesq system of two equations that describe the wave motion in shallow water when the wavelength is small compared to wave amplitude. The Lie point symmetries of the PDE are utilized to reduce it to an ordinary differential equation (ODE) the travelling wave solutions of which are obtained. These solutions are of trigonometric, hyperbolic and exponential types. Graphical representation of some solutions exhibiting solitary and cnoidal wave phenomena are presented.

The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method

<p>The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.</p>

The Efficient Method to Solve the Conformable Time Fractional Benney Equation

The current study employed the innovative conformable fractional method to analyze the nonlinear Benney equations involving the conformable fractional derivative. Conformable fractional Benney equations have been examined by the conformable q‐Shehu analysis transform method. By including nonlinear factors, it offers a more precise depiction of wave propagation compared to linear models. Various natural phenomena, including ocean waves, plasma waves, and some forms of solitons, display nonlinear behavior that cannot be precisely explained by linear equations. The fractional Benney equation is important because it extends the classical Benney equation, which describes the evolution of weakly nonlinear and weakly dispersive long waves in shallow water. By incorporating fractional calculus operators, the fractional Benney equation provides a more accurate description of wave propagation phenomena in certain physical systems characterized by nonlocal or memory‐dependent behavior. The utilization of the Benney equation enables researchers to simulate these occurrences with greater realism. This study investigates the convergence and inaccuracy of the future scheme. The conformable q‐Shehu homotopy analysis transform method (Cq‐SHATM) generates h‐curves that demonstrate the convergence interval of the series solution obtained. In order to determine the effectiveness and suitability of the Cq‐SHATM, uniqueness and convergence theorems have been proven. This study presents an application that showcases the potential advantages and efficacy of the suggested method. Moreover, an error analysis is conducted to validate the precision of the scheme. Computational simulations are performed to verify the accuracy of the upcoming method. This study presents the results gained from the numerical and graphical analysis. The method presented in this work demonstrates a high level of computational accuracy and simplicity in analyzing and solving complex phenomena associated with conformable fractional nonlinear partial differential equations in the fields of science and technology.

Exploring the new soliton solutions to the nonlinear M-fractional evolution equations in shallow water by three analytical techniques

This paper explores the new soliton solutions of the evolution equations named as truncated M-fractional (1+1)-dimensional non-linear Kaup-Boussinesq system by utilizing the expa function, modified simplest equation and Sardar sub-equation techniques. This system is used in the analysis of long waves in shallow water. The attained results involving trigonometric, hyperbolic and exponential functions. The effect of fractional order derivative is also discussed. Obtained results are very close to the approximate results due to the use of M-fractional derivative. Achieved results are verified by Mathematica tool. Few of the gained results are also explained through 2-D, 3-D and contour graphs. At the end, these techniques are straight forward, useful and effective to deal with non-linear FPDEs.

Exact Solutions of the Stochastic Conformable Broer–Kaup Equations

In this article, the exact solutions of the stochastic conformable Broer–Kaup equations with conformable derivatives which describe the bidirectional propagation of long waves in shallow water are obtained using the modified exponential function method and the generalized Kudryashov method. These exact solutions consist of hyperbolic, trigonometric, rational trigonometric, rational hyperbolic, and rational function solutions, respectively. This shows that the proposed methods are competent and sufficient. In addition, it is aimed to better understand the physical properties by drawing two- and three-dimensional graphics of the exact solutions according to different parameter values. When these exact solutions obtained by two different methods are compared with the solutions attained by other methods, it can be said that these two methods are competent.

On the traveling wave solutions of the (1 + 1)-dimensional Broer–Kaup system for modeling the bi-directional propagation of long waves in shallow waters

The development of an effective and efficient local discontinuous Galerkin algorithm for the solution of the coupled ( 1 + 1 ) -dimensional Broer–Kaup system is the primary focus of this research. In this algorithm, spatial discretization is attained using the local discontinuous Galerkin method, while temporal discretization is handled using the explicit total variation diminishing higher-order Runge-Kutta method. However, the ( G ′ G ) -expansion method is also implemented to produce the exact solutions for the coupled ( 1 + 1 ) -dimensional Broer–Kaup system. The obtained solutions are the traveling wave solutions. The efficiency and reliability of the proposed method are analyzed by comparing the generated numerical simulations to various traveling wave solutions using several tables and figures. The exact solutions and the numerical simulations of the coupled ( 1 + 1 ) -dimensional Broer–Kaup system are shown to correspond exceptionally well.

A Time Two-Mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time two-mesh (TT-M) decoupled finite difference numerical scheme is proposed in this paper to improve the efficiency of solving the SRLW equation. Based on the time two-mesh technique and two time-level finite difference method, the proposed scheme can calculate the velocity u(x,t) and density ρ(x,t) in the SRLW equation simultaneously. The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation to obtain solutions. By using the discrete energy and mathematical induction methods, the convergence results with O(τC2+τF+h2) in the discrete L∞-norm for u(x,t) and in the discrete L2-norm for ρ(x,t) are proved, respectively. The stability of the scheme was also analyzed. Finally, some numerical examples, including error estimate, computational time and preservation of conservation laws, are given to verify the efficiency of the scheme. The numerical results show that the new method preserves conservation laws of four quantities successfully. Furthermore, by comparing with the original two-level nonlinear finite difference scheme, the proposed scheme can save the CPU time.

On a generalized Broer-Kaup-Kupershmidt system for the long waves in shallow water

On a generalized Broer-Kaup-Kupershmidt system for the long waves in shallow water

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.

A new generalized KdV equation: Its lump-type, complexiton and soliton solutions

A new generalized KdV equation, describing the motions of long waves in shallow water under the gravity field, is considered in this paper. By adopting a series of well-organized methods, the Bäcklund transformation, the bilinear form and diverse wave structures of the governing model are formally extracted. The exact solutions listed in this paper are categorized as lump-type, complexiton, and soliton solutions. To exhibit the physical mechanism of the obtained solutions, several graphical illustrations are given for particular choices of the involved parameters. As a direct consequence, diverse wave structures given in this paper enrich the studies on the KdV-type equations.

Numerical and analytical investigations for solution of fractional Ito equation describing motions of long waves in shallow water under gravity

In this manuscript, a meshfree numerical scheme based on radial basis function-finite difference (RBF-FD) method is proposed to solve the time-fractional Ito equation. First, the temporal derivative is approximated through the finite difference technique. Then, the spatial derivatives are discretized via the multiquadric RBF. Furthermore, some useful theorems are discussed to establish the stability and convergence of the proposed numerical scheme. Finally, some test problems are presented and the results are compared with the acquired Kudryashov method and tanh method solutions to establish the computational efficiency and applicability of the proposed method.

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.

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.

Interaction of two long waves in shallow water using Hirota-Satsuma model and similarity transformations

Interaction of two long waves in shallow water using Hirota-Satsuma model and similarity transformations

Lump, lump-stripe, and breather wave solutions to the (2 + 1)-dimensional Sawada-Kotera equation in fluid mechanics

The present study investigates the lump, one-stripe, lump-stripe, and breather wave solutions to the (2+1)-dimensional Sawada-Kotera equation using the Hirota bilinear method. For lump and lump-stripe solutions, a quadratic polynomial function, and a quadratic polynomial function in conjunction with an exponential term are assumed for the unknown function f giving the solution to the mentioned equation, respectively. On the other hand, only an exponential function is considered for one-stripe solutions. Besides, the homoclinic test approach is adopted for constructing breather wave solutions. The propagations of the attained lump, lump-stripe, and breather wave solutions are shown through some graphical illustrations. The graphical outputs demonstrate that the lump wave moves along the straight line y=−x and exponentially decreases away from the origin of the spatial domain. On the other hand, lump-kink solutions illustrate the fission and fusion interaction behaviors upon the selection of the free parameters. Fission and fusion processes show that the stripe soliton splits into a stripe soliton and a lump soliton, and then the lump soliton merges into one stripe soliton. In addition, the achieved breather waves illustrate the periodic behaviors in the xy-plane. The outcomes of the study can be useful for a better understanding of the physical nature of long waves in shallow water under gravity.

Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

Under investigation in this paper is the higher-order Broer-Kaup(HBK) system, which describes the bidirectional propagation of long waves in shallow water. Via the standard truncated Painlevé expansion method, the residual symmetry of this system is derived. By introducing an appropriate auxiliary-dependent variable, the residual symmetry is successfully localized to Lie point symmetries. Via solving the initial value problems, the finite symmetry transformations are presented. However, the solution which obtained from the residual symmetry is a special group invariant solutions. In order to find more general solution of HBK system, we further generalize the residual symmetry method to the consistent tanh expansion (CTE) method and prove that the HBK system is CTE solvable, then the resonant soliton solutions and interaction solutions among different nonlinear excitations are obtained by the CET method.

Dynamics of Lump-periodic, breather and two-wave solutions with the long wave in shallow water under gravity and 2D nonlinear lattice

A lump solution is a rational function solution which is real analytic and decays in all directions of space variables. The equation under consideration in this study is the (2 + 1)-dimensional generalized fifth-order KdV equation which demonstrates long wave movements under the gravity field and in a two-dimensional nonlinear lattice in shallow water. The collisions between lump and other analytic solutions is studied in this work. Using Hirota bilinear approach, lump-periodic, breather and two-wave solutions are successfully reported. In order to shade more light on the characteristics of the acqured solutions, numerical simulations have been performed by means of the 3-dimensional and contour profiles under careful choice of the values of the parameters involved.

Dynamics of interaction between solitary and rogue wave of the space-time fractional Broer–Kaup models arising in shallow water of harbor and coastal zone

Under inquisition, we consider the space-time fractional Broer–Kaup model that replicate the real coastal profile to describe bidirectional wave transmission of long waves in shallow water. We integrate of the model via the unified scheme to derive exact solitary wave solutions. The achieved results present the hyperbolic, trigonometric, and rational functions solutions. All of these solutions convey the periodic and solitonic natures. In particular, we present combo waves that imply a curvy periodic waves in which rogue waves occur in both sides of each waves. The dynamics of obtained nonlinear wave solutions are demonstrated in 3-D and 2-D shapes with definite parametric values. It is shown that the obtained solutions of the model pact a very rich dynamical behavior and can be used to describe seismic type nonlinear wave motion in shallow water waves of the coastal and harbor region.

Piston-Driven Numerical Wave Tank Based on WENO Solver of Well-Balanced Shallow Water Equations

A numerical wave tank equipped with a piston type wave-maker is presented for long-duration simulations of long waves in shallow water. Both wave maker and tank are modelled using the nonlinear shallow water equations, with motions of the numerical piston paddle accomplished via a linear mapping technique. Three approaches are used to increase computational efficiency and accuracy. First, the model satisfies the exact conservation property (C-property), a stepping stone towards properly balancing each term in the governing equation. Second, a high-order weighted essentially non-oscillatory (WENO) method is used to reduce accumulation of truncation error. Third, a cut-off algorithm is implemented to handle contaminated digits arising from round-off error. If not treated, such errors could prevent a numerical scheme from satisfying the exact C-property in long-duration simulations. Extensive numerical tests are performed to examine the well-balanced property, high order accuracy, and shock-capturing ability of the present scheme. Correct implementation of the wave paddle generator is verified by comparing numerical predictions against analytical solutions of sinusoidal, solitary, and cnoidal waves. In all cases, the model gives satisfactory results for small-amplitude, low frequency waves. Error analysis is used to investigate model limitations and derive a user criterion for long wave generation by the model.