Residual symmetry, CRE integrability and interaction solutions of two higher-dimensional shallow water wave equations

Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.

Collisions Between Lumps/Rogue Waves and Solitons for A(3+1)-Dimensional Generalized Variable-Coefficient Shallow Water Wave Equation

In this paper,we investigate a(3+1)-dimensional generalized variable-coefficient shallow water wave equation,which can be used to describe the flow below a pressure surface in oceanography and atmospheric science.Employing the Kadomtsev−Petviashvili hierarchy reduction,we obtain the semi-rational solutions which describe the lumps and rogue waves interacting with the kink solitons.We find that the lump appears from one kink soliton and fuses into the other on the x−y and x−t planes.However,on the x−z plane,the localized waves in the middle of the parallel kink solitons are in two forms:lumps and line rogue waves.The effects of the variable coefficients on the two forms are discussed.The dispersion coefficient influences the speed of solitons,while the background coefficient influences the background’s height.

Exact Traveling Wave Solutions for the System of Shallow Water Wave Equations and Modified Liouville Equation Using Extended Jacobian Elliptic Function Expansion Method

In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the system of shallow water wave equations and modified Liouville equation which play an important role in mathematical physics.

Exact solutions of a(2+1)-dimensional extended shallow water wave equation

We give the bilinear form and n-soliton solutions of a（2+1）-dimensional [（2+1）-D] extended shallow water wave（eSWW） equation associated with two functions v and r by using Hirota bilinear method. We provide solitons, breathers,and hybrid solutions of them. Four cases of a crucial φ（y）, which is an arbitrary real continuous function appeared in f of bilinear form, are selected by using Jacobi elliptic functions, which yield a periodic solution and three kinds of doubly localized dormion-type solution. The first order Jacobi-type solution travels parallelly along the x axis with the velocity（3k12+ α, 0） on（x, y）-plane. If φ（y） = sn（y, 3/10）, it is a periodic solution. If φ（y） = cn（y, 1）, it is a dormion-type-Ⅰ solutions which has a maximum（3/4）k1p1 and a minimum-（3/4）k1p1. The width of the contour line is ln■. If φ（y） = sn（y, 1）, we get a dormion-type-Ⅱ solution（26） which has only one extreme value-（3/2）k1p1. The width of the contour line is ln■. If φ（y） = sn（y, 1/2）/（1 + y2）, we get a dormion-type-Ⅲ solution（21） which shows very strong doubly localized feature on（x, y） plane. Moreover, several interesting patterns of the mixture of periodic and localized solutions are also given in graphic way.

The Exact Rational Solutions to a Shallow Water Wave-Like Equation by Generalized Bilinear Method

A Shallow Water Wave-like nonlinear differential equation is considered by using the generalized bilinear equation with the generalized bilinear derivatives D3,x and D3,t, which possesses the same bilinear form as the standard shallow water wave bilinear equation. By symbolic computation, four presented classes of rational solutions contain all rational solutions to the resulting Shallow Water Wave-like equation, which generated from a search for polynomial solutions to the corresponding generalized bilinear equation.

Analytical Solution of Two Extended Model Equations for Shallow Water Waves By Adomian’S Decomposition Method

In this paper, we consider two extended model equations for shallow water waves. We use Adomian’s decomposition method (ADM) to solve them. It is proved that this method is a very good tool for shallow water wave equations and the obtained solutions are shown graphically.

Some New Nonlinear Wave Solutions for a Higher-Dimensional Shallow Water Wave Equation

In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended F-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.

Explicit Kinetic Flux Vector Splitting Scheme for the 2-D Shallow Water Wave Equations

Originally, the kinetic flux vector splitting (KFVS) scheme was developed as a numerical method to solve gas dynamic problems. The main idea in the approach is to construct the flux based on the microscopical description of the gas. In this paper, based on the analogy between the shallow water wave equations and the gas dynamic equations, we develop an explicit KFVS method for simulating the shallow water wave equations. A 1D steady flow and a 2D unsteady flow are presented to show the robust and accuracy of the KFVS scheme.

Analytical Solution of Two Extended Model Equations for Shallow Water Waves by He’s Variational Iteration Method

In this paper, we consider two extended model equations for shallow water waves. We use He’s variational iteration method (VIM) to solve them. It is proved that this method is a very good tool for shallow water wave equations and the obtained solutions are shown graphically.

Modified (2+1)-dimensional displacement shallow water wave system and its approximate similarity solutions

Recently, a new （2＋1）-dimensional shallow water wave system, the （2＋1）-dlmenslonal displacement shallow water wave system （2DDSWWS）, was constructed by applying the variational principle of the analytic mechanics in the Lagrange coordinates. The disadvantage is that fluid viscidity is not considered in the 2DDSWWS, which is the same as the famous Kadomtsev-Petviashvili equation and Korteweg-de Vries equation. Applying dimensional analysis, we modify the 2DDSWWS and add the term related to the fluid viscidity to the 2DDSWWS. The approximate similarity solutions of the modified 2DDSWWS （M2DDSWWS） is studied and four similarity solutions are obtained. For the perfect fluids, the coefficient of kinematic viscosity is zero, then the M2DDSWWS will degenerate to the 2DDSWWS.

Applications of sediment sudden deposition model based on the third-generation numerical model for shallow water wave

The existing numerical models for nearshore waves are briefly introduced, and the third-generation numerical model for shallow water wave, which makes use of the most advanced productions of wave research and has been adapted well to be used in the environment of seacoast, lake and estuary area, is particularly discussed. The applied model realizes the significant wave height distribution at different wind directions. To integrate the model into the coastal area sediment, sudden deposition mechanism, the distribution of average silt content and the change of sediment sudden deposition thickness over time in the nearshore area are simulated. The academic productions can give some theoretical guidance to the applications of sediment sudden deposition mechanism for stormy waves in the coastal area. And the advancing directions of sediment sudden deposition model are prospected.

EQUAITON ON SPATIAL VARIATION OF SHALLOW WATER WAVES AMPLITUDE

Boussinesq’s theory are used in this study on watr waves entering shallow water showing that the spacial variation of the wave amplitude is nonlinear, and is governed by the Duffing equation usually applied to describs nonlinear oscillation in nature.

Dispersive soliton solutions for shallow water wave system and modified Benjamin-Bona-Mahony equations via applications of mathematical methods

We have utilized three novel methods,called generalized direct algebraic,modified F-expansion and improved simple equation methods to construct traveling wave solutions of the system of shallow water wave equations and modified Benjamin-Bona-Mahony equation.After substituting particular values of the parameters,different solitary wave solutions are derived from the exact traveling wave solutions.It is shown that these employed methods are more powerful tools for nonlinear wave equations.

Darboux Transformation and Soliton Solutions for the (2+1)-Dimensional Generalization of Shallow Water Wave Equation with Symbolic Computation

In this paper, the （2＋l）-dimensional generalization of shallow water wave equation, which may be used to describe the propagation of ocean waves, is analytically investigated. With the aid of symbolic computation, we prove that the （2＋ l）-dimensional generalization of shallow water wave equation possesses the Palnlev6 property under a certain condition, and its Lax pair is constructed by applying the singular manifold method. Based on the obtained Lax representation, the Darboux transformation （DT） is constructed. The first iterated solution, second iterated solution and a special N-soliton solution with an arbitrary function are derived with the resulting DT. Relevant properties are graphically illustrated, which might be helpful to understanding the propagation processes for ocean waves in shallow water.

Exact Periodic Wave Solution of Extended(2+1)-Dimensional Shallow Water Wave Equation with Generalized D_-operators

With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.

Wind-Driven Ocean Circulation in Shallow Water Lattice Boltzmann Model

A lattice Boltzmann (LB) model with overall second-order accuracy is applied to the 1.5-layer shallow water equation for a wind-driven double-gyre ocean circulation. By introducing the second-order integral approximation for the collision operator, the model becomes fully explicit. In this case, any iterative technique is not needed. The Coriolis force and other external forces are included in the model with second-order accuracy, which is consistent with the discretized accuracy of the LB equation. The numerical results show correct physics of the ocean circulation driven by the double-gyre wind stress with different Reynolds numbers and different spatial resolutions. An intrinsic low-frequency variability of the shallow water model is also found. The wind-driven ocean circulation exhibits subannual and interannual oscillations, which are comparable to those of models in which the conventional numerical methods are used.

Applying the New Extended Direct Algebraic Method to Solve the Equation of Obliquely Interacting Waves in Shallow Waters

In this study,the potential Kadomtsev-Petviashvili(pKP)equation,which describes the oblique interaction of surface waves in shallow waters,is solved by the new extended direct algebraic method.The results of the study show that by applying the new direct algebraic method to the pKP equation,the behavior of the obliquely interacting surface waves in two dimensions can be analyzed.This article fairly clarifies the behaviors of surface waves in shallow waters.In the literature,several mathematical models have been developed in attempt to study these behaviors,with nonlinear mathematics being one of the most important steps;however,the investigations are still at a level that can be called‘baby steps’.Therefore,every study to be carried out in this context is of great importance.Thus,this study will serve as a reference to guide scientists working in this field.

Study of abundant explicit wave solutions of the Drinfeld-Sokolov-Satsuma-Hirota(DSSH)equation and the shallow water wave equation

In this article,the two variable(G'G,1/G)-expansion method is suggested to investigate new and further general multiple exact wave solutions to the Drinfeld-Sokolov-Satsuma-Hirota(DSSH)equation and the shallow water wave equation which arise in mathematical physics with the aid of computer algebra software,like Mathematica.Three functions and the rational functions solution are found.The method demonstrates power,reliability and efficiency.Indeed,the method is the generalization of the well-known(G/G)-expansion method established by Wang et al.and the method also presents a wider applicability for conducting nonlinear wave equations.

Wave Numerical Model for Shallow Water

The history of forecasting wind waves by wave energy conservation equation Is briefly described. Several currently used wave numerical models for shallow water based on different wave theories are discussed. Wave energy conservation models for the simulation of shallow water waves are introduced, with emphasis placed on the SWAN model, which takes use of the most advanced wave research achievements and has been applied to several theoretical and field conditions. The characteristics and applicability of the model, the finite difference numerical scheme of the action balance equation and its source terms computing methods are described in detail. The model has been verified with the propagation refraction numerical experiments for waves propagating in following and opposing currents; finally, the model is applied to the Haian Gulf area to simulate the wave height and wave period field there, and the results are compared with observed data.

Dispersive Shallow Water Wave Modelling.Part I:Model Derivation on a Globally Flat Space

In this paperwe reviewthe history and current state-of-the-art in modelling of long nonlinear dispersive waves.For the sake of conciseness of this review we omit the unidirectional models and focus especially on some classical and improved BOUSSINESQ-type and SERRE-GREEN-NAGHDI equations.Finally,we propose also a unified modelling framework which incorporates several well-known and some less known dispersive wave models.The present manuscript is the first part of a series of two papers.The second part will be devoted to the numerical discretization of a practically important model on moving adaptive grids.