Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.

Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation with Symbolic Computation

In this paper, we put our focus on a variable-coe^cient fifth-order Korteweg-de Vries （fKdV） equation, which possesses a great number of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived, such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out.

Infinite Sequence of Conservation Laws and Analytic Solutions for a Generalized Variable-Coefficient Fifth-Order Korteweg-de Vries Equation in Fluids

In this paper, an infinite sequence of conservation laws for a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids are constructed based on the Backlund transformation. Hirota bilinear form and symbolic computation are applied to obtain three kinds of solutions. Variable coefficients can affect the conserved density, associated flux, and appearance of the characteristic lines. Effects of the wave number on the soliton structures are also discussed and types of soliton structures, e.g., the double-periodic soliton, parallel soliton and soliton complexes, are presented.

In the Atmosphere and Oceanic Fluids:Scaling Transformations,Bilinear Forms,Bäcklund Transformations and Solitons for A Generalized Variable-Coefficient Korteweg-de Vries-Modified Korteweg-de Vries Equation

The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-de Vriesmodified Korteweg-de Vries equation for the atmosphere,oceanic fluids and plasmas.With symbolic computation,beginning with a presumption,we work out certain scaling transformations,bilinear forms through the binary Bell polynomials and our scaling transformations,N solitons(with N being a positive integer)via the aforementioned bilinear forms and bilinear auto-Bäcklund transformations through the Hirota method with some solitons.In addition,Painlevé-type auto-Bäcklund transformations with some solitons are symbolically computed out.Respective dependences and constraints on the variable/constant coefficients are discussed,while those coefficients correspond to the quadratic-nonlinear,cubic-nonlinear,dispersive,dissipative and line-damping effects in the atmosphere,oceanic fluids and plasmas.

Exact Periodic and Solitary-Wave Solutions of Multi-component Modified Korteweg-de Vries Equations

The present work extends the search of Jacobi elliptic function solutions for the multi-component modified Korteweg-de Vries equations. When the modulum m →1, those periodic solutions degenerate as the corresponding solitary wave and shock wave ones. Especially, exact solutions for the three-component system are presented in detail and graphically.

Generalized Wronskian Solutions to Modified Korteweg-de Vries Equation via Its Bcklund Transformation

In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation （mKdV） via the Backlund transformation （BT） and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the GN （t） for the original diagonal one.

Existence for a Higher Order Coupled System of Korteweg-de Vries Equations

Consider the following system of coupled Korteweg-de Vries equations, where u, v ⊆ W2,2, 2≤N≤7 and λi,β > 0, β denotes a real coupling parameter. Firstly, we prove the existence of the solutions of a coupled system of Korteweg-de Vries equations using variation approach and minimization techniques on Nehari manifold. Then, we show the multiplicity of the equations by a bifurcation theory which is rare for studying higher order equations.

The Dispersion Relation of Internal Wave Extended-Korteweg-de Vries Equation in a Two-Layer Fluid

To understand the characteristics of ocean internal waves better, we study the dispersion relation of extended-Korteweg-de Vries (EKdV) equation with quadratic and cubic nonlinear terms in a two-layer fluid by using the Poincaré-Lighthill-Kuo (PLK) method which is one of the perturbation methods. Starting from the partial differential equation, the PLK method can be used to solve the dispersion relation of the equation. In this paper, we use PLK method to solve the equation and derive the dispersion relation of EKdV equation which is related to wave number and amplitude. Based on the dispersion relation obtained in this paper, the expressions of group velocity and phase velocity of the equation are obtained. Under the actual hydrological data, the influence of hydrological parameters on the dispersion relation for descending internal wave is discussed. It is hope that the obtained results will be helpful to the study of energy transfer and other internal wave parameters in the future.

Artificial perturbation for solving the Korteweg-de Vries equation

A perturbation method is introduced in the context of dynamical system for solving the nonlinear Korteweg-de Vries (KdV) equation. Best efficiency is obtained for few perturbative corrections. It is shown that, the question of convergence of this approach is completely guaranteed here, because a limited number of term included in the series can describe a sufficient exact solution. Comparisons with the solutions of the quintic spline, and finite difference are presented.

Discrete Singular Convolution Method for Numerical Solutions of Fifth Order Korteweg-De Vries Equations

A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.

Travelling Solitary Wave Solutions to Higher Order Korteweg-de Vries Equation

The travelling solitary wave solutions to the higher order Korteweg-de Vries equation are obtained by using tanh-polynomial method. The method is effective and concise, which is also applied to various partial differential equations to obtain traveling wave solutions. The numerical simulation of the solutions is given for completeness. Numerical results show that the tanh-polynomial method works quite well.

Martingale Solution to Stochastic Extended Korteweg-de Vries Equation

The deterministic extended Korteweg-de Vries equation plays an essential role in the description of the creation and propagation of nonlinear waves in many fields. We study a stochastic extended Korteweg-de Vries equation driven by a multiplicative noise in the form of a cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied for all physically relevant initial conditions. The proof of the solution is based on two approximations of the problem considered and the compactness method.

A second-order convergent and linearized difference schemefor the initial-boundary value problemof the Korteweg-de Vries equation

To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is constructed for the system.The new variable introduced can be separated from the difference scheme to obtain another difference scheme containing only the original variable.The energy method is applied to the theoretical analysis of the difference scheme.Results show that the difference scheme is uniquely solvable and satisfies the energy conservation law corresponding to the original problem.Moreover,the difference scheme converges when the step ratio satisfies a constraint condition,and the temporal and spatial convergence orders are both two.Numerical examples verify the convergence order and the invariant of the difference scheme.Furthermore,the step ratio constraint is unnecessary for the convergence of the difference scheme.Compared with a known two-level nonlinear difference scheme,the proposed difference scheme has more advantages in numerical calculation.

An improved element-free Galerkin method for solving the generalized fifth-order Korteweg-de Vries equation

In this paper, an improved element-free Galerkin （IEFG） method is proposed to solve the generalized fifth-order Korteweg-de Vries （gfKdV） equation. When the traditional element-free Galerkin （EFG） method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares （MLS） shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares （IMLS） approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.

Application of Hyperbola Function Method to the Family of Third Order Korteweg-de Vries Equations

In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions and trigonometric functions. The method used is a promising method to solve other nonlinear evaluation equations.

Integrability and solutions of a nonsymmetric discrete Korteweg-de Vries equation

In this paper,we present Lax pairs and solutions for a nonsymmetric lattice equation,which is a torqued version of the lattice potential Korteweg-de Vries equation.This nonsymmetric equation is special in the sense that it contains only one spacing parameter but consists of two consistent cubes with other integrable lattice equations.Using such a multidimensionally consistent property we are able to derive its two Lax pairs and also construct solutions using B?cklund transformations.

New lump interaction complexitons to the(2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential in plasmas

A soliton is a packet of self-reinforcing waves that maintains its structure when moving at a constant speed.Solitons are caused by the cancellation of the medium’s nonlinear and dispersive effects.In plas-mas,the bilinear form of Hirota will be utilized to investigate the(2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential.Solutions for complexiton lump interaction have been devel-oped.To throw further light on the physical qualities of the recorded data,certain 3-dimensional and contour plots are presented to illustrate the interaction elements of these solutions.

CONTROL AND STABILIZATION OF THE KORTEWEG-DE VRIES EQUATION:RECENT PROGRESSES

The study of the control and stabilization of the KdV equation began with the work ofRussell and Zhang in late 1980s.Both exact control and stabilization problems have been intensivelystudied since then and significant progresses have been made due to many people's hard work andcontributions.In this article,the authors intend to give an overall review of the results obtained so farin the study but with an emphasis on its recent progresses.A list of open problems is also providedfor further investigation.

Effect of Coriolis constant on Geophysical Korteweg-de Vries equation

The present article investigates the effect of Coriolis constant on the solution of the Geophysical Korteweg-de Vries(gKdV)equation.As such,the Homotopy Perturbation Method(HPM)has been applied here for solving the nonlinear gKdV equation.Present results are compared with existing results that are available in the literature and they are found to be in good agreement.Then the Coriolis term has been considered in terms of the interval to form interval Geophysical Korteweg-de Vries(IgKdV)equation.IgKdV equation has been solved by HPM to analyse the effect of Coriolis.From this analysis,it has been concluded that the constant of Coriolis is directly proportional to wave height and inversely proportional to wavelength.The presence of the Coriolis term in gKdV equation has a remarkable change in the shape of the solution.

CRE Solvability, Nonlocal Symmetry and Exact Interaction Solutions of the Fifth-Order Modified Korteweg-de Vries Equation

This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion(CTE) method, the nonlocal symmetry related to the consistent tanh expansion(CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlev′e method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.