All traveling wave exact solutions of three kinds of nonlinear evolution equations

In this article, we employ the complex method to obtain all meromorphic exact solutions of complex Klein–Gordon (KG) equation, modified Korteweg‐de Vries (mKdV) equation, and the generalized Boussinesq (gB) equation at first, then find all exact solutions of the Equations KG, mKdV, and gB. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions w_2r,2(z) and simply periodic solutions w_1s,2(z),w_2s,1(z) in these equations such that they are not only new but also not degenerated successively by the elliptic function solutions. We have also given some computer simulations to illustrate our main results. Copyright © 2014 John Wiley & Sons, Ltd.     

Symmetry analysis of the nonlinear two‐dimensional Klein–Gordon equation with a time‐varying delay

The group analysis method is applied to the two‐dimensional nonlinear Klein–Gordon equation with time‐varying delay. Determining equations for equations with a time‐varying delay are derived. A complete group classification of the studied equation with respect to the function involved into the equation is obtained. All admitted Lie algebras are classified. By using the classifications, representations of all invariant solutions are found. Copyright © 2017 John Wiley & Sons, Ltd.     

A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation

Based on the homogeneous balance method,the Jacobi elliptic expansion method and the auxiliary equation method,the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations.New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple.The method is also valid for other (1+1)-dimensional and higher dimensional systems.     

New periodic wave solutions for the shallow water equations and the generalized Klein–Gordon equation

The periodic wave solutions and the corresponding solitary solutions for the shallow water equations and the generalized Klein–Gordon equation are obtained by means of mapping method. The solutions obtained in this paper include as well the shock wave solution, complex line period, complex line soliton and rational solutions. Moreover, the obtained solutions are degenerated in terms of hyperbolic function solutions and trigonometric function solutions when the modulus m of the Jacobi elliptic function is driven to 1 and 0, respectively. The previously known periodic and solitary wave solutions are recovered. Many new results are presented.     

Integral transform approach to solving Klein–Gordon equation with variable coefficients

In this paper we describe the integral transform that allows to write solutions of the time‐dependent partial differential equation via solution of a simpler equation. This transform was suggested by the author in the case when the last equation is a wave equation, and then it was used to investigate several well‐known equations such as Tricomi‐type equation, the Klein–Gordon equation in the de Sitter and Einstein‐de Sitter spacetimes. A generalization given in this paper allows us to consider also the Klein–Gordon equations with coefficients depending on the spatial variables.     

On the complete group classification of the one‐dimensional nonlinear Klein–Gordon equation with a delay

This research gives a complete Lie group classification of the one‐dimensional nonlinear delay Klein–Gordon equation. First, the determining equations are derived and their complete solutions are found. Then the complete group classification and representations of all invariant solutions are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Modified ‐expansion method for finding exact wave solutions of the coupled Klein–Gordon–Schrödinger equation

The coupled Klein–Gordon–Schrödinger equation is reduced to a nonlinear ordinary differential equation (ODE) by using Lie classical symmetries, and various solutions of the nonlinear ODE are obtained by the modified ‐expansion method proposed recently. With the aid of solutions of the nonlinear ODE, more explicit traveling wave solutions of the coupled Klein–Gordon–Schrödinger equation are found out. The traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. Copyright © 2012 John Wiley & Sons, Ltd.     

New generalized Jacobi elliptic function expansion method

A new generalized Jacobi elliptic function expansion method is described and used for constructing many new exact travelling wave solutions for nonlinear partial differential equations (PDEs) in a unified way. We obtain many new Jacobi and Weierstrass double periodic elliptic function solutions for (3 + 1)-dimensional Kadmtsev–Petviashvili (KP) equation. This method can be applied to many other equations.     

A conservative compact difference scheme for the coupled Klein–Gordon–Schrödinger equation

In this article, a conservative compact difference scheme is presented for the periodic initial‐value problem of Klein–Gordon–Schrödinger equation. On the basis of some inequalities about norms and the priori estimates, convergence of the difference solution is proved with order O(h⁴ +τ ²) in maximum norm. Numerical experiments demonstrate the accuracy and efficiency of the compact scheme. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

用改进的代数方法构造(2+1)维ZK-MEW方程的精确行波解

利用一种改进的统一代数方法将构造(2+1)维ZK MEW((2+1)-dimensionalZakharov-Kuznetsovmodifiedequalwidth)方程精确行波解的问题转化为求解一组非线性的代数方程组．再借助于符号计算系统Mathematica求解所得到的非线性代数方程组,最终获得了方程的多种形式的精确行波解．其中包括有理解,三角函数解,双曲函数解,双周期Jacobi椭圆函数解,双周期Weierstrass椭圆形式解等．并给出了部分解的图形.     

Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

We considered the inverse problem of scattering theory for a boundary value problem on the half line generated by Klein–Gordon differential equation with a nonlinear spectral parameter‐dependent boundary condition. We defined the scattering data, and we proved the continuity of the scattering function S(λ); in a special case, the relation for the difference of the logarithm of the scattering function, which is called the Levinson‐type formula, was obtained. Copyright © 2012 John Wiley & Sons, Ltd.     

A coupled method of Laplace transform and Legendre wavelets for nonlinear Klein–Gordon equations

Klein–Gordon equation models many phenomena in both physics and applied mathematics. In this paper, a coupled method of Laplace transform and Legendre wavelets, named (LLWM), is presented for the approximate solutions of nonlinear Klein–Gordon equations. By employing Laplace operator and Legendre wavelets operational matrices, the Klein–Gordon equation is converted into an algebraic system. Hence, the unknown Legendre wavelets coefficients are calculated in the form of series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence analysis of the LLWM is discussed. The results show that LLWM is very effective and easy to implement. Copyright © 2013 John Wiley & Sons, Ltd.     

Rational Jacobi elliptic solutions for nonlinear differential–difference lattice equations

In this work, we present a direct new method for constructing the rational Jacobi elliptic solutions for nonlinear differential–difference equations, which may be called the rational Jacobi elliptic function method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential–difference equations in mathematical physics via the lattice equation. The proposed method is more effective and powerful for obtaining the exact solutions for nonlinear differential–difference equations.     

(3+1)-维广义随机KP方程的精确解

利用统一方式构造非线性偏微分方程行波解的广义Jacobi椭圆函数展开法和Hermite变换来研究(3+1)-维广义随机KP方程,给出了它的随机对偶周期和多孤子解.     

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.     

Time evolution of discrete fourth‐order elliptic operators

The evolution equation is considered. A discrete parabolic methodology is developed, based on a discrete elliptic (fourth‐order) calculus. The main ingredient of this calculus is a discrete biharmonic operator (DBO). In the general case, it is shown that the approximate solutions converge to the continuous one. An “almost optimal” convergence result (O(h^{4 ? ?})) is established in the case of constant coefficients, in particular in the pure biharmonic case. Several numerical test cases are presented that not only corroborate the theoretical accuracy result, but also demonstrate high‐order accuracy of the method in nonlinear cases. The nonlinear equations include the well‐studied Kuramoto–Sivashinsky equation. Numerical solutions for this equation are shown to approximate remarkably well the exact solutions. The numerical examples demonstrate the great improvement achieved by using the DBO instead of the standard (five‐point) discrete bilaplacian.     

New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation

In this work, a new generalized Jacobi elliptic functions expansion method based upon four new Jacobi elliptic functions is described and abundant new Jacobi-like elliptic functions solutions for the variable-coefficient mKdV equation are obtained by using this method, some of these solutions are degenerated to solitary-like solutions and triangular-like functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m→1 or 0, which shows that the new method can be also used to solve other nonlinear partial differential equations in mathematical physics.     

Doubly periodic solutions of the coupled scalar field equations

Some doubly periodic (Jacobi elliptic function) solutions of the coupled Schrödinger–Boussinesq (KdV) equations are presented in closed form. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solution to construct doubly periodic solutions of the coupled equations. When the module m→1, these solutions degenerate to the exact solitary wave solutions of the coupled equations.     

Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space‐time fractional partial differential equations. The fractional derivative used here is the modified Riemann‐Liouville derivative. For illustrating the validity of this method, we apply it to solve the space‐time fractional Fokas equation and the the space‐time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd.