Exact traveling wave solutions to nonlinear diffusion and wave equations

By the introduction of some ansatz equations, we have obtained several new classes of traveling (solitary) wave solutions to the nonlinear diffusion equation $$f_1 (u)u_t + f_2 (u)u_x + f_3 (u)u_{xx} + f_4 (u)u_x^2 = f_5 (u)$$ and the nonlinear wave equation $$f_1 (u)u_u + f_2 (u)u_t + f_3 (u)u_{xx} + f_4 (u)u_x + f_5 (u)u_x^2 + \cdots = f_6 (u)$$ Some applications of these solutions are discussed.     

一类非线性波动方程的精确行波解

利用改进的G’/G展开方法,借助于计算机代数系统Mathematica成功获得了一大类非线性波动方程一系列新的含有多个参数的精确行波解.这些解包括孤立波解、双曲函数解、三角函数解.     

Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method

Based on the Exp-function method, exact solutions for some nonlinear evolution equations are obtained. The KdV equation, Burgers' equation and the combined KdV–mKdV equation are chosen to illustrate the effectiveness of the method.     

强非线性发展方程孤波近似解

研究了一个强非线性发展方程.利用变分原理，首先构造了相应的泛函.选取Lagrange乘子，再用广义变分迭代方法得到了孤波的任意次精度的近似解. 关键词： 发展方程 非线性 孤立子 近似方法     

Exact solutions for nonlinear evolution equations using Exp-function method

In this Letter, the Exp-function method is used to construct solitary and soliton solutions of nonlinear evolution equations. The Klein-Gordon, Burger-Fisher and Sharma-Tasso-Olver equations are chosen to illustrate the effectiveness of the method. The method is straightforward and concise, and its applications are promising. The Exp-function method presents a wider applicability for handling nonlinear wave equations.     

Exact periodic wave solutions to the generalized Nizhnik-Novikov-Veselov equation

The extended mapping method with symbolic computation is developed to obtain exact periodic wave solutions to the generalized Nizhnik-Novikov-Veselov equation. Limit cases are studied and new solitary wave solutions and triangular periodic wave solutions are obtained. The method is applicable to a large variety of non-linear partial differential equations, as long as odd-and even-order derivative terms do not coexist in the equation under consideration.     

Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method

In this paper,an approximate function for the Galerkin method is composed using the combination of the exponential B-spline functions.Regularized long wave equation(RLW)is integrated fully by using an exponential B-spline Galerkin method in space together with Crank–Nicolson method in time.Three numerical examples related to propagation of single solitary wave,interaction of two solitary waves and wave generation are employed to illustrate the accuracy and the efficiency of the method.Obtained results are compared with some early studies.     

非线性“loop”孤子方程的确定解

提出一种求解非线性“loop”孤子方程确定解的新方法，即以“行波”为因子，利用幂级数直接求解该方程解析函数U(ξ)，用MatLab绘制c→光速和c→声速的U(ξ)-ξ图像，直 接观察该方程解的变化规律，找出该方程的确定解(含孤波解）. 该方法为求解难度大的非 线性孤子方程提供借鉴. 关键词： 非线性孤子方程 确定解 MatLab图像     

Exact solutions of variable coefficient nonlinear diffusion-reaction equations with a nonlinear convective term

Attempts have been made to look for the exact solutions of certain types of nonlinear diffusion-reaction equations which involve not only the quadratic and quartic nonlinearities but also a time-dependent nonlinear convective flux term. In particular, the solitary wave solutions are found. Such equations arise in a variety of contexts in physical and biological problems.     

Numerical simulation of the regularized long wave equation by He's homotopy perturbation method

In this Letter, we present the homotopy perturbation method (shortly HPM) for obtaining the numerical solution of the RLW equation. We obtain the exact and numerical solutions of the Regularized Long Wave (RLW) equation for certain initial condition. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions.     

Exact travelling wave solutions to some fifth order dispersive nonlinear wave equations

Exact travelling wave solutions to some nonlinear equations of fifth order derivatives are derived by using some accurate ansatz methods.     

两个非线性发展方程的双向孤波解与孤子解

采用分步确定拟解的原则, 对齐次平衡法求非线性发展方程孤子解的关键步骤作了进一步改 进. 以广义Boussinesq方程和bidirectional Kaup-Kupershmidt方程为应用实例, 说明使用 该方法可有效避免“中间表达式膨胀”的问题, 除获得标准Hirota形式的孤子解外, 还能获 得其他形式的孤子解. 关键词： 齐次平衡法 孤子解 孤波解 广义Boussinesq方程 bidirectional Kaup-Kupershmi dt方程     

Construction of solitary solutions for nonlinear dispersive equations by variational iteration method

Variational iteration method is implemented to construct solitary solutions for nonlinear dispersive equations. In this scheme the solution takes the form of a convergent series with easily computable components. The chosen initial solution or trial function plays a major role in changing the physical structure of the solution. Many models are approached and the obtained results reveal that the method is very effective and convenient for constructing solitary solutions.     

两类非线性方程的精确解

利用行波约化方法，并借助于一维立方非线性Klein-Gordon方程的精确解，求出了（1+1）维Zakharov方程组、变系数Korteweg-de Vries方程的一些精确解- 关键词： 行波约化方法 一维立方非线性Klein-Gordon方程 （1+1）维Zakharov方程组 变系数Korteweg-de Vries方程     

Exact periodic solution in coupled nonlinear Schodinger equations

The coupled nonlinear Schodinger equations （CNLSEs） of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.     

Traveling wave solutions for two nonlinear evolution equations with nonlinear terms of any order

In this paper, based on the known first integral method and the Riccati sub-ordinary differential equation (ODE) method, we try to seek the exact solutions of the general Gardner equation and the general Benjamin-Bona-Mahoney equation. As a result, some traveling wave solutions for the two nonlinear equations are established successfully. Also we make a comparison between the two methods. It turns out that the Riccati sub-ODE method is more effective than the first integral method in handling the proposed problems, and more general solutions are constructed by the Riccati sub-ODE method.     

Traveling wave solutions for two nonlinear evolution equations with nonlinear terms of any order

In this paper, based on the known first integral method and the Riccati sub-ordinary differential equation (ODE) method, we try to seek the exact solutions of the general Gardner equation and the general Benjamin-Bona-Mahoney equation. As a result, some traveling wave solutions for the two nonlinear equations are established successfully. Also we make a comparison between the two methods. It turns out that the Riccati sub-ODE method is more effective than the first integral method in handling the proposed problems, and more general solutions are constructed by the Riccati sub-ODE method.     

非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了Burgers方程，得到了其扭结形孤立波的近似解析解，该解非常接近于相应的精确解.结果表明，同伦分析法可用来求解非线性演化方程的孤立波解.同时，也对所用方法进行了一定扩展，得到了Kadomtsev-Petviashvili(KP)方程的钟形孤立子解.经过扩展后的方法能够更方便地用于求解更多非线性演化方程的高精度近似解析解. 关键词： Burgers方程 同伦分析法 KP方程 孤立波解     

Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations

In this Letter, He's homotopy perturbation method (HPM) is implemented for finding the solitary-wave solutions of the regularized long-wave (RLW) and generalized modified Boussinesq (GMB) equations. We obtain numerical solutions of these equations for the initial conditions. We will show that the convergence of the HPM is faster than those obtained by the Adomian decomposition method (ADM). The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.