非线性Klein-Gordon方程新的精确解

将行波变换替换为更一般的函数变换,推广了修正的Jacobi椭圆函数展开方法.给出了非线性 Klein-Gordon方程新的周期解.当模m→1或m→0时,这些解退化成相应的孤立波解、三 角函数解和奇异的行波解.对于某些非线性方程,在一定条件下一般变换退化为行波约化. 关键词： Jacobi椭圆函数 非线性发展方程 精确解     

New exact solutions for two nonlinear equations

In this Letter, we investigate two nonlinear equations given by ut−uxxt+3u²ux=2uxuxx+uuxxx and ut−uxxt+4u²ux=3uxuxx+uuxxx. Through some special phase orbits we obtain four new exact solutions for each equation above. Some previous results are extended.     

New exact solutions of some nonlinear evolution equations of pseudoparabolic type

This paper aims to conduct an analytical study into some nonlinear models of pseudoparabolic type, including the Oskolkov, Oskolkov–Benjamin–Bona–Mahony–Burgers, and Benjamin–Bona–Mahony–Peregrine–Burgers equations. A number of new exact solutions for these pseudoparabolic type equations have been derived based on the modified Kudryashov method that its calculations are performed in a symbolic computation system known as Maple.     

一类非线性演化方程的新多级准确解

在Lamé方程和新的Lamé函数的基础上，应用小扰动方法和Jacobi椭圆函数展开法求解一类非线性演化方程（如mKdV方程，非线性Klein-Gordon方程Ⅱ等），获得多种新的多级准确解 .这些多级准确解对应着不同形式的周期波解.这些解在极限条件下可以退化为多种形式的孤 立波解，如带状孤立子、钟形孤立子等. 关键词： Jacobi椭圆函数 Lam函数 多级准确解 非线性演化方程 扰动方法     

Explicit exact solitary wave solutions for generalized symmetric regualrized long—wave equations with high—order nonlinear terms

In this paper,we have obtained the bell-type and kink-type solitary wave solutions of the generalized symmetric regularized long-wave equations with high-order nonlinear terms by meas of proper transformation and undeterined assumption method.     

New variable separation solutions for the generalized nonlinear diffusion equations

The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u, ux)uxx +B(u, ux) is studied by using the conditional Lie–Ba¨cklund symmetry method. The variant forms of the considered equations,which admit the corresponding conditional Lie–Ba¨cklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.     

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding Bäcklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.     

New exact solutions of nonlinear Klein--Gordon equation

New exact solutions, expressed in terms of the Jacobi elliptic functions, to the nonlinear Klein--Gordon equation are obtained by using a modified mapping method. The solutions include the conditions for equation's parameters and travelling wave transformation parameters. Some figures for a specific kind of solution are also presented.     

几种辅助方程与非线性发展方程的无穷序列精确解

本文为了获得非线性发展方程新的无穷序列精确解,给出了几种辅助方程的Böcklund变换和解的非线性叠加公式,并构造了一些非线性发展方程新的无穷序列精确解,其中包括无穷序列Jacobi椭圆函数解、无穷序列双曲函数解和无穷序列三角函数解.该方法在构造非线性发展方程无穷序列精确解方面具有普遍意义. 关键词： 辅助方程法 解的非线性叠加公式 无穷序列解 非线性发展方程     

New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method

In this paper, the nonlinear Boussinesq equations with the conformable time-fractional derivative are solved analytically using the well-established modified Kudryashov method. As a consequence, a number of new exact solutions for this type of equations are formally derived. It is believed that the method is one of the most effective techniques for extracting new exact solutions of nonlinear fractional differential equations.     

Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W(±∞,t)=0 and the sharp initial condition W(x,0)=δ(x).     

A procedure to construct exact solutions of nonlinear evolution equations

In this paper, we implemented the functional variable method for the exact solutions of the Zakharov?CKuznetsov-modified equal-width (ZK-MEW), the modified Benjamin?CBona?CMahony (mBBM) and the modified KdV?CKadomtsev?CPetviashvili (KdV?CKP) equations. By using this scheme, we found some exact solutions of the above-mentioned equations. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. The functional variable method presents a wider applicability for handling nonlinear wave equations.     

New exact solutions to some difference differential equations

In this paper, we use our method to solve the extended Lotka--Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of \sinh and \cosh, periodic solutions presented by trigonometric functions of \sin and \cos, and rational solutions. This method can be used to solve some other nonlinear difference--differential equations.     

New exact static solutions of einsteins field equations

We describe a method for deriving new solutions for an ideal fluid from old and give some new solutions which may be of interest in astrophysics.     

sine-Gordon型方程的无穷序列新精确解

为了获得sine-Gordon型方程的无穷序列精确解,给出三角函数型辅助方程和双曲函数型辅助方程及其Bäcklund变换和解的非线性叠加公式,借助符号计算系统Mathematica,构造了sine-Gordon方程、mKdV-sine-Gordon方程、(n+1)维双sine-Gordon方程和sinh-Gordon方程的无穷序列新精确解.其中包括无穷序列三角函数解、无穷序列双曲函数解、无穷序列Jacobi椭圆函数解和无穷序列复合型解. 关键词： sine-Gordon型方程 解的非线性叠加公式 辅助方程 无穷序列精确解     

New exact solutions of the Yang-Mills field equations

In this paper new exact solutions of the Yang-Mills SU(2) gauge field equations are obtained using the Carmeli-Charach-Kaye null-tetrad formalism. The solutions are classified and briefly discussed.     

New axially symmetric solutions of the Einstein-Maxwell equations

New exact solutions of the algebraic form for the static Einstein-Maxwell equations representing the exterior gravitational field of a massive magnetic dipole are derived. They are then used for construction of the stationary electrovacuum solutions reducing to the Schwarzschild metric in a pure vacuum limit.     

用Riccati方程构造非线性差分微分方程新的精确解

把Riccati方程应用到非线性差分微分方程求解领域,并相结合与一种函数变换,借助符号计算系统Mathematica构造了修正的Volterra方程和一般格子方程新的精确孤立波解和三角函数解. 关键词： Riccati方程 函数变换 非线性差分微分方程 孤立波解     

The relation among the hyperbolic-function-type exact solutions of nonlinear evolution equations

First, we investigate the solitary wave solutions of the Burgers equation and the KdV equation, which are obtained by using the hyperbolic function method. Then we present a theorem which will not only give us a clear relation among the hyperbolic-function-type exact solutions of nonlinear evolution equations, but also provide us an approach to construct new exact solutions in complex scalar field. Finally, we apply the theorem to the KdV–Burgers equation and obtain its new exact solutions.