试探方程法及其在非线性发展方程中的应用

提出了一种比较系统的求解非线性发展方程精确解的新方法, 即试探方程法. 以一个带5阶 导数项的非线性发展方程为例, 利用试探方程法化成初等积分形式，再利用三阶多项式的完 全判别系统求解，由此求得的精确解包括有理函数型解, 孤波解, 三角函数型周期解, 多项 式型Jacobi椭圆函数周期解和分式型Jacobi椭圆函数周期解 关键词： 试探方程法 非线性发展方程 孤波解 Jacobi椭圆函数 周期解     

三类非线性演化方程新的Jacobi椭圆函数精确解

应用修正影射法分别求解三类非线性演化方程,即非线性Klein-Gordon方程,mKdV方程和广义Boussinesq方程,得到了一些新的Jacobi椭圆函数展开解,包括Jacobi椭圆函数解、混合Jacobi椭圆函数解、孤子解和三角函数解. 关键词： Klein-Gordon方程 mKdV方程 广义Boussinesq方程 Jacobi椭圆函数     

组合KdV方程的显式精确解

借助计算机代数系统Mathematica，利用双曲函数法找到了组合KdV方程(Combined KdV Equation)的精确孤立波解，包括钟型孤立波解和扭结型孤立波解.在此基础上又对双曲函数法的思想进行了推广，从而获得了其更多的显式精确解，包括间断型激波解和指数函数型解.这种方法也适用于求解其他非线性发展方程(组). 关键词： 组合KdV方程 双曲函数法 孤立波解 精确解     

(2+1) 维Broer-Kau-Kupershmidt方程一系列新的精确解

借助于符号计算软件Maple，通过一种构造非线性偏微分方程(组)更一般形式精确解的直接方法即改进的代数方法，求解(2+1) 维 Broer-Kau-Kupershmidt方程，得到该方程的一系列新的精确解，包括多项式解、指数解、有理解、三角函数解、双曲函数解、Jacobi 和 Weierstrass 椭圆函数双周期解. 关键词： 代数方法 (2+1) 维 Broer-Kau-Kupershmidt 方程 精确解 行波解     

两类2+1维非线性波动方程的线性叠加解

借助于计算机软件Maple将线性叠加方法应用于2+1维广义非线性Schr?dinger和Boussinesq 方程，给出此两类方程不同周期的线性叠加解以及与这些周期解相应的速度值，求解过程严 格基于一类新近发现的Jacobi椭圆函数周期循环特性及其推论. 关键词： Jacobi椭圆函数 线性叠加方法 周期循环特性 非线性Schrdinger方程 Boussine sq方程     

非线性LC电路方程的显式精确行波解

理论上考察了具有耗散的非线性LC电路中的行波. 借助于作者最近发展的精确求解非线性偏微分方程的扩展的双曲函数方法解析地研究了模拟非线性电路中冲击波的四阶耗散非线性波动方程. 一致地获得了丰富的显式精确解析行波解, 包括精确冲击波解和奇异的行波解, 和三角函数有理形式的周期波解. 关键词： LC电路')" href="#">非线性LC电路 非线性耗散波动方程 冲击波 周期波     

推广的B-BBM方程和B-BBM方程的显式精确解

研究了BBM方程和推广的B-BBM方程解之间的形变理论,通过给出的一类新的形变关系可从BBM方程的已知解形变出推广的B-BBM方程的显式精确解,并附有这种方法的应用举例.还对B-BBM方程运用齐次平衡法获得其解的显式表示 关键词： 形变理论 齐次平衡法 显式精确解 推广的B-BBM方程 B-BBM方程     

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     

(2+1)维色散长波方程的新的类孤子解

应用一种新的修改的代数方法去求解(2+1)维色散长波方程,获得方程的大量新的精确解.这些解包括类孤子解、类周期解、类有理解、类双曲函数解、类Jacobi椭圆函数解等等. 关键词： (2+1)维色散长波方程 类孤子解 类有理解 类双曲函数解 类Jacobi椭圆 函数解     

一般变换下的Jacobi椭圆函数展开法及应用

将在行波变换下的Jacobi椭圆函数展开法推广到范围非常广泛的一般函数变换下进行，利用这一方法求得了一些非线性发展方程的精确周期解，这些解包括了在行波变换下所求得的周期解. 证明了一些非线性发展方程的周期解一定是行波解. 关键词： 非线性发展方程 周期解 行波解 Jacobi椭圆函数     

Exact solutions to the generalized lienard equation and its applications

Some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to solve nonlinear wave equations with nonlinear terms of any order directly. The generalized one-dimensional Klein-Gordon equation, the generalized Ablowitz (A) equation and the generalized Gerdjikov-Ivanov (GI) equation are investigated and abundant new exact travelling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions.      

New Multiple Soliton-like and Periodic Solutions for (2+1)-Dimensional Canonical Generalized KP Equation with Variable Coefficients

In this paper, the generalized tanh function method is extended to (2 1)-dimensional canonical generalized KP (CGKP) equation with variable coefficients. Taking advantage of the Riccati equation, many explicit exact solutions,which contain multiple soliton-like and periodic solutions, are obtained for the (2 1)-dimensional CGKP equation with variable coefficients.     

A General Mapping Approach and New Travelling Wave Solutions to (2+1)-Dimensional Boussinesq Equation

A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.     

New Multiple Soliton-like and Periodic Solutions for （2＋l）-Dimensional Canonical Generalized KP Equation with Variable Coefficients

In this paper, the generalized ranch function method is extended to （2＋1）-dimensianal canonical generalized KP （CGKP） equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit exact solutions, which contain multiple soliton-like and periodic solutions, are obtained for the （2＋1）-dimensional OGKP equation with variable coetffcients.     

Comparison between the generalized tanh–coth and the (<Emphasis Type="Italic">G</Emphasis>′/<Emphasis Type="Italic">G</Emphasis>)-expansion methods for solving NPDEs and NODEs

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G ^′/G )-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.     

Dielectric constant of graphene-on-polarized substrate: A tight-binding model study

By using the bifurcation theory of planar dynamical systems and the qualitative theory of differential equations, we studied the dynamical behaviours and exact travelling wave solutions of the modified generalized Vakhnenko equation (mGVE). As a result, we obtained all possible bifurcation parametric sets and many explicit formulas of smooth and non-smooth travelling waves such as cusped solitons, loop solitons, periodic cusp waves, pseudopeakon solitons, smooth periodic waves and smooth solitons. Moreover, we provided some numerical simulations of these solutions.     

New explicit exact solutions for a generalized Hirota—Satsuma coupled KdV system and a coupled MKdV equation

In this paper, we make use of a new generalized ansatz in the homogeneous balance method, the well-known Riccati equation and the symbolic computation to study a generalized Hirota--Satsuma coupled KdV system and a coupled MKdV equation, respectively. As a result, numerous explicit exact solutions, comprising new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.     

Explicit Solutions for Generalized (2+1)-Dimensional NonlinearZakharov-Kuznetsov Equation

The exact solutions of the generalized (2+1)-dimensional nonlinearZakharov-Kuznetsov (Z-K) equation are explored by the method of the improved generalized auxiliary differential equation. Many explicit analytic solutions of the Z-K equation are obtained. The methods used to solve the Z-K equation can be employed in further work to establish new solutions for other nonlinear partial differential equations.     

2N+1阶KdV型方程的孤波解

获得了２Ｎ＋１阶ＫｄＶ型方程的显式精确孤波解．作为特例，讨论了高阶广义ＫｄＶ型方程、高阶广义ＭＫｄＶ型方程和高阶广义Ｓｃｈａｍｅｌ的ＭＫｄＶ型方程．还研究了２Ｎ＋１阶ＫＰ型方程 关键词：     

Lienard Equation and Exact Solutions for Some Soliton-Producing Nonlinear Equations

In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then, explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq, generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.