求解非线性薛定谔方程的简便方法

通过引入变换和选准试探函数,把难于求解的非线性偏微分方程化为易于求解的代数方程,然后用待定系数法确定相应的常数,从而简洁地求得了非线性薛定谔方程的解析解.     

非线性传输线方程的几类显式精确解

应用试探函数方法求解非线性传输线方程.通过引入变换和选准试探函数,把难于求解的非线性偏微分方程化为易于求解的代数方程,然后用待定系数法确定相应的常数,从而简洁地求得了方程的精确解.     

对“求一类非线性偏微分方程解析解的一种简洁方法”一文的一点注记

利用文献中所引入的变换，将一个非线性偏微分方程化为一个非线性常微分方程，再直接求解该常微分方程，从而简洁地求得了Burgers方程的几个精确解析解.所得结果与已有结果完全符合. 关键词： 非线性偏微分方程 非线性常微分方程 解析解     

Volterra差分微分方程和KdV差分微分方程新的精确解

辅助方程法和试探函数法为基础，给出函数变换与辅助方程相结合的一种方法，借助符号计算系统Mathematica构造了Volterra差分微分方程和KdV差分微分方程新的精确孤立波解和三角函数解.该方法也适合求解其他非线性差分微分方程的精确解. 关键词： 辅助方程 函数变换 非线性差分微分方程 孤立波解     

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     

非线性波方程尖峰孤子解的一种简便求法及其应用

根据尖峰孤子解的特点，提出了一种待定系数法求非线性波方程尖峰孤子解的思路和方法，并利用该方法求解了5个非线性波方程，即CH（Camassa-Holm)方程、五阶KdV-like 方程、广义Ostrovsky方程、组合KdV-mKdV方程和Klein-Gordon方程，比较简便地得到了这些方程的尖峰孤子解.文献中关于CH方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.简要说明了非线性波方程存在尖峰孤子解所须满足的特定条件.该方法也适用于求其他非线性波方程的尖峰孤子解. 关键词： 非线性波方程 尖峰孤子解 待定系数法     

非线性偏微分方程边值问题的优化算法研究与应用

针对椭圆类非线性偏微分方程边值问题,以差分法和动态设计变量优化算法为基础,以离散网格点未知函数值为设计变量,以离散网格点的差分方程组构建为复杂程式化形式的目标函数.提出一种求解离散网格点处未知函数值的优化算法.编制了求解未知离散点函数值的通用程序.求解了具体算例.通过与解析解对比,表明了本文提出求解算法的有效性和精确性,将为更复杂工程问题分析提供良好的解决方法. 关键词： 非线性偏微分方程 边值问题 动态设计变量优化算法 程序设计     

Ti：LiNbO3光波导导模的场分布

给出了Ti扩散LiNbO_3条波导任意阶导模场分布的试探解.通过变分法分析,不但可以合理地确定其中的待定参数,而且也得到了相应导模传播常数的近似值.与扩展的有效折射率方法比较:这种解不但在函数形式上简单,待定常数确定方便,而且具有精度高的优点.还可以得到等效一维波导折射率分布的解析表达式.     

一类带限定变换的二阶耦合线性微分方程组的解析解

用函数和方程变换将二阶耦合线性微分方程组转化为一阶非线性类椭圆方程，并给出了一次和二次限定变换下方程组的Jacobi椭圆函数解析解，所得结果修正了文献中超导特例的近似解，进一步肯定了超导边界层电场的存在性. 关键词： 微分方程 Jacobi椭圆函数 解析解 超导     

边界耦合的Marangoni对流边界层问题的近似解析解

研究了由温度梯度引起的Marangoni对流边界层问题.由于动量方程和能量方程的边界条件耦合，利用相似变换将偏微分方程组转化为常微分方程非线性边界值问题.通过巧妙引入摄动小参数对速度和温度边界层方程同时渐近展开求解，得到了问题的近似解析解，并对相应的动量、能量传递特性进行了讨论. 关键词： Marangoni对流 近似解析解 渐近展开     

Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.     

非线性偏微分方程的多孤子解

对齐次平衡法的一些关键步骤进行拓宽,获得了一系列非线性方程的多孤子解,使得对非线性方程的多孤子解的求解方法更加直接,且许多步骤可以利用计算机完成. 关键词： 齐次平衡法 非线性方程 多孤子解     

Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations

By means of the modified extended tanh-function (METF) method the multiple traveling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. The solutions for the nonlinear equations such as variants of the RLW and variant of the PHI-four equations are exactly obtained and so the efficiency of the method can be demonstrated.     

Bäcklund transformations as nonlinear Dirac equations

It is pointed out that the Bäcklund transformations for a physically interesting class of nonlinear partial differential equations can be interpreted as generalisations of the Cauchy Riemann equations or as nonlinear Dirac equations. The generalisations are inhomogenisations of the Cauchy Riemann equations (or their hyperbolic analogue), whose condensed form makes the transformations easy to remember, which suggests ways to generalise to more than 2 dimensions, and which suggest that complex analysis techniques may be helpful in understanding the transformations.     

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the (2 1)-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.     

New explicit and exact travelling wave solutions for a system of dispersive long wave equations

A method to construct the new exact solutions of nonlinear partial differential equations (NLPDEs) in a unified way is presented, which is named an improved sine-cosine method. This method is more powerful than the sine-cosine method. Systems of dispersive long wave equations in (1+1) and (2+1) dimensions are chosen to illustrate the method and several types of explicit and exact travelling wave solutions are obtained. These solutions contain Wang's results and other types of solitary wave solutions and new solutions. The method presented here is general and can also be applied to solve more systems of nonlinear partial differential equations, such as the coupled KdV equations.     

The modified extended tanh-function method for solving Burgers-type equations

By means of the modified extended tanh-function (METF) method the multiple travelling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. Solutions for the nonlinear equations such as one-dimensional Burgers, KDV–Burgers, coupled Burgers and two-dimensional Burgers’ equations are obtained precisely and so the efficiency of the method can be demonstrated.     

New analytical solutions for nonlinear physical models of the coupled Higgs equation and the Maccari system via rational exp <Emphasis Type="BoldItalic">(−φ(η))</Emphasis>-expansion method

In this article, a variety of solitary wave solutions are found for some nonlinear equations. In mathematical physics, we studied two complex systems, the Maccari system and the coupled Higgs field equation. We construct sufficient exact solutions for nonlinear evolution equations. To study travelling wave solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into the corresponding partial differential equation and the rational exp (?φ(η))-expansion method is implemented to find exact solutions of nonlinear equation. We find hyperbolic, trigonometric, rational and exponential function solutions using the above equation. The results of various studies show that the suggested method is very effective and can be used as an alternative for finding exact solutions of nonlinear equations in mathematical physics. A comparative study with the other methods gives validity to the technique and shows that the method provides additional solutions. Graphical representations along with the numerical data reinforce the efficacy of the procedure used. The specified idea is very effective, pragmatic for partial differential equations of fractional order and could be protracted to other physical phenomena.     

On spectral methods for solving nonlinear acoustics equations

Two very efficient methods for obtaining approximate solutions to nonlinear acoustics equations are discussed. I proposed these methods earlier, but they are still little known. The first method is based on expanding an unknown function into a Taylor series with respect to the coordinate (evolution variable) and on approximate summation of the terms of this series in all orders up to the infinite order. This series can be summed completely only in particular cases, e.g., for a simple wave. It has been noted that the partial summation technique is implemented more easily if all the terms of the series are represented as corresponding topological diagrams. The second method is based on introducing a “nonlinear” phase delay (proportional to the wave amplitude) for the temporal variable in linear solutions of the problem. The application technique of these methods is illustrated by obtaining approximate solutions of the Burgers equation.