一类非线性非局部扰动LGH方程的孤子行波解

利用经过改进的泛函分析变分迭代方法讨论了一类非线性非局部Landau-Ginzburg-Higgs(LGH)微分方程.首先,做行波变换,引入泛函,并求出其变分,令其为0,得到了Lagrange(拉格朗日)算子应满足的条件,并求出它.然后,引入一个经过改进的变分迭代式,选取初始迭代函数为对应的无扰动LGH方程的孤子解.最后,利用迭代式依次得到非线性非局部LGH扰动方程求出各次孤子行波的渐近解和LGH扰动方程的精确解.通过一个例子说明了用经过改进的泛函分析变分迭代方法得到求解是有效的方法.     

非线性扰动广义NNV系统的孤立子渐近行波解

采用了一个简单而有效的技巧,研究一类非线性扰动广义NNV(Nizhnik-Novikov-Veselov)系统.首先用待定系数法得到一个相应典型系统的孤立子解.其次构造一个广义泛函式,并对它进行变分计算,利用变分原理求出对应的Lagrange乘子,并由此构造一个特殊的变分迭代关系式.然后依次求出原非线性扰动广义NNV系统的孤立子渐近行波解.最后通过举例,说明了使用该方法得到的近似解具有简单而有效的优点.     

非线性扰动广义NNV微分系统的孤子研究

采用了一个简单而有效的技巧,研究了一类非线性扰动广义NNV微分系统.首先引入一个行波变换,将NNV系统转化为一组非线性常微分方程系统.其次用双曲函数待定系数法得到一个相应的典型系统的孤子解·然后构造一个广义泛函迭代同伦映射,由此构造一个特殊的渐近解的迭代关系式.并依次地求出原非线性扰动广义NNV微分系统的孤子渐近行波解.最后通过举例,说明了使用本方法得到的近似解简单而有效.     

一个组合方程的单孤子解和周期尖波解

构造一个组合方程的单孤子解和周期尖波解.应用格林函数的性质,以及求一个非线性偏微分方程(简称PDE)弱解的方法.求出了这个组合方程的单孤子解和周期尖波解,推广了前人的研究成果.     

变系数浅水波方程的精确解

均衡作用法给出了一种求非线性发展方程孤波解的有效方法.利用该方法,运用计算机符号计算,求出了变系数的一般浅水波方程的孤子解.     

非线性强迫扰动Klein-Gordon方程的孤波渐近解法

研究了一类非线性强迫扰动Klein-Gordon方程.首先利用双曲正切待定系数法求得了典型的方程孤波解.然后利用泛函变分迭代原理得到了强迫扰动Klein-Gordon方程的一个近似解,并论述了解的一致有效性.所得到的近似解是一个解析式,它还可对近似解进行解析运算,而使用简单的模拟方法所得到的近似解是达不到这种效果的.     

厄尔尼诺/拉尼娜-南方海涛模型的变分迭代解法

研究了-类厄尔尼诺-南方海涛（ENSO）振子的模型，利用变分迭代方法求出了ENSO模型的近似解。     

一类非线性双曲型发展方程的孤子解

研究了一类非线性发展方程.首先在无扰动情形下,利用待定函数和泛函同伦映射方法得到了非扰动发展方程的孤子精确解和扰动方程的任意次近似行波孤子解.接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出非线性双曲型发展扰动方程孤子解的各次近似解析解.再利用摄动理论举例说明了用该方法得到的近似解析解的有效性和各次近似解的近似度.最后,简述了用同伦映射方法得到的近似解的意义,指出了用上述方法得到的各次近似解具有便于求解、精度高等优点.     

非线性Duffing扰动振子共振机制的研究

研究了一类广义Duffing扰动共振机制.利用泛函分析同伦映射方法,构造了求得问题渐近解的迭代关系式.首先求出了Duffing模型的初始近似函数;其次利用迭代关系依次求出了模型的各次渐近解;然后通过举例,说明了用泛函同伦映射方法得到的广义Duffing扰动振子随机共振机制的近似解简单而有效.讨论了得到的渐近解的意义.     

用Hirota双线性方法构造一种(3+1)维高维孤子方程的多孤子解

通过两种方法构造了一种(3+1)维高维孤子方程的孤子解.第一种方法是利用对数函数变换,将其化成双线性形式的方程,在用级数扰动法求解双线性方程的单孤子解、双孤子解和N-孤子解.第二种方法是用广义有理多项式与试探法相结合,构造了(3+1)维高维孤子方程的怪波解.     

A Method for Constructing Traveling Wave Solutions to Nonlinear Evolution Equations

In this paper, an analytical method is proposed to construct explicitly exact and approximate solutions for nonlinear evolution equations. By using this method, some new traveling wave solutions of the Kuramoto-Sivashinsky equation and the Benny equation are obtained explicitly. These solutions include solitary wave solutions, singular traveling wave solutions and periodical wave solutions. These results indicate that in some cases our analytical approach is an effective method to obtain traveling solitary wave solutions of various nonlinear evolution equations. It can also be applied to some related nonlinear dynamical systems.     

Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

Three new iteration methods, namely the squared-operator method, the modified squared-operator method, and the power-conserving squared-operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast-converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared-operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher-gap vortex solitons in the two-dimensional nonlinear Schrödinger equations with periodic potentials, and isolated solitons in Ginzburg–Landau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared-operator method delivers the best performance among the methods proposed in this article.     

非线性波方程准确孤立波解的符号计算

该文将机械化数学方法应用于偏微分方程领域，建立了构造一类非线性发展方程孤立波解的一种统一算法，并在计算机数学系统上加以实现，推导出了一批非线性发展方程的精确孤立波解．算法的基本原理是利用非线性发展方程孤立波解的局部性特点，将孤立波表示为双曲正切函数的多项式．从而将非线性发展方程（组）的求解问题转化为非线性代数方程组的求解问题．利用吴文俊消元法在计算机代数系统上求解非线性代数方程组，最终获得非线性发展方程（组）的准确孤立波解.     

Numerical simulation of the modified regularized long wave equation by He's variational iteration method

The modified regularized long wave (MRLW) equation, with some initial conditions, is solved numerically by variational iteration method. This method is useful for obtaining numerical solutions with high degree of accuracy. The variational iteration solution for the MRLW equation converges to its exact solution. Moreover, the conservation laws properties of the MRLW equation are also studied. Finally, interaction of two and three solitary waves is shown. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

关于《保辛水波动力学》的一个注记

研究完全非线性水波的数值模拟方法，将《保辛水波动力学》中提出的保辛摄动方法，扩展到非线性水波压强的分析.数值算例表明，该文方法可用于水波的非线性演化分析，也可用于模拟孤立波、尖锐波峰的涌波等非线性水波，并给出水波的压强分布.     

Construction of exact periodic wave and solitary wave solutions for the long–short wave resonance equations by VIM

In this paper, He’s variational iteration method is employed to construct periodic wave and solitary wave solutions for the long–short wave resonance equations. The chosen initial solution can be in soliton form with some unknown parameters, which can be determined in the solution procedure. Some examples are given. The results reveal that the method is very effective and convenient.     

A numeric–analytic method for time-fractional Swift–Hohenberg (S–H) equation with modified Riemann–Liouville derivative

In this paper, the fractional variational iteration method (FVIM) was applied to obtain the approximate solutions of time-fractional Swift–Hohenberg (S–H) equation with modified Riemann–Liouville derivative. A new application of fractional variational iteration method (FVIM) was extended to derive analytical solutions in the form of a series for these equations. Numerical results showed the FVIM is powerful, reliable and effective method when applied strongly nonlinear equations with modified Riemann–Liouville derivative.     

Harmonic wave generation in non linear thermoelasticity by variational iteration method and Adomian's method

This paper applies the variational iteration method and Adomian's decomposition method to solve numerically the harmonic wave generation in a nonlinear, one-dimensional elastic half-space model subjected initially to a prescribed harmonic displacement. The results show that the variational iteration method is much easier, more convenient, and more stable and efficient than Adomian decomposition method.