非线性发展方程的Jacobi椭圆函数解

借助齐次平衡原则,提出了一种新的构造非线性发展方程的Jacobi椭圆函数精确解的方法. 并利用之得到了KdV方程,Boussinesq方程,KGS方程组的新形式 Jacobi椭圆函数解.     

应用F展开法求KdV方程的周期波解

提出了求非线性数学物理演化方程周期波解的F展开法,该方法可看作最近提出的扩展的Jacobi椭圆函数展开方法的浓缩.直接利用F展开法而不计算Jacobi椭圆函数,我们可同时得到著名的KdV方程的多个用Jacobi椭圆函数表示的周期波解.当模数m→1 时,可得到双曲函数解(包括孤立波解).     

关于Jacobi椭圆函数展开法

主要利用Jacobi椭圆函数所满足的方程并用其解代替Jacobi椭圆函数以求非线性偏微分方程的周期解,并举例说明该方法的应用.     

五阶变系数模型方程新的Jacobi椭圆函数解

我们给出了一种统一的Jacobi椭圆函数方法来构造非线性偏微分方程精确行波解的新方法．借助于Mathematica，我们获得了五阶变系数模型方程的24种Jacobi椭圆函数解．     

一类具波动算子非线性Schr?dinger方程的精确解

研究一类具波动算子非线性Schr?dinger方程的精确解问题.引入Jacobi椭圆函数组合及双曲函数组合方法,将其应用于求解具有波动算子的非线性Schr?dinger方程中.通过简单代数运算,可以得到具有波动算子非线性Schr?dinger方程的许多新解,并在极限情况下,给出了该方程对应的双曲函数解.同时得出了双曲函数组合解是Jacobi椭圆函数组合解情况下的极限解的结论.该方法可以推广到更多非线性偏微分方程精确解求解问题.     

一类非线性演化方程的周期解

通过直接积分和映射方法,得到了一类非线性演化方程的一系列Jacobi椭圆函数周期解.在极限情况下,得到三角函数解.     

Davey StewartsonⅠ的周期波解

利用新近提出的F展开法，导出了Davey StewartsonⅠ方程的由Jacobi椭圆函数表示的周期波解；并且在极限的情况下，得到了Davey StewartsonⅠ方程的孤波解以及其他形式解.     

广义变形的DP-CH方程的显式精确解

通过构造一个辅助方程,利用动力系统方法,Jacobi椭圆函数,来讨论广义变形的DP-CH方程,得到了广义变形的DP-CH方程更丰富的行波解的显示精确表示.     

广义随机KP方程的椭圆周期解

利用Hermite变换和Jacobi椭圆函数展开法研究(2+1)-维广义随机Kadomtsev-Petviashvili方程,并给出了它的随机椭圆周期解及随机孤立波解.     

(3+1)维Klein-Gordon方程的新求解方法

给出第一种椭圆方程与函数变换相结合的方法,通过几个步骤,构造了(3+1)维Klein-Gordon方程的多种新解.步骤一、根据Jacobi椭圆函数的性质,获得了第一种椭圆方程的几种新解.步骤二、用第一种椭圆方程与函数变换相结合的方法,将(3+1)维Klein-Gordon方程的求解问题转化为非线性代数方程的求解问题.步骤三、借助符号计算系统Mathematica求出该方程组的解,并构造了由Riemannθ函数、Jacobi椭圆函数、双曲函数和三角函数两两组合的双周期解和双孤子解等多种复合型新解.     

NEW EXACT TRAVELLING WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS USING THE DYNAMICAL SYSTEM METHOD

By the method of dynamical system,we construct the exact travelling wave solutions of a new Hamiltonian amplitude equation and the Ostrovsky equation.Based on this method,the new exact travelling wave solutions of the new Hamiltonian amplitude equation and the Ostrovsky equation,such as solitary wave solutions,kink and anti-kink wave solutions and periodic travelling wave solutions,are obtained,respectively.     

2+1维Ginzburg-Landau方程的精确波解

分析研究了一个具有三次增益效应和五次耗散项的2+1维Ginzburg-Landau方程.利用同解变型法并结合一个高阶辅助方程的解,成功地取得了该方程的一些新的精确行波解.     

The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation

We obtain new exact solutions to generalized Sawada-Kotera equation. Using the variational iteration method combined with the improved generalized tanh-coth method, we construct new traveling wave solutions for the standard Sawada-Kotera equation and, by means of scaling, we obtain new solutions to general Sawada-Kotera equation. Periodic and soliton solutions are formally derived for both models.     

一类Burgers-KPP方程新的显式精确行波解

In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which include new singular solitary wave solutions and periodic solutions. Particular important cases of the equation, such as the generalized Burgers-Fisher equation, Burgers-Chaffee infante equation and KPP equation, the corresponding solutions can be obtained also. The method can also solve other nonlinear partial differential equations.     

A new lattice Boltzmann model for the laplace equation

In this paper, a new lattice Boltzmann equation which is independent of time is proposed. Based on the new lattice Boltzmann equation, some steady problems can be modeled by the lattice Boltzmann method. In the further study, the Laplace equation is investigated with the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different space scales. The numerical results show that the new method is effective.     

求解热传导反问题的一种正则化Newton型迭代法

讨论热传导方程求解系数的一个反问题．把问题归结为一个非线性不适定的算子方程后,考虑该方程的Newton型迭代方法．对线性化后的Newton方程用隐式迭代法求解,关键的一步是引入了一种新的更合理的确定(内)迭代步数的后验准则．对新方法及对照的Tikhonov方法和Bakushiskii方法进行了数值实验,结果显示了新方法具有明显的优越性．     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

Interval expansion method for nonlinear equation in several variables

An interval expansion method is presented to find the solution of nonlinear equation in several variables with guaranteed accuracy. In this method, the existence theorem for the solution of nonlinear equation is obtained by using two newly defined operators. A new algorithm for solving nonlinear equation in several variables is given. Furthermore, we prove that the new method is global convergent under mild conditions.     

Note on exact solutions for a class of nonlinear diffusion equations

An effective characterization is given for a class of generalized nonlinear diffusion equations with power law dependent terms. Further, a new auxiliary equation ansatz is derived. Consequently, new exact traveling wave trigonometric function, solitary-like and Weierstrass elliptic solutions to a subclass are obtained by means of an auxiliary equation method and a generalized Riccati equation expansion method.     

求解2—D波动方程系数反问题的迭代方法及收敛性

本文针对地震勘探中提出一类重要的２－Ｄ波动方程反演问题，通过定义一个新的非线性算子将２－Ｄ波动方程的反演问题归结为一个新的非线性算子方程，详细讨论了非线性算子的性质，给出了求解反问题的迭代方法，并证明了这种迭代方法的收敛性。