广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方法,即改进的广义射影Ricccati方程方法, 求解(2+1)维色散长波方程, 得到该方程的新的更一般形式的行波解, 包括扭状孤波解, 钟状解,孤子解和周期解. 并对部分新形式孤波解画图示意.     

一类高阶非线性波方程的子方程与精确行波解

结合子方程和动力系统分析的方法研究了一类五阶非线性波方程的精确行波解.得到了这类方程所蕴含的子方程, 并利用子方程在不同参数条件下的精确解, 给出了研究这类高阶非线性波方程行波解的方法, 并以Sawada Kotera方程为例, 给出了该方程的两组精确谷状孤波解和两组光滑周期波解.该研究方法适用于形如对应行波系统可以约化为只含有偶数阶导数、一阶导数平方和未知函数的多项式形式的高阶非线性波方程行波解的研究.     

非线性热传导方程的几类精确解

通过引入恰当的试探函数,将非线性热传导方程化为易于求解的常微分方程组并对其求解,进而得到非线性热传导方程的孤波解、奇异行波解、三角函数周期波解等一些不同形式的行波解.     

一类非线性发展方程的精确孤波解

本文首先求出了非线性常微分方程ｕ″（ξ）＋ｍｕ２（ξ）＋ｎｕ３（ξ）＋ｐｕ（ξ）＝ｃ（Ⅰ）和ｕ″（ξ）＋ｒｕ′（ξ）＋ｍｕ２（ξ）＋ｎｕ３（ξ）＋ｐｕ（ξ）＝ｃ（Ⅱ）的显式精确解．进而求出了组合ＢＢＭ方程、Ｂｕｒｇｅｒｓ方程与组合ＢＢＭ方程混合型的钟状孤波解和扭状孤波解，同时还求出了广义Ｂｏｕｓｓｉｎｅｓｑ方程和广义ＫＰ方程的钟状和扭状孤波解．文中指出了其行波解可化为（Ⅰ）的发展方程既有钟状又有扭状孤波解，而其行波解可化为（Ⅱ）的发展方程没有钟状孤波解．     

非线性波方程新的显式精确解

利用Darboux和一个可化为标准Bernoulli方程的4阶常微分方程,统一地处理了三个著名方程KdV方程,Kadomtsev-Petviashvili(KP)方程和Hirota-Satsuma(HS)方程的求解问题.给出了这些方程一批新的具有更为丰富形式的精确解,其中包括孤波解和行波解.     

Kadomtsev-Petviashvili方程的精确解

齐次平衡法是把非线性偏微分方程转换成带约束条件的线性偏微分方程的一种很好的方法 .本文在齐次平衡法的基础上具体讨论了KP方程的精确解 ,包括孤波解 ,一般的行波解 ,有理函数解和一种新类型的解 .     

Gardner-Kadomtsev-Petviashvili方程的精确行波解与分支

本文讨论Gardner-Kadomtsev-Petviashvili方程的行波解,该方程在物理中有广泛应用.我们运用动力系统分支理论,首先得到了方程的分支和相图,然后通过讨论参数的范围得到了精确行波解的所有形式,其中包括孤波解,周期波解,扭波解和爆破解     

利用有理形式的指数函数法解决非线性Jaulent-Miodek方程的一系列复孤波解(英文)

本文研究了非线性Jaulent-Miodek方程的行波解.利用有理形式的指数函数法,得到了一系列包括由三角周期函数和有理函数组合而成的新复孤波解。     

Dodd-Bullough-Mikhailov方程的精确解

利用(G'/G)法求解了Dodd-Bullough-Mikhailov的精确解,得到了Dodd-Bullough-Mikhailov方程的用双曲函数,三角函数和有理函数表示的三类精确行波解.由于方法中的G为某个二阶常系数线性ODE的通解,故方法具有直接、简洁的优点;更重要的是,方法可用于求得其它许多非线性演化方程的行波解.如果对其中双曲函数表示的行波解中的参数取特殊值,那么可得已有的孤波解.     

强非线性广义 Boussinesq 方程孤波解的波形分析及求解

上海理工大学理学院\quad 上海 200093该文建立了强非线性广义 Boussinesq 方程的耗散项、波速、渐进值与波形函数的导数之间的关系.利用适当变换和待定假设方法,作者求出了上述广义 Boussinesq 方程的扭状或钟状孤波解,还求出了以前文献中未曾提到过的余弦函数的周期波解.进一步给出了波速对波形影响的结论,即:``好'广义 Boussinesq 方程的行波当波速由小变大时,波形由钟状孤波变成余弦函数周期波解;``坏'广义 Boussinesq 方程的行波当波速由小变大时,波形由余弦函数周期波解变成钟状孤波.     

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.     

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.     

Local Discontinuous Galerkin Methods for Three Classes of Nonlinear Wave Equations

In this paper, we further develop the local discontinuous Galerkin method to solve three classes of nonlinear wave equations formulated by the general KdV-Burgers type equations, the general fifth-order KdV type equations and the fully nonlinear K(n, n, n) equations, and prove their stability for these general classes of nonlinear equations. The schemes we present extend the previous work of Yan and Shu [30, 31] and of Levy, Shu and Yan [24] on local discontinuous Galerkin method solving partial differential equations with higher spatial derivatives. Numerical examples for nonlinear problems are shown to illustrate the accuracy and capability of the methods. The numerical experiments include stationary solitons, soliton interactions and oscillatory solitary wave solutions.The numerical experiments also include the compacton solutions of a generalized fifthorder KdV equation in which the highest order derivative term is nonlinear and the fully nonlinear K (n, n, n) equations.     

Propagation of traveling wave solutions for nonlinear evolution equation through the implementation of the extended modified direct algebraic method

In this work, different kinds of traveling wave solutions and uncategorized soliton wave solutions are obtained in a three dimensional(3-D) nonlinear evolution equations(NEEs)through the implementation of the modified extended direct algebraic method. Bright-singular and dark-singular combo solitons, Jacobi's elliptic functions, Weierstrass elliptic functions, constant wave solutions and so on are attained beside their existing conditions. Physical interpretation of the solutions to the 3-D modified KdV-Zakharov-Kuznetsov equation are also given.     

Asymptotic Stability of Traveling Wave Solutions for Perturbations with Algebraic Decay

For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation of the traveling wave profile decays at an algebraic rate, then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate, and optimal intermediate results that combine temporal and spatial decay are obtained. The proofs are based on a general interpolation principle which says that algebraic decay results of this form always follow if exponential temporal decay holds for perturbation with exponential spatial decay and the wave profile is stable for general perturbations.     

基于Riccati-Bernoulli辅助常微分方程的Davey-Stewartson方程的行波解

Riccati-Bernoulli辅助常微分方程方法可以用来构造非线性偏微分方程的行波解.利用行波变换,将非线性偏微分方程化为非线性常微分方程, 再利用Riccati-Bernoulli方程将非线性常微分方程化为非线性代数方程组, 求解非线性代数方程组就能直接得到非线性偏微分方程的行波解.对Davey-Stewartson方程应用这种方法, 得到了该方程的精确行波解.同时也得到了该方程的一个Backlund变换.所得结果与首次积分法的结果作了比较.Riccati-Bernoulli辅助常微分方程方法是一种简单、有效地求解非线性偏微分方程精确解的方法.     

Different physical structures of solutions for a generalized Boussinesq wave equation

A technique based on the reduction of order for solving differential equations is employed to investigate a generalized nonlinear Boussinesq wave equation. The compacton solutions, solitons, solitary pattern solutions, periodic solutions and algebraic travelling wave solutions for the equation are expressed analytically under several circumstances. The qualitative change in the physical structures of the solutions is highlighted.     

Binary Darboux transformation and soliton solutions for the coupled complex modified Korteweg-de Vries equations

This paper considers the coupled complex modified Korteweg-de Vries (mKdV) equations and presents a binary Darboux transformation for the equations. As a direct application, we give a classification of general soliton solutions derived from vanishing and non-vanishing backgrounds, on the basis of the dynamical behavior of the solutions. Special types of solutions in the presented solutions include breathers, bright-bright solitons, bright-dark solitons, bright-W-shaped solitons, and rogue wave solutions. Furthermore, dynamics and interactions of vector bright solitons are exhibited.     

Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics

In this paper, we devise a new unified algebraic method to construct a series of explicit exact solutions for general nonlinear equations. Compared with most existing methods such as tanh method, Jacobi elliptic function method and homogeneous balance method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solutions according to the values of some parameters. The solutions obtained in this paper include (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic, and soliton solutions, (f) Jacobi, and Weierstrass doubly periodic wave solutions. The efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper, combined KdV–MKdV, Camassa–Holm, Kaup–Kupershmidt, Jaulent–Miodek, (2+1)-dimensional dispersive long wave, new (2+1)-dimensional generalized Hirota, (2+1)-dimensional breaking soliton and double sine-Gordon equations. In addition, the links among our proposed method, the tanh method, the extended method and the Jacobi function expansion method are also clarified generally.