M.Wadati可积非线性发展方程的精确行波解

用平面动力系统方法研究由M.Wadati提出的一类可积非线性发展方程的精确行波解,获得了该方程的扭波、反扭波解,周期波解和不可数无穷多光滑孤立波解的精确的参数表达式,以及上述解存在的参数条件.     

(2+1)-维广义Benney-Luke方程的精确行波解

用平面动力系统方法研究（2＋1）＋维，“义Benney-Luke方程的精确行波解，获得了该方程的扭波解，不可数无穷多光滑周期波解和某些无界行波解的精确的参数表达式，以及上述解存在的参数条件．     

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.     

一类广义四阶非线性Camassa-Holm方程的行波解

用动力系统的分支理论研究了一类广义四阶非线性Camassa-Holm方程的动力学行为和行波解,发现方程存在一些孤立波解,周期波解和一些诸如Compacton类型的非光滑行波解.在不同的参数条件下,给出了这些解存在的条件和一些特殊条件下的精确解.     

非线性演化方程显式精确解的新算法

本文给出了一种求解非线性演化方程的新算法 .将这种算法运用于变形浅水波方程 ,获得了八组显式精确解 ,其中包括新的孤波解和周期解 .借助于 Mathematica软件 ,这种算法能够在 Computer上实现 .     

广义带导数的非线性Schrdinger方程的动态分析和精确解

利用动力系统方法,针对广义带导数的非线性Schrdinger方程的精确解问题进行研究分析.采用行波变换,将其化为常微分方程动力系统;计算出该方程动力系统的首次积分,讨论了系统在不同参数条件下的奇点与相图,得到对应的精确解,包括孤立波解、周期波解、扭结波解和反扭结波解.运用数值模拟的方法,对方程的光滑孤立波解和周期波解等进行了数值模拟.分析计算获得的结果完善了相关文献已有的研究成果.     

F展开法结合指数函数法求解Zhiber-Shabat 方程的精确解

利用F展开法与指数函数法相结合的方法,在相关文献的基础上,重新研究了Zhiber-Shabat方程,获得了许多与现有文献中解的表达式不相同的各种精确解.这些解同样具有孤立波解,纽子波解和周期波解的各种动力学特征.从而丰富了相关文献中关于Zhiber-Shabat波方程的孤立子解和周期解的种类.     

耦合Higgs方程和Maccari系统的行波解分支

利用动力系统方法,对耦合Higgs方程和Maccari系统的定性行为和行波解进行了研究.基于这种方法,给出了系统在不同参数条件下的相图,得到了包括孤立波解和周期波解在内的行波解.运用数值模拟的方法,对方程的光滑孤立波解和周期波解进行了数值模拟.获得的结果完善了相关文献已有的研究成果.     

应用拓展双曲函数方法求KP方程的新精确解

本文引入行波解,并应用拓展双曲函数方法,求得（2＋1）维Kadomtsev-Petviashvili（KP）方程的精确解.通过应用拓展双曲函数方法,可以得到关于方程的一类有理函数形式的孤立波,行波以及三角函数周期波的精确解,并且此方法适用于求解一大类非线性偏微分进化方程.     

广义Camassa-Holm方程的行波解

运用平面动力系统理论和方法给出了广义Camassa-Holm方程在各种参数条件下的相图与分支,分析了奇线对其行波解的影响,获得了广义Camassa-Holm方程光滑、非光滑孤立波解和周期波解的存在性及个数,求出了它的两组新周期尖波解的显式表达式.     

(3+1)-维广义随机KP方程的精确解

利用统一方式构造非线性偏微分方程行波解的广义Jacobi椭圆函数展开法和Hermite变换来研究(3+1)-维广义随机KP方程,给出了它的随机对偶周期和多孤子解.     

New periodic wave solutions via extended mapping method

In this paper, an extended mapping method with a computerized symbolic computation is used for constructing new periodic wave solutions for two nonlinear evolution equations arising in mathematical physics, namely, generalized nonlinear Schroedinger equation and generalized-Zakharov equations. As a result, many exact travelling wave solutions are obtained which include new periodic wave solutions, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also applied to other nonlinear evolution equations.     

Exact travelling wave solutions for nonlinear Schr\"{o}dinger equation with variable coefficients

In this paper, two nonlinear Schr\"{o}dinger equations with variable coefficients in nonlinear optics are investigated. Based on travelling wave transformation and the extended $(\frac{G''}{G})$-expansion method, exact travelling wave solutions to nonlinear Schr\"{o}dinger equation with time-dependent coefficients are derived successfully, which include bright and dark soliton solutions, triangular function periodic solutions, hyperbolic function solutions and rational function solutions.     

New periodic wave solutions for nonlinear evolution equations with variable coefficients via mapping method

An extended mapping method with a computerized symbolic computation is used for constructing a new exact travelling wave solutions for nonlinear evolution equations arising in physics, namely, generalized Zakharov Kuznetsov equation with variable coefficients. As a result, many exact travelling wave solutions are obtained which include new periodic wave solution, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations with variable coefficients arising in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

BIFURCATIONS AND NEW EXACT TRAVELLING WAVE SOLUTIONS OF THE COUPLED NONLINEAR SCHRDINGER-KdV EQUATIONS

By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrdinger-KdV equations are studied. Based on this method, all phase portraits of the system in the parametric space are given. All possible bounded travelling wave solutions such as solitary wave solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, the numerical simulations are conducted for solitary wave solutions and periodic travelling wave solutions to the coupled nonlinear Schrdinger-KdV equations. The results show that the presented findings improve the related previous conclusions.     

Construction of periodic and solitary wave solutions by the extended Jacobi elliptic function expansion method

In this paper, an extended Jacobi elliptic function expansion method is used with a computerized symbolic computation for constructing the exact periodic solutions of some polynomials or nonlinear evolution equations. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in nonlinear mathematical physics. As a result, many exact travelling wave solutions are obtained which include new solitary or shock wave solution and envelope solitary and shock wave solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

New abundant exact solutions for the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov system

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) system. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the soliton solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.     

Bifurcations of exact travelling wave solutions for the generalized R-K-L equation

Using the method of dynamical systems for the the generalized Radhakrishnan, Kundu, Lakshmanan equation, the existence of soliton solutions, uncountably infinite many periodic wave solutions and unbounded wave solution are obtained. Exact explicit parametric representations of the above travelling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.     

New exact travelling wave solutions of some complex nonlinear equations

In this paper, we establish exact solutions for complex nonlinear equations. The tanh–coth and the sine–cosine methods are used to construct exact periodic and soliton solutions of these equations. Many new families of exact travelling wave solutions of the coupled Higgs and Maccari equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems.     

Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method

In this paper, we establish exact solutions for (2 + 1)-dimensional nonlinear evolution equations. The sine-cosine method is used to construct exact periodic and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Boussinesq, breaking soliton and BKP equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.