有限变形弹性杆中的几何非线性波

利用有限变形理论的Lagrange描述,借助非保守系统的Hamilton型变分原理,导出了描述弹性杆中几何非线性波的波动方程．为了使非线性波动方程有稳定的行波解,计及了粘性效应引入的耗散和横向惯性效应导致的几何弥散．运用多重尺度法将非线性波动方程简化为KdV-Bergers方程,这个方程在相平面上对应着异宿鞍-焦轨道,其解为振荡孤波解．如果略去粘性效应或横向惯性,方程将分别退化为KdV方程或Bergers方程,由此得到孤波解或冲击波解,它们在相平面上对应着同宿轨道或异宿轨道．     

含变系数或强迫项的KdV方程的新解

Jacobi椭圆函数展开法被推广并用于求解另一种形式的KdV方程的新的精确解,所求解的这类KdV方程包括一种典型的变系数的KdV方程和具有强迫项（随机项）的KdV方程．用这种方法得到的新的类周期解在极限条件下可以退化为类孤立波解或类冲击波解．     

非线性演化方程的显式行波解

本文系统归纳了重要的非线性演化方程的各类显式行波解.说明常微分方程中的同宿轨道和异宿轨道分别和非线性演化方程中的孤立波(或称脉冲波)和波前相对应.耗散系统也存在孤立波,二维(x,y)演化方程中的孤立波可以显示出模式(Pattern)结构.三维相空间的鞍点同宿轨道和鞍—焦点同宿轨道(称Silnikov同宿轨道)常和混沌相联系,     

非线性演化方程的孤立波解

用齐次平衡原则和辅助微分方程方法得到了6个重要的n次非线性演化方程的孤立波解.辅助微分方程方法的主要思想是借助简单的可解微分方程的解去构造复杂的非线性演化方程的行进波解.这里简单的可解微分方程称为辅助微分方程.本文使用的辅助方程有双曲正割幂型解或双曲正切幂型解.     

有限变形弹性圆杆中的孤波

在同时引入横向惯性和横向剪切应变的情况下,导出了有限变形弹性圆杆的非线性纵向波动方程,方程中包含了二次和三次的非线性项以及由横向剪切与横向惯性导致的两种几何弥散效应．借助Mathematica软件,利用双曲正割函数的有限展开法,对该方程和对应的截断的非线性方程进行求解,得到了非线性波动方程的孤波解,同时给出了这些解存在的必要条件．     

有限挠度下Timoshenko梁中的非线性弯曲波及其混沌行为

以Timoshenko梁理论为基础,引入了有限挠度和轴向惯性,建立了支配梁运动的非线性偏微分方程组,采用行波法求解,通过某些积分技巧,将其转化为一个非线性常微分方程．常微分方程的定性分析表明,在一定条件下,系统存在异宿轨道,预示着有冲击波解存在．借助Jacobi椭圆函数展开求解,得到了非线性波动方程的准确周期解及其当模数m→1退化情况下的冲击波解．进而考虑阻尼和外加横向载荷对系统的摄动,利用Melnikov函数给出了横截异宿点出现的阈值条件,从而表明系统具有Smale马蹄意义下的混沌性质．     

长短波演化方程的孤波解、周期波解及它们之间的演变关系

本文运用定性分析与首次积分相结合的方法研究了长短波演化方程的精确孤波解、周期波解以及这两种解之间的演变关系.揭示出所研方程之所以会出现周期波解和孤波解,本质上是由该方程解中短波u的模对应的Hamilton系统的能量取不同的值所决定的.     

两个具有相同Hirota双线性方程的(3+1)-维演化方程的拟周期波解

应用Hirota双线性方法,构造了一个用Riemannθ函数表示的双线性方程的拟周期波解.应用到两个(3+1)-维演化方程:一个是与AKNS可积方程族相关的可积模型,另一个是著名的Jimbo-Miwa方程,分别得到了这两个演化方程的拟周期波解.     

由一类非线性常微分方程的抛物线解所确定的扭波

通过求解与平面动力系统的两个平衡点相连接的抛物线解,获得了6种非线性行波方程的扭波解存在条件,并给出了这些扭波解的参数表达式,以及上述解存在的参数条件．     

带色散项的Degasperis-Procesi方程的孤立尖波解

用动力系统的定性分析理论研究了带有色散项的Degasperis-Procesi方程的孤立尖波解.在一定的参数条件下,利用Degasperis-Procesi方程对应行波系统的相图分支从两种不同方式给出了孤立尖波解的表达式.     

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.     

Dynamical Behavior and Exact Traveling Wave Solutions for Three Special Variants of the Generalized Tzitzeica Equation

The dynamics and bifurcations of traveling wave solutions are studied for three nonlinear wave equations. A new phenomenon, such as a composed orbit, which consists of two or three heteroclinic orbits, may correspond to a solitary wave solution, a periodic wave solution or a peakon solution, is found for the equations. Some previous results are extended.     

Solitary wave solution of higher-order Korteweg–de Vries equation

An attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg–de Vries (KdV) equation by using tanh-method or hyperbolic method. The higher-order equation can be derived for magnetized plasmas by using the reductive perturbation technique. It is found that the exact solitary wave solution of higher-order KdV equation is obtained by tanh-method. Using this method, different kinds of nonlinear wave equations can be evaluated. The higher-order nonlinearity and higher-order dispersive effect can be observed from the solutions of the equations. The method is applicable for other nonlinear wave equations.     

Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III)

In this paper, the integral bifurcation method was used to study the higher order nonlinear wave equations of KdV type (III), which was first proposed by Fokas. Some new travelling wave solutions with singular or nonsingular character are obtained. In particular, we obtain a peculiar exact solution of parametric type in this paper. This one peculiar exact solution has three kinds of wave-form including solitary wave, cusp wave and loop solion under different wave velocity conditions. This phenomenon has proved that the loop soliton solution is one continuous solution, not three breaking solutions though the loop soliton solution “is not in agreement with the Poincaré phase analysis”.     

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.     

TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic travelingwaves for a class of nonlinear dispersive partial differential equations.By using the bifurcationtheory of dynamical systems to do qualitative analysis,all possible phase portraits in theparametric space for the traveling wave systems are obtained.It can be shown that the existenceof a singular straight line in the traveling wave system is the reason why smooth solitary wavesolutions converge to solitary cusp wave solution when parameters are varied.The differentparameter conditions for the existence of solitary and periodic wave solutions of different kindsare rigorously determined.     

TRAVELINGWAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.     

Methods of judging shape of solitary wave and solution formulae for some evolution equations with nonlinear terms of high order

In this paper, we present several methods of judging shape of the solitary wave and solution formulae for some nonlinear evolution equations by means of Lienard equations. Then, using the judgement methods and solution formulae, we obtain solutions of the solitary wave for some of important nonlinear evolution equations, which include generalized modified Boussinesq, generalized nonlinear wave, generalized Fisher, generalized Klein-Gordon and generalized Zakharov equations. Some new solitary-wave solutions are found for the equations.     

齐次平衡法若干新的应用

齐次平衡法是求非线性发展方程孤波解的一种有效方法．该文将以KdV方程为例把齐次平衡法向三个方面拓广应用：1）获得非线性发展方程新的具有更为丰富形式的精确解;2）寻找非线性发展方程的Backlund变换、Lax表示;3）求非线性发展方程的对称性约化和相似解．     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.