EXACT SOLITRAY WAVE SOLUTIONS AND SINGULAR SOLUTIONS TO THE TWO-DIMENSIONAL NONLINEAR DISSIPATIVE-DISPERSIVE SYSTEM

IIntroductlonIntegrMle systems;both classical ajnd quantum me山anies,are aMclnatingsubject·Decades ofresearch In this areahave led to mathematical developme血s that are quite beau－tiful．However,not ail systems posed in physics are Integrable,M Instajnce,the Korteweg－deVies－Burgers(KdV－Burgers),Kur。ot。SI、hinsky(KS)and ifth－order dispersi、Kortevegde Vries(ifth－order KdV)eqUatio。 Therefore the direct methods to sol。nonlinear systemsppear to be more important．In this paper…     

EXACT SOLITARY WAVE SOLUTIONS OF THE TWO NONLINEAR EVOLUTION EQUATIONS

The solitary wave solutions of the combined KdV-mKdV-Burgers equation and the Kolmogorov-Petrovskii-Piskunov equation are obtained by means of the direct algebra method, which can be generalized to deal with high dimensional nonlinear evolution equations.     

几类具5次强非线性项的发展方程的显式精确孤波解

本文道德求出了具５次强非线性项的Ｌｉｅｎａｒｄ方程的二类显式精确解。     

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．     

EXACT SOLUTIONS TO NONLINEAR WAVE EQUATION

In this paper,we use an invariant set to construct exact solutions to a nonlinear wave equation with a variable wave speed. Moreover,we obtain conditions under which the equation admits a nonclassical symmetry. Several different nonclassical symmetries for equations with different diffusion terms are presented.     

Fisher方程和Burgers-Fisher方程新的精确行波解

本文研究了Fisher方程和Burgers-Fisher方程.运用一种辅助微分方程方法,得到了这两种非线性偏微分方程新的精确行波解.     

EXACT TRAVELLING WAVE SOLUTIONS TO BBM EQUATION

Abundant new travelling wave solutions to the BBM (Benjamin-Bona-Mahoni) equation are obtained by the generalized Jacobian elliptic function method. This method can be applied to other nonlinear evolution equations.     

共振长短波方程的孤波解

本文研究了共振长短波方程的孤波解.利用扩展映射法和符号计算,得到许多新的孤波解.这些孤波解能很好地模拟水波,辅助方程用更一般方程代替的扩展映射法能更有效找到这些孤波解.

Abstract：

In this article,soliton solutions of the long-short wave resonance equations are investigated.By the extended mapping method and symbolic computation,many new exact soliton solutions are obtained.These soliton solutions are fascinating in modeling water waves.The extended mapping method,with the auxiliary ordinary equations replaced by more general ones,is more effective to find these soliton solutions.     

含频散项的K(2,3)方程的精确行波解

本文研究了包含频散项的K(2,3)方程u_t+(u²)_x-(u³)_xxx=0的分支问题.利用动力系统的定性分析,并且借助Maple软件进行数值模拟得到行波解系统相应的相图,然后通过积分计算得到周期尖波解、类扭波和类反扭波的精确解的函数表达式,以及孤立波精确解的隐函数表达式.     

RAPID EXACT CONTROLLABILITY OF THE SEMILINEAR WAVE EQUATION

1.IntroductionConsiderthefollowingsemilinearwaveequationwithalocallydistributedcofltroller:whereI~g,11~$,nCR"isaboundeddomainwithaboundaryoffECI)andforeachtE[0,co),G(t)isasubdomainoffi.Intheabove,y(t,x)isthestateandXG(t)(x)u(t,x)isthecolltrol.Thus,u(t,x)istheintensityofthecontrolactionandG(t)isthelocationandtheshapeofthecontroller.Wewillallowthelocationandtheshapeofthecontrollertochange.LetU=Lfo.(0'co;L'(fl))andletQbeafamilyofset--valuedfunctionsG(t)definedon[0,co)takingsubdomainsoffias…     

一类非线性波动方程的显式精确解

本文用直接方法和假设的一种结合求出了一类较广泛的非线性波动方程ｕｔｔ－ａ１ｕｘｘ＋ａ２ｕｔ＋ａ３ｕ＋ａ４ｕＳ＾２＋ａ５ｕ＾３＝０的一些显式精确行波解,贱个有重要的非线性数学物理方程,如φ＾４方程,Ｋｌｅｉｎ－Ｇｏｒｄｏｎ方程,Ｓｉｎｅ－Ｇｏｒｄｏｎ方程,及Ｓｉｎｈ－Ｇｏｒｄｏｎ方程的近似,Ｌａｎｄａｕ－Ｇｉｎｚｂｕｒｇ－Ｈｉｇｇｓ方程,Ｄｕｆｆｉｎｇ方程,非线性电报方程等都可作为该方程的特殊情形得     

一般非线性Schrödinger方程的显式孤立波解

本文考虑推广的导数非线性Schrodinger方程;运用动力系统的几何理论、分支理论和直接方法,得到其指定形式的所有显式精确孤立波解.本文的结果包含和推广了关于这一方程的显式精确孤立波解的所有已知结果.     

EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHR(O)DINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION

A system comprised of the nonlinear Schrodinger equation coupled to theBoussinesq equation (S-B equations) which dealing with the stationary propagation of cou-pled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed.To examine its solitary wave solutions, a reduced set of ordinary differential equations areconsidered by a simple traveling wave transformation. It is then shown that several newsolutions (either functional or parametrical) can be obtained systematically, in addition torederiving all known ones by means of our simple and direct algebra method with the helpof the computer algebra system Maple.     

基于吴方法的孤波自动求解软件包及其应用

基于非线性代数方程组的吴特征列方法,在计算机代数系统Maple上实现了非线性微分方程孤波解的自动求解,编制了一个小型实用的软件包。作为应用,考虑了一个一般的五阶模型方程,利用该软件包获得了此方程新的孤波解以及孤子解存在的条件。     

广义特殊Tzitzeica-Dodd-Bullough类型方程的行波解

本文研究了广义特殊Tzitzeica-Dodd-Bullough类型方程,利用动力系统分支理论方法,证明该方程存在周期行波解,无界行波解和破切波解,并求出了一些用参数表示的显示精确行波解.     

一般非线性Schr(o)dinger方程的显式孤立波解

本文考虑推广的导数非线性Schrodinger方程;运用动力系统的几何理论、分支理论和直接方法,得到其指定形式的所有显式精确孤立波解.本文的结果包含和推广了关于这一方程的显式精确孤立波解的所有已知结果.     

BIFURCATIONS OF THE TRAVELLING WAVE SOLUTIONS TO A GENERALIZED CAMASSA-HOLM EQUATION

In this paper, we study some generalized Camassa-Holm equation. Through the analysis of the phase-portraits, the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave and compactons were discussed. In some certain parametric conditions, many exact solutions to the above travelling waves were given. Further-more, the 3D and 2D pictures of the above travelling wave solutions are drawn using Maple software.     

一类广义Zakharov方程的精确行波解

本文研究了一类广义Zakharov方程的精确解行波解的问题.利用改进的G/G展开方法,借助于计算机代数系统Mathematica,获得了具有重要物理背景的广义Zakharov方程一系列新的含有多个参数的精确行波解,这些解包括孤立波解,双曲函数解,三角函数解,以及有理函数解.