Burgers与组合KdV混合型方程的精确解

该文求出了组合ＫｄＶ方程的渐近值不为零的钟状孤波解和扭状孤波解；求出了Ｂｕｒｇｅｒｓ与组合ＫｄＶ混合型方程ｕｔ＋ａｕｕｘ＋ｂｕ２ｕｘ＋ｒｕ（ｘｘ）＋ｕ（ｘｘｘ）＝０的二类扭状孤波解．作为推论，还求出了波方程ｕ（ｔｔ）－ｋｕ（ｘｘ）＋ｐｕ十ｑｕ２＋ｓｕ３＝０的钟状和扭状孤波解．     

组合Zakharov-Kuznetsov方程的显式孤波解

借助于Ｍａｔｈｅｍａｔｉｃａ是吴消元法,本文通过用一个新的假设,获得了组合Ｚａ－ｋｈａｒｏｖ－Ｋｕｚｎｅｔｓｏｖ方程的１２种孤波解,其中包括钟状与扭状组合型孤波解和周期型孤波解。这种假设也能用于其他的非线性演化方程（组）。     

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．     

耗散对广义的WBK型耗散方程行波解的影响

运用平面动力系统理论对广义的WBK型耗散方程所对应的动力系统作了定性分析,给出了其在不同参数条件下的全局相图.研究了该方程行波解的性态与耗散系数r之间的关系,得到当耗散作用较大时行波表现为扭状孤波,当耗散作用较小时行波表现为衰减振荡解的结论.     

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.     

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．     

广义Pochhammer-Chree方程的显式精确孤波解

首先对广义Pochhammer-Chre方程(PC方程)u_tt-u_ttxx+ru_xxt-(a₁u+a₂u2+a₃u3)_xx=0(r≠0)(Ⅰ)的孤波解u(ξ)建立了公式∫_-∞+∞[u'(ξ)]2dξ=1/12rv(C₊-C_-)3[3a₃(C₊+C_-)+2a₂]。由此推知:广义PC方程(Ⅰ)不可能有钟状孤波解,只可能有扭状孤波解;而广义PC方程u_tt-u_ttxx-(a₁u+a₂u2+a₃u3)_xx=0(Ⅱ)可能既有钟状孤波解又有渐近值满足3a₃(C₊+C_-)+2a₂=0的扭状孤波解。进一步求出了广义PC方程(Ⅰ)的扭状孤波解,求出了广义PC方程(Ⅱ)的钟状孤波解和渐近值满足2a₃(C₊+C_-)+2a₂=0的扭状孤波解。最后给出了广义PC方程u_tt-u_ttxx-(a₁u+a₃u3+a₅u5)_xx=0(Ⅲ)的显式孤波解。     

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.     

具任意次非线性项的Lienard方程的精确解及其应用

该文推导了具任意次非线性项的Liénard方程a″(ξ)+la(ξ)+ma\+q(ξ)+na\+\{2q-1\}(ξ)=0和\{a″(ξ)\}+ra′(ξ)+la(ξ)+ma\+q(ξ)+na\+\{2q-1\}(ξ)=0解的若干性质，通过适当变换，并结合假设待定法求出了它们的钟状和扭状显式精确解.据此，求出了一批具任意次非线性项的发展方程的钟状和扭状显式精确孤波解，其中包括广义BBM型方程、二维广义Klein Gordon方程、广义Pochhammer Chree方程和非线性波方程等.     

含任意次正幂项的广义五阶KdV方程的精确解

利用齐次平衡原理,通过引进含非线性辅助微分方程（sub-ODE）,获得了含任意次正幂项的广义五阶KdV方程的精确解,包括钟状孤波解,扭状孤波解和三角函数表示的周期波解.所得精确解与前人用其它方法所获得一致,并包含了以往文献未提供的部分解,扩充并完善了以往文献的相关结果.     

KK方程和改进的Boussinesq方程的新精确解

应用改进的Fan's代数方法,得到了KK方程和改进的Boussinesq方程的一系列新精确解,包括孤立波解、类孤立波解、纽结波解、奇异纽结波解和三角函数周期解.     

Wick型随机广义Burgers方程的确切解

借助白噪声分析、Hermite变换和扩展的双曲函数法,研究了Wick型随机广义Burgers'方程,求出了一些精确的Wick型孤立波解和周期波解.由于Wick型函数难以赋值,为此,我们得到一些特殊情形下的Wick型随机广义Burgers'方程的非Wick型解.     

一类变形的Boussinesq方程组的行波解分支

在Boussinesq方程组求解方面,用平面动力系统的分支理论研究了一类变形的Boussinesq方程组的行波解分支．得到了不同参数条件下的分支集、相图及所有孤立波和扭波的精确公式．     

Geometric Properties and Exact Travelling Wave Solutions for the Generalized Burger-Fisher Equation and the Sharma-Tasso-Olver Equation

In this paper, we study the dynamical behavior and exact parametric representations of the traveling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation under different parametric conditions, the exact monotonic and non-monotonic kink wave solutions, two-peak solitary wave solutions, periodic wave solutions, as well as unbounded traveling wave solutions are obtained.     

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．     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.     

Explicit Traveling Wave Solutions and Their Dynamical Behaviors for the Coupled Higgs Field Equation

In this paper, we focus on the traveling wave solutions of the coupled Higgs field equation from the perspective of dynamical systems. Through the phase portraits, in addition to periodic wave solutions and solitary wave solutions, we also obtain explicit periodic singular wave solutions, singular wave solutions and kink wave solutions, which were not found in the previous works. The dynamical behavior of these solutions and their internal relations are revealed through asymptotic analysis. The results will help supplement the study of field equation.     

Boussinesq-Burgers方程的分岔及一些有界行波解

利用平面动力系统理论研究了Boussinesq-Burgers方程,讨论了方程在行波变换后所对应的平面动力系统的分岔行为,并基于相平面上特定的相轨道求出了该方程的扭结波、孤立波及周期波的解析表达式.数值模拟进一步验证了所得结论的正确性.     

Interaction Between Kink Solitary Wave and Rogue Wave for (2+1)-Dimensional Burgers Equation

Based on a suitable ansätz approach and Hirota’s bilinear form, kink solitary wave, rogue wave and mixed exponential–algebraic solitary wave solutions of (2+1)-dimensional Burgers equation are derived. The completely non-elastic interaction between kink solitary wave and rogue wave for the (2+1)-dimensional Burgers equation are presented. These results enrich the variety of the dynamics of higher dimensional nonlinear wave field.