一类非线性演化方程新的显式行波解

借助Mathematica软件,采用三角函数法和吴文俊消元法,获得了一类非线性演化方程u_tt+au_xx+bu+cu³=0的三组行波解,其中包括新的行波解、扭状孤波解和钟状孤波解.从而作为该方程的特例,如Duffing方,Klein-Gordon方程、Landau-Ginburg-Higgs方程和⁴方程等也都获得了相应的若干行波解.这种方法也适用于其他非线性方程. 关键词：     

一类五阶非线性演化方程的新孤波解

利用齐次平衡法给出了一类较广泛的五阶非线性演化方程的孤波解，数学物理中著名的Kaup-Kupershmidt方程、Caudrey-Dodd-Gibbon-Sawada-Kotera方程和五阶Korteweg-de-Vries方程等都可作为该方程的特殊情形而得到相应的孤波解. 关键词：     

具5次强非线性项的波方程新的孤波解

提出一种新的函数变换法，并与直接积分法相结合简便地求出了Lienard方程、广义PC方程以及力学中重要的一类非线性波方程等几类具5次强非线性项的波方程的四类显示精确孤波解.本方法同样适用于求解其他具有更高次非线性项的非线性方程. 关键词： 函数变换法 孤波解 直接积分法 非线性波方程     

非线性孤子方程的齐次平衡法

齐次平衡法是求非线性孤子方程孤波解的一种十分有效的方法.对齐次平衡法的一些关键步骤进行拓广使用，获得了非线性孤子方程一批新的具有更为丰富形式的精确解，孤波解仅是其中的一种特殊情形，使得结果更加完美. 关键词：     

一类非线性演化方程的新多级准确解

在Lamé方程和新的Lamé函数的基础上，应用小扰动方法和Jacobi椭圆函数展开法求解一类非线性演化方程（如mKdV方程，非线性Klein-Gordon方程Ⅱ等），获得多种新的多级准确解 .这些多级准确解对应着不同形式的周期波解.这些解在极限条件下可以退化为多种形式的孤 立波解，如带状孤立子、钟形孤立子等. 关键词： Jacobi椭圆函数 Lam函数 多级准确解 非线性演化方程 扰动方法     

推广的KdV方程的孤波解

本文研究了推广的KdV方程 u_t+2μuu_x+v_3x+δu_5x=0(μvδ≠0) (1)的精确孤子解,得到了(1)式的一些新的孤波解,对文献[10]的若干结论作了补充与修正。 关键词：     

一类相对论性非球谐振子系统的束缚态

给出了具有形式为12r²+A2r²的非球谐振子型标量势和矢量势 的相对论系 统在两种势相等的条件下三维Klein-Gordon方程，二维和三维Dirac方程的s波束缚态解. 关键词： 三维非球谐振子势 Klein-Gordon方程 Dirac方程 束缚态     

非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了Burgers方程，得到了其扭结形孤立波的近似解析解，该解非常接近于相应的精确解.结果表明，同伦分析法可用来求解非线性演化方程的孤立波解.同时，也对所用方法进行了一定扩展，得到了Kadomtsev-Petviashvili(KP)方程的钟形孤立子解.经过扩展后的方法能够更方便地用于求解更多非线性演化方程的高精度近似解析解. 关键词： Burgers方程 同伦分析法 KP方程 孤立波解     

扩展的混合指数方法及其应用

改进了Hereman提出的构造非线性发展方程孤波解的混合指数方法,通过将非线性发展方程孤波解的表示形式推广到实指数解或复指数解的无穷级数,得到了扩展的混合指数方法.以正则长波方程为例,说明通过扩展的混合指数方法可获得包括正则孤波解、奇异孤波解及周期解在内的诸多精确解 关键词： 孤波解 混合指数法 正则长波方程     

求解非线性差分方程孤立波解的直接代数法

推广了求解非线性差分方程孤立波解的直接代数法.用此方法研究了Hybrid晶格方程，借助于符号计算Maple,得到它的新孤波解.这种方法也可用于求解其他的差分方程. 关键词： 微分-差分方程 Hybrid晶格方程 行波解 孤     

具有阻尼项的非线性波动方程的相似约化

利用Clarkson和Kruskal引入的直接约化法,给出了具有阻尼项的非线性波动方程u_tt-2bu_xxt+αu_xxxx=β(uⁿ_x)x(α>0,β≠0,n≥2)三种类型的相似约化.从这些约化方程的Painlevé分析表明该方程在Ablowitz的猜测意义下是不可积的.此外还获得了该方程(n=2)的4种精确类孤波解. 关键词： 波动方程 相似约化 Painlevé分析 精确解     

Calculation of molecular constants for the X 1Σ g + , A 1Σ u + , B 1Πu, and a 3Σ u + electronic states of 39K2

The molecular constants are calculated for the X ¹Σ _g ⁺ , A ¹Σ _u ⁺ , B ¹Π_u, and a ³Σ _u ⁺ and electronic states of a potassium dimer. The wave functions and vibrational energies necessary for calculating the molecular constants are determined by solving the radial wave equation with the use of potential energy curves constructed by the semiempirical method. The vibrational terms, the rotational constants, and the centrifugal distortion constants calculated from the potential curves are compared with those determined from the experimental data.     

Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a ₀+a ₁tan ξ+a ₂tan ² ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method.     

拍波激光加速器中的频率匹配

本文从广义协变的运动方程和麦克斯韦方程出发,导出了电子等离子体波各量的解析表达式。指出△ω=2ω_p的等离子体波是完全简谐的。完全共振的条件由△ω=2ω_(p0)[1+(e²(A₂⁽⁽¹⁾²⁾+(A₂^((2)2))/(2m²c⁴)+(3e²A₂⁽¹⁾A₂⁽²⁾)/(m²c⁴)]^-1/2 给出。 关键词：     

Bäcklund transformations for diffusion equations

A class of Bäcklund transformations is deduced for the diffusion equation (S(u))_t = (C(u,u_x))_x by taking advantage of the conservative form of the equation. The transformations are used for obtaining new relations connecting various equations of the type u_t = (u^au_x)_x and u_t = u^au_xx.     

Numerical study on characteristics of nitrogen discharge at high pressure with induced plasma

Based on the fluid theory of plasma, a model is built to study the characteristics of nitrogen discharge at high pressure with induced argon plasma. In the model, the spices such as electron, N₂⁺, N₄⁺, Ar⁺, and two metastable states (N₂ (A³∑_u⁺), N₂ (a¹∑_u^-)) are taken into account. The model includes particle's continuity equations, electron's energy balance equation, and Poisson equation. The model is solved with a finite difference method. The numerical results are obtained and used to investigate the effect of time taken to add nitrogen gas and initially-induced argon plasma pressure. It is found that lower speeds of adding the nitrogen gas and varying the gas pressure can induce higher plasma density, and inversely lower electron temperature. At high-pressure discharge, the electron density increases when the proportion of nitrogen component is below 40%, while the electron density will keep constant as the nitrogen component further increases. It is also shown that with the increase of initially-induced argon plasma pressure, the density of charged particles increases, and the electron temperature as well as the electric field decrease.     

Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation

We prove that the set of solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in two dimensions, (u _t+(u^m+1)_x+u_xxx)_x=u_yy is stable for 0<m<4/3.     

Zeitliche Entwicklung des Reflexionskoeffizienten einer Plasmaschicht bei starker elektromagnetischer Einstrahlung

We have calculated numerically the temporal evolution of the nonlinear reflection coefficient R of an overdense plasma layer by solving the system of partial differential equations consisting of the wave equation for the slowly varying amplitude of the electric field and the hydrodynamic equations for the ion motion, including a ponderomotive force term. In dependence of the (normalized) amplitude of the incident wave u_A two regimes exist: Below a critical amplitude u_A^* ? 1 the reflection coefficient is approximately independent on the amplitude u_A and temporally constant. In the opposite case u_A>u_A^*, on the other hand, ‖R‖² decreases slowly with time down to a minimum value and after that it increases rapidly to the initial value. We think, that our results are important to interpret the anomalous reflectivity observed in some experiments when strong electromagnetic waves are incident on an overdense plasma.     

On Integration of the Nonlinear d’Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations

Abstract

We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d’Alembert equation □u = F (u) and nonlinear eikonal equation u _xµ u _x µ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct a general solution of the d’Alembert-eikonal system for all cases when it is compatible. The results obtained can be applied, in particular, to construct principally new (non-Lie, non-similarity) solutions of the non-linear d’Alembert, Dirac, and Yang-Mills equations. Solutions found in this way are shown to correspond to conditional symmetry of the equations enumerated above. Using the said approach, we study in detail conditional symmetry of the nonlinear wave equation □w = F ₀(w) in the four-dimensional Minkowski space. A number of new (non-Lie) reductions of the above equation are obtained giving rise to its new exact solutions which contain arbitrary functions.