两个非线性发展方程的双向孤波解与孤子解

采用分步确定拟解的原则, 对齐次平衡法求非线性发展方程孤子解的关键步骤作了进一步改 进. 以广义Boussinesq方程和bidirectional Kaup-Kupershmidt方程为应用实例, 说明使用 该方法可有效避免“中间表达式膨胀”的问题, 除获得标准Hirota形式的孤子解外, 还能获 得其他形式的孤子解. 关键词： 齐次平衡法 孤子解 孤波解 广义Boussinesq方程 bidirectional Kaup-Kupershmi dt方程     

一类非线性微分方程的近似解析解

从理论上研究了一类广义扩散方程的求解问题. 利用相似变换和解析拆分技巧给出了求解该类非线性微分方程近似解的一种有效方法, 方程的解可以表示为一个收敛的幂级数. 近似解结果和数值结果非常符合，证明了所提出的方法的准确性和可靠性, 该方法可以用于解决其他科学和工程技术问题. 关键词： 广义扩散方程 非线性边界值问题 解析拆分 近似解析解     

对称性及多群中子扩散方程数值解

在多群中子扩散方程解析解的基础上,利用方程及求解域的对称性建立了新的数值求解中子扩散方程的理论模型.该模型显著的优点是适用于各种对称区域(二维、三维区域)尤其是非正方形区域中子扩散方程的求解,它彻底避免了常规节块法应用于非正方形几何时所出现的奇异性问题,且所得的解在求解域内任意点上均满足扩散方程.以二、三维六角形几何为例建立了数学模型,并用基准问题校核了模型的正确性. 关键词： 中子扩散方程 对称群 数值解 解析     

求解结构动力响应的卷积型DQ半解析法

通过卷积将原始控制方程构造成包含初始条件的新的具有完整初值问题特征的控制方程.该方程既与Gurtin变分原理一样有合理的数学内涵,又避免了卷积型Gurtin变分原理泛函和计算的繁复.对新的控制方程在时间域取解析函数,在空间域采用离散的DQ法,经对梁的动力响应问题的计算表明,该方法是一种精度好效率高的求解动力响应问题的计算方法.     

阻尼谐振子波函数的解

对一维阻尼受迫谐振子,根据其哈密顿量,通过正则量子化、变量代换等方法求解其薛定谔方程,得到其波函数和能量的表达式,同时分析讨论了该解的零点量子涨落问题。     

Fredholm积分方程的数值解

积分方程是泛函分析的一个重要分支，它是研究数学及其它学科，如偏微分方程边值问题和各种物理问题的重要数学方法，为此，我们研制了求解Ｆｒｅｄｈｏｌｍ积分方程的代码ＩＴＧＭＯ，并给出了该代码实验应用的例子。     

Gerdjikov-Ivanov方程的精确解

研究在量子场理论、弱非线性色散水波、非线性光学等领域中出现的Gerdjikov-Ivanov方程.对Gerdjikov-Ivanov方程的研究会导出具有高次非线性项的非线性数学物理方程.选取Liénard方程作为辅助常微分方程,借助于它并根据齐次平衡原则,求解了Gerdjikov-Ivanov方程,得到了该方程的包络孤立波解和包络正弦波解. 关键词： 齐次平衡原则 F展开法 Gerdjikov-Ivanov方程 包络孤立波解     

mKdV-sine-Gordon方程丰富的相互作用解

基于mKdV-sine-Gordon方程的Wronsk解的形式和结构,提出了Wronsk形式展开法,通过这一方法求得了该方程的丰富的相互作用解,该方法的主要特征是不要求Wronsk行列式元素满足线性偏微分方程组。     

非线性演化方程椭圆函数解的一种简单求法及其应用

非线性演化方程的许多行波解可以写成满足投影Riccati方程的两个基本函数的多项式形式.利用这一性质，通过建立一般的椭圆方程与投影Riccati方程解之间的关系，导出了一个构造这些解的新方法.该方法对类型Ⅰ的方程和类型Ⅱ的方程均有效，同时也回答了如何求出非线性演化方程分式形式椭圆函数解的问题. 关键词： 非线性演化方程 椭圆函数解     

Burgers方程新的精确解及初始值问题解的封闭形式

利用扩展齐次平衡法,求出了Burgers方程无穷多个单孤子解和无穷多个有理函数解,特别是得到了Hopf-Cole’s变换和方程初始值问题解的封闭形式.方法简单直接,并且可以推广到其它方程.     

Soliton Molecule and Breather-Soliton Molecule Structures for a General Sixth-Order Nonlinear Equation

Starting from a general sixth-order nonlinear wave equation,we present its multiple kink solutions,which are related to the famous Hirota form.We also investigate the restrictions on the coefficients of this wave equation for possessing multiple kink structures.By introducing the velocity resonance mechanism to the multiple kink solutions,we obtain the soliton molecule solution and the breather-soliton molecule solution of the sixth-order nonlinear wave equation with particular coefficients.The three-dimensional image and the density map of these soliton molecule solutions with certain choices of the involved free parameters are well exhibited.After matching the parametric restrictions of the sixth-order nonlinear wave equation for having three-kink solution with the coefficients of the integrable bidirectional Sawada-Kotera-Caudrey-Dodd-Gibbons(SKCDG) equation,the breather-soliton molecule solution for the bidirectional SKCDG equation is also illustrated.     

Spatio-Temporal Deformation of Kink-Breather to the (2+1)-Dimensional Potential Boiti–Leon–Manna–Pempinelli Equation

In the paper, the rational breather soliton and kink solitary wave solution of the (2+1)-dimensional PBLMP equation are obtained by adopting Hirota bilinear method and selecting different test functions. Furthermore, it has been found that the fusion and degeneration of the kink solitary wave occur when interaction between the rational breather soliton and the kink solitary wave happens. These phenomena are very helpful in researching soliton dynamical complexity in the higher dimensional systems.     

Role of defects in the kink mechanism of state switching

State switching occurring via the kink mechanism in linear systems has been described considering the effect of defects forming centers of enhanced kink pair nucleation and obstacles for kink propagation. An equation describing the switching kinetics has been derived considering the stochastic nature of kink pair formation in time and the random distribution of defects in space. Using this equation, the average switching time has been calculated as a function of the defect density and delay times caused by defects, and the ranges of parameters with domination of this or that defect type have been determined. The theory has been applicable to magnetic nanowires, dislocations, biological macromolecules, and many other systems.     

组合KdV方程的双扭结孤立波及其稳定性研究

利用扩展的双曲函数法得到了combined KdV-mKdV (cKdV)方程的几类精确解,其中一类为具有扭结—反扭结状结构的双扭结单孤子解.在不同的极限情况下,该解分别退化为cKdV方程的扭结状或钟状孤波解.理论分析表明,cKdV方程既有传播型孤立波解,也有非传播型孤立波解.文中对双扭结型孤立波解的稳定性进行了数值研究,结果表明,cKdV方程既存在稳定的双扭结型孤立波,也存在不稳定的双扭结型孤立波. 关键词： cKdV方程 双扭结单孤子 稳定性     

耦合KdV方程的双峰孤立子及其稳定性

利用扩展双曲函数法求解了耦合KdV方程,得到了6类精确解,其中一类为具有双峰状结构的单孤子解.在不同的极限情况下,该解分别退化为耦合KdV方程的扭结状或钟状孤波解.文中对双峰孤立波的稳定性进行了数值研究,结果表明:耦合KdV方程的双峰孤立波在长波小振幅扰动和小振幅钟型孤立波扰动下,均稳定. 关键词： 耦合KdV方程 双峰孤立子 稳定性     

Asymptotics for kink propagation in the discrete Sine-Gordon equation

The evolution of a propagating kink in a Sine-Gordon lattice is studied asymptotically using an averaged Lagrangian formulation appropriately coupled to the effect of the radiation. We find that unlike the continuum case the interaction with the Goldstone mode is important to explain the acceleration of the kink as it hops along the lattice. We develop a discrete WKB type solution to study the interaction of the kink and the radiation. In particular using this solution we show how to calculate the effect of the Peyrard and Kruskal resonant radiation in the energy loss of the kink. We obtain a set of modulation equation which explains qualitatively the evolution of the kink with remarkable quantitative agreement.     

非线性延时修正光纤孤子方程的稳态解问题Ⅰ.静态扭结孤波解

严格求解含非线性延时修正光纤孤立子方程,得到一类完全不同于光纤中已知的亮孤子和暗孤子的新型光学孤波解,并讨论了其物理含义及在光纤实验中观察这种扭结孤波的可能性。     

盐对DNA相变影响的非线性特性研究

在Prohofsky，Peyrard-Bishop等提出的描述DNA双螺旋分子结构模型以及实验测量的基础上 , 给出了与盐(指NaCl)浓度有关的哈密顿模型, 得到了非线性动力学方程及扭结孤波解.并 由此求出了DNA变性相变所需要的Peierls相变力. 进一步讨论了盐浓度对相界面和相变力的 影响, 得到的结果与实验测量一致. 关键词： DNA 盐浓度 相界面 相变力     

The window Josephson junction: a coupled linear nonlinear system

We investigate the interface coupling between the two-dimensional sine-Gordon equation and the two-dimensional wave equation in the context of a Josephson window junction using a finite volume numerical method and soliton perturbation theory. The geometry of the domain as well as the electrical coupling parameters are considered. When the linear region is located at each end of the nonlinear domain, we derive an effective one-dimensional model, and using soliton perturbation theory, compute the fixed points that can trap either a kink or antikink at an interface between two sine-Gordon media. This approximate analysis is validated by comparing with the solution of the partial differential equation and describes kink motion in the one-dimensional window junction. Using this, we analyze steady-state kink motion and derive values for the average speed in the one- and two-dimensional systems. Finally, we show how geometry and the coupling parameters can destabilize kink motion.     

Near integrability of kink lattice with higher order interactions

We make use of Manton's analytical method to investigate the force between kinks and anti-kinks at large distances in 1+1 dimensional field theory.The related potential has infinite order corrections of exponential pattern,and the coefficients for each order are determined.These coefficients can also be obtained by solving the equation of the fluctuations around the vacuum.At the lowest order,the kink lattice represents the Toda lattice.With higher order correction terms,the kink lattice can represent one kind of generic Toda lattice.With only two sites,the kink lattice is classically integrable.If the number of sites of the lattice is larger than two,the kink lattice is not integrable but is a near integrable system.We make use of Flaschka's variables to study the Lax pair of the kink lattice.These Flaschka's variables have interesting algebraic relations and non-integrability can be manifested.We also discuss the higher Hamiltonians for the deformed open Toda lattice,which has a similar result to the ordinary deformed Toda.