变系数Whitham-Broer-Kaup方程组的对称、约化及精确解

利用修正的Clarkson-Kruskal直接法对变系数Whitham-Broer-Kaup(VCWBK)方程组进行等价转化,建立了VCWBK方程组与常系数WBK方程组解之间的关系,并得到了常系数WBK方程组的一些对称和相似约化.借助辅助函数法得到了VCWBK方程组的一些新精确解,包括有理函数解、双曲函数的解、三角函数解和Jacobi椭圆函数解.     

(2+1)维Camassa-Holm方程的相似约化与解析解

将Clarkson和Krushal引入的直接约化方法推广并应用到(2+1)维CamassaHolm方程组,获得了该方程的若干相似约化和解析解,其中包括Logistic方程和Bernoulli方程.约化结果得到了Peakon解、Cuspon解和关于时间t的奇异解.该方法也适用于其他有重要物理背景的非线性演化方程 关键词： Camassa-Homl方程 相似约化 直接方法 解析解     

两类非线性方程的精确解

利用行波约化方法，并借助于一维立方非线性Klein-Gordon方程的精确解，求出了（1+1）维Zakharov方程组、变系数Korteweg-de Vries方程的一些精确解- 关键词： 行波约化方法 一维立方非线性Klein-Gordon方程 （1+1）维Zakharov方程组 变系数Korteweg-de Vries方程     

Kaup-Kupershmidt方程的非局域对称及新的精确解

利用机械化算法得到了Kaup-Kupershmidt方程的非局域对称、约化,通过解约化方程得到了该方程的一些新的精确解.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

基于直线段对应的相机位姿估计直接最小二乘法

为了实现高精度的基于直线段对应的相机位姿估计,提出一种直接最小二乘法。通过提出并利用一种直线段之间的距离测度,将原问题转化为最小化一个姿态旋转矩阵的二次目标函数,该距离测度综合考虑了线段的端点距离、中点距离、夹角和线段的长度,通过旋转矩阵的CGR参数表达获得一个修正的目标函数,此修正目标函数的最优解条件组成了一个三元三次方程组,利用代数多项式方法在不需要迭代的情况下直接求解这个方程组,从而得到了旋转矩阵的全局最优解。该算法的计算复杂度为O(n)。仿真和真实实验验证了该方法的有效性和高精度。     

非线性Klein-Gordon方程新的精确解

将行波变换替换为更一般的函数变换,推广了修正的Jacobi椭圆函数展开方法.给出了非线性 Klein-Gordon方程新的周期解.当模m→1或m→0时,这些解退化成相应的孤立波解、三 角函数解和奇异的行波解.对于某些非线性方程,在一定条件下一般变换退化为行波约化. 关键词： Jacobi椭圆函数 非线性发展方程 精确解     

变形对称双阱势的束缚态

从第一性原理出发，得到了变形对称双阱势模型的Schroedinger方程束缚态的精确解。应用得到的精确解，给出了对称双阱势和修正Roeschl—Tdler势的能谱和波函数。     

修正的BBM方程的一些新的精确解

用修正影射法解修正的BBM(mBBM)方程,得到了一些新的精确解.这个方法的优点在于:①待定函数f(ξ)的指数i的范围从N扩大到-N;②可以不必给出函数f(ξ)的具体表达式求解方程,这样便于寻找更多的解.本文就是利用了这一特点,选择合适的参数,得到一些mBBM方程新的精确解.我们相信;这个方法还可以推广到含有更多维和更高阶的求导项的方程.     

耦合KdV方程的约化及求解

利用Clarkson和Kruskal(CK)直接方法,对耦合KdV方程进行相似约化,同时从李群出发对该约化方程作了群论解释.进一步地,借助Ablowitz-Ramani-Segur(ARS)算法对耦合方程展开Painlevé测试,找到了3个Painlevé可积模型.最后通过形变映射法,求得耦合KdV方程的准确解析解. 关键词： 耦合KdV方程 CK直接法 Painlevé分析法 准确解析解     

Peakons and compactons on the background of periodic wave

In this paper, the extended tanh-function method (ETM) based on the mapping equation is further improved by generalizing the Riccati equation. The new variable separation solutions of the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system are derived. From the periodic wave solution and by selecting appropriate functions, the evolutional behaviours of peakons and compactons on the background of Jacobian elliptic wave are investigated.     

Lump and rogue wave solutions for the Broer-Kaup-Kupershmidt system

This paper retrieves lump solution for (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) system by the aid of Hirota bilinear method (HBM). We also obtain rogue wave solutions formed by the interaction of lump solution and a pair of stripe solitons. The dynamics of these solutions are figured out graphically by selecting suitable values to parameters.     

New exact excitations and soliton fission and fusion for the （2＋1）-dimensional Broer-Kaup-Kupershmidt system

With the help of an extended mapping approach, a series of new types of exact excitations with two arbitrary functions of the (2 1)-dimensional Broer-Kaup-Kupershmidt (BKK) system is derived. Based on the derived solitary wave excitation, some specific soliton fission and fusion solutions of the higher-dimensional BKK system are also obtained.     

Approximate Homotopy Direct Reduction Method: Infinite Series Reductions to Perturbed mKdV Equations

An approximate homotopy direct reduction method is proposed and applied to two perturbed modified Korteweg- de Vries （mKdV） equations with fourth-order dispersion and second-order dissipation. The similarity reduction equations are derived to arbitrary orders. The method is valid not only for single soliton solutions but also for the Painlevd Ⅱ waves and periodic waves expressed by Jacobi elliptic functions for both fourth-order dispersion and second-order dissipation. The method is also valid for strong perturbations.     

Symmetry Reduction of the (2+1)-Dimensional Modified Dispersive Water-Wave System

Using the standard truncated Painlevé expansion, the residual symmetry of the (2+1)-dimensional modified dispersive water-wave system is localized in the properly prolonged system with the Lie point symmetry vector. Some different transformation invariances are derived by utilizing the obtained symmetries. The symmetries of the system are also derived through the Clarkson-Kruskal direct method, and several types of explicit reduction solutions relate to the trigonometric or the hyperbolic functions are obtained. Finally, some special solitons are depicted from one of the solutions.     

Quasi-Periodic Structures Based on Symmetrical Lucas Function of(2+1)-Dimensional Modified Dispersive Water-Wave System

By introducing the Lucas--Riccati method and a linear variable separationmethod, new variable separation solutions with arbitrary functions arederived for a (2+1)-dimensional modified dispersive water-wave system. Themain idea of this method is to express the solutions of this system aspolynomials in the solution of the Riccati equation that the symmetricalLucas functions satisfy. From the variable separation solution and byselecting appropriate functions, some novel Jacobian elliptic wave structurewith variable modulus and their interactions with dromions and peakons are investigated.     

Application of the (G^{\prime}/G)-expansion method for the Burgers, Burgers–Huxley and modified Burgers–KdV equations

In this work, we present travelling wave solutions for the Burgers, Burgers–Huxley and modified Burgers–KdV equations. The (G′/G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.     

New Similarity Reductions and Compacton Solutions for Boussinesq-Like Equations with Fully Nonlinear Dispersion

In this paper, similarity reductions of Boussinesq-like equations with nonlinear dispersion (simply called B(m,n) equations) u_tt=(uⁿ)_xx+(u^m)_xxxx, which is a generalized model of Boussinesq equation u_tt=(u²)_xx+u_xxxx and modified Bousinesq equation u_tt=(u³)_xx+u_xxxx, are considered by using the direct reduction method. As a result, several new types of similarity reductions are found. Based on the reduction equations and some simple transformations, we obtain the solitary wave solutions and compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they re-emerge with the same coherent shape) of B(1,n) equations and B(m,m) equations, respectively.     

The extended Fan's sub-equation method and its application to KdV–MKdV,BKK and variant Boussinesq equations

An extended Fan's sub-equation method is used for constructing exact travelling wave solutions of nonlinear partial differential equations (NLPDEs). The key idea of this method is to take full advantage of the general elliptic equation involving five parameters which has more new solutions and whose degeneracies can lead to special sub-equations involving three parameters. More new solutions are obtained for KdV–MKdV, Broer–Kaup–Kupershmidt (BKK) and variant Boussinesq equations. Then we present a technique which not only gives us a clear relation among this general elliptic equation and other sub-equations involving three parameters (Riccati equation, first kind elliptic equation, auxiliary ordinary equation, generalized Riccati equation and so on), but also provides an approach to construct new exact solutions to NLPDEs.     

mKdV方程的双扭结单孤子及其稳定性研究

基于双曲函数法的思想,通过选择新的展开函数,得到了modified Korteweg-de Vries(mKdV)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mKdV方程的扭结状或钟状孤波解.文中对双扭结型孤子解的稳定性进行了数值研究,结果表明:在长波和短波简谐波扰动、钟型孤立波扰动与随机扰动下,该孤子均稳定.