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Integrable peakon systems with weak kink and kink-peakon interactional solutions

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Frontiers of Mathematics in China 2013年第5期8卷 1185-1196页

作者： Zhijun QIAO Baoqiang XIA Department of Mathematics University of Texas-Pan American Edinburg TX 78541 USA School of Mathematical Sciences Jiangsu Normal University Xuzhou 221116 China

We report two integrable peakon systems that have weak kink and kink-peakon interactional solutions. Both peakon systems are guaranteed integrable through providing their Lax pairs. The peakon and multi-peakon solutions of both equations are studied. In particular, the two-peakon dynamic systems are explicitly presented and their collisions are investigated. The weak kink solution is studied, and more interesting, the kink-peakon interactional solutions are proposed for the first time.

关键词： Integrable system Lax pair peakon weak kink kink-peakon

Compacton, peakon, and Foldon Structures in the （2＋1）-DimensionalNizhnik-Novikov-Veselov Equation

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Communications in Theoretical Physics 2004年第1期41卷 7-14页

作者： ZHANGJie-Fang MENGJian-Ping WUFeng-Min SIJian-Qing InstituteofNonlinearPhysics ZhejiangNormalUniversityJinhua321004ChinaDepartmentofMathematicalScienceLoughboroughUniversityLoughboroughLeicestershireLEll3TUUKInstituteofAppliedMathematicsandMechanics InstituteofNonlinearPhysics ZhejiangNormalUniversityJinhua321004China DepartmentofPhysics ZhejiangUniversityofTechnologyHangzhou310014China

By the use of the extended homogenous balance method,the B(?)cklund transformation for a (2+1)- dimensional integrable model,the(2+1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation,is obtained,and then the NNV equation is transformed into three equations of linear,bilinear,and tri-linear forms,*** the above three equations,a rather general variable separation solution of the model is *** novel class localized structures of the model are founded by the entrance of two variable-separated arbitrary functions.

关键词： compacton peakon foldon Nizhnik-Novikov-Veselov equation

GeometricNumerical Integration for peakon b-Family Equations

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Communications in Computational Physics 2016年第1期19卷 24-52页

作者： Wenjun Cai Yajuan Sun Yushun Wang Key Laboratory of Computational Geodynamics University of Chinese Academy of SciencesBeijing 100049China LSEC Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China Jiangsu Provincial Key Laboratory for NSLSCS School of Mathematical SciencesNanjing Normal University Nanjing 210023China

In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi *** two equations are in the family of integrable peakon equations,and both have very rich geometric *** on these geometric structures,we construct the geometric numerical integrators for simulating their soliton *** Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi *** using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two *** illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition *** the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.

关键词： Symplectic integrator splitting method WENO scheme multisymplectic integrator peakon shockpeakon

Interaction between compacton and anticompacton,peakon and antipeakon in （2 ＋ 1）-dimensional spaces

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Chinese Physics B 2003年第10期12卷 1166-1171页

作者：韩平张解放孟剑平 DepartmentofPhysics ZhejiangOceanUniversityZhoushan316004China ShanghaiInstituteofAppliedMathematicsandMechanics ShanghaiUniversityShanghai200072China InstituteofNonlinearPhysics ZhejiangNormalUniversityJinhua321004China

Starting from the variable separation solution obtained by using the extended homogenous balance method, a class of novel localized coherent structures such as the multi-peakon-antipeakons solution and the multi-compacton-anticompactons solution of the (2+1)-dimensional dispersive long wave equation are found by selecting appropriate functions. These new structures exhibit some novel interaction features that are different from one of the known results. Their interaction behaviour is very similar to the completely elastic collisions between two classical particles.

关键词： homogeneous balance method compacton peakon (2+1)-dimensional dispersive long wave equation

四次非线性Camassa-Holm方程peakon和周期peakon的轨道稳定性

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四次非线性Camassa-Holm方程peakon和周期peakon的轨道稳定性

作者：邓同杰桂林电子科技大学

学位级别：硕士

本文主要证明了四次非线性Camassa-Holm方程的peakon和周期peakon的轨道稳定性.首先,我们验证所获得的peakon和周期peakon解是方程的弱解.然后根据方程的守恒量并构造合适的辅助函数,将轨道稳定性的证明过程转化为守恒量和辅助函数的相... 详细信息

本文主要证明了四次非线性Camassa-Holm方程的peakon和周期peakon的轨道稳定性.首先,我们验证所获得的peakon和周期peakon解是方程的弱解.然后根据方程的守恒量并构造合适的辅助函数,将轨道稳定性的证明过程转化为守恒量和辅助函数的相关研究,最终可证明方程的peakon和周期peakon解均是轨道稳定的.全文共分为五章,每章的主要内容描述如下:第一章是绪论,主要介绍四次非线性Camassa-Holm方程轨道稳定性的研究背景,研究现状以及本文主要工作.第二章,主要介绍了双曲函数和动力系统理论的一些相关知识,用来作为预备知识.第三章,验证四次非线性Camassa-Holm方程的peakon解和周期peakon解均是方程的弱解.第四章,证明四次非线性Camassa-Holm方程的peakon和周期peakon是轨道稳定的.首先考察方程轨道稳定性与三个守恒量的关系,根据守恒量相关性质,构造合适的辅助函数,利用辅助函数完成轨道稳定性的证明.第五章,总结本文的研究结果,并预估将来可能进行的研究工作.

关键词：四次非线性Camassa-Holm方程 peakon 周期peakon 守恒量轨道稳定性

一个mCH方程的椭圆周期peakon的轨道稳定性

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一个mCH方程的椭圆周期peakon的轨道稳定性

作者：陆鑫慧桂林电子科技大学

学位级别：硕士

本文利用平面动力系统分支理论研究了修正的Camassa-Holm(mCH)方程行波解的具体参数表达式以及椭圆周期peakon解的轨道稳定性.首先,我们将mCH方程转化为一个平面系统,然后得到了该系统的首次积分和代数曲线.利用首次积分和代数曲线,我... 详细信息

本文利用平面动力系统分支理论研究了修正的Camassa-Holm(mCH)方程行波解的具体参数表达式以及椭圆周期peakon解的轨道稳定性.首先,我们将mCH方程转化为一个平面系统,然后得到了该系统的首次积分和代数曲线.利用首次积分和代数曲线,我们得到了由双曲函数表示的新的peakon解.除此之外,我们还得到了由椭圆函数和三角函数表示的新的周期peakon解.对于该方程椭圆周期peakon解的轨道稳定性,我们主要运用椭圆函数和椭圆积分理论进行计算,利用方程的不变性和控制解的极值,证明了该方程的椭圆周期peakon解在能量空间的小扰动下是轨道稳定的.据我们所知,这是第一个关于椭圆周期peakon解轨道稳定性的结果.本文将分为五个部分,它的主要框架安排如下:第一章,我们介绍了mCH方程的研究背景、研究现状以及本文的主要研究内容.第二章,给出了非线性波方程行波解的分支理论、椭圆积分与椭圆函数等相关知识.第三章,应用动力系统分支理论讨论了mCH方程的行波解.首先将该方程转换成一个平面系统,然后,获得了这个方程的光滑周期波解、孤立波解、以及新的peakon解和周期peakon解的具体参数表达式.第四章,应用椭圆函数和椭圆积分理论对mCH方程进行计算,证明了该方程的椭圆周期peakon解是轨道稳定的.第五章,总结本文研究内容,并展望了今后有待研究的方向及课题.

关键词：修正的Camassa-Holm方程 peakon 周期peakon 轨道稳定椭圆函数和椭圆积分

Interactions Among peakons, Dromions, and Compactons for a （2＋1）-Dimensional Soliton System

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Communications in Theoretical Physics 2004年第4期41卷 513-520页

作者： ZHENGChun-Long DepartmentofPhysics ZhejiangLishuiNormalCollegeLishui323000China

Starting from the known variable separation excitations of a(2 + 1)-dimensional generalized Ablowitz-Kaup-Newell-Segur system,rich coherent structures can be *** interactions among different types of solitary waves like peakons,dromions,and compactons are investigated and some novel features or interesting behaviors are *** results show that the interactions for peakon-dromion,compacton-dromion,and peakon-compacton may be completely nonelastic or completely elastic.

关键词： interaction peakon dromion compacton

Orbital Stability of peakons for a Generalized Camassa-Holm Equation

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Journal of Applied Mathematics and Physics 2019年第10期7卷 2200-2211页

作者： Xinhui Lu Aiyong Chen Tongjie Deng Department of Mathematics and Computing Science Guilin University of Electronic Technology Guilin China Department of Mathematics Hunan First Normal University Changsha China

We investigate the orbital stability of the peakons for a generalized Camassa-Holm equation (gCH). Using variable transformation, a planar dynamical system is obtained from the gCH equation. It is shown that the planar system has two heteroclinic cycles which correspond two peakon solutions. We then prove that the peakons for the gCH equation are orbitally stable by using the method of Constantin and Strauss.

关键词： Camassa-Holm Equation peakon Heteroclinic Orbit Orbital Stability

On Camassa-Holm Equation with Self-Consistent Sources and Its Solutions

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Communications in Theoretical Physics 2010年第3期53卷 403-412页

作者：黄晔辉姚玉芹曾云波 Department of Mathematical Science Tsinghua University Department of Applied Mathematics China Agricultural University

Regarded as the integrable generalization of Camassa-Holm (CH) equation, the CH equation with selfconsistent sources (CHESCS) is derived. The Lax representation of the CHESCS is presented. The conservation laws for CHESCS are constructed. The peakon solution, N-soliton, N-cuspon, N-positon, and N-negaton solutions of CHESCS are obtained by using Darboux transformation and the method of variation of constants.

关键词： Camassa Holm equation with self-consistent sources Lax representation conservation laws peakon soliton positon negaton