【作者】 孙聪；

【导师】 冀书关；

【作者基本信息】 吉林大学 ， 基础数学， 2017， 博士

【摘要】 本文主要是在总结前人工作的基础上,对一类Zakharov方程,Klein-Gordon-Zakharov方程、以及Zakharov-Rubenchik方程精确周期解的求法以及这些周期解的周期性质进行了研究.同时,我们还研究了(n + 1)维耦合的非线性Klein-Gordon方程组精确周期解的求法及其轨道稳定性.首先,本文受文献[1]的启发,结合Jacobian椭圆函数方法,我们求出了 Zakharov方程,Klein-Gordon-Zakharov方程及Zakharov-Rubenchik方程精确周期解.同时,我们分别证明出在相应波速c的某个邻域内,Zakharov方程,Klein-Gordon-Zakharov方程周期解的周期是波速c的函数.此外,我们还得到,Zakharov-Rubenchik方程周期解的周期是波速c的函数.其次,我们还研究了(n + 1)维耦合的非线性Klein-Gordon方程组精确周期解的求法.通过Jacobian椭圆函数方法,我们获得了此方程组的一类精确周期解.同时,我们还证明出在波速c的某个邻域内,上述精确周期解的周期是波速c的函数.最后,我们采用由M.Grillakis,J.Shatah和W.Strauss等人[2]提出的轨道稳定性理论,研究了(n + 1)维耦合的非线性Klein-Gordon方程组一类精确周期解的轨道稳定性.本论文共分为五章:第一章为绪论,主要是介绍非线性科学及孤立子的发展概况、求解非线性发展方程的一些主要方法以及非线性发展方程解的稳定性研究现状.最后陈述了本论文的主要内容.第二章为预备知识.在第三章,我们求出了 Zakharov方程Klein-Gordon-Zakharov 方程以及 Zakharov-Rubenchik 方程的精确周期解.同时,我们分别证明出在波速c的某个邻域内,Zakharov方程,Klein-Gordon-Zakharov方程精确周期解的周期是波速c的函数.此外,我们通过分析方法得到,Zakharov-Rubenchik方程精确周期解的周期是波速c的函数.在第四章,我们研究了(n + 1)维耦合的非线性Klein-Gordon方程组首先,我们求出了此方程组的一类精确周期解同时,我们还证明出在波速c的某个邻域内,上述精确周期解的周期也是波速c的函数.最后,证明出上述解具有轨道稳定性.第五章是对本文的总结及未来工作的一些展望.更多还原

【Abstract】 On the basis of summarizing predecessors’ work,we study a method to obtain exact periodic solutions for a series of Zakharov equation,including Klein-Gordon-Zakharov equation and Zakharov-Rubenchik equation.And the properties of period of those peri-odic solutions were also studied in this paper.Meanwhile,we consider(n+1)dimension coupled nonlinear Klein-Gordon equations.We mainly study how to obtain a series of exact periodic solutions and orbital stability of this periodic solution in this system.Firstly,inspired by[1],in combination with Jacobian ellipse function method,we get the exact periodic solutions for a class Zakharov equation,Klein-Gordon-Zakharov equation and Zakharov-Rubenchik equation.Moreover,in a neighborhood of the relevant wave velocity c,we prove that the periods of the periodic solutions of Zakhaxov equation and Klein-Gordon-Zakharov equation are function of wave velocity c,respectively.Next,we can obtain that the period of the periodic solution of Zakharov-Rubenchik equation is also function of wave velocity c.Secondly,we consider(n + 1)dimension coupled nonlinear Klein-Gordon equations.By Jacobian ellipse function method,we acquire the periodic solution of this system.Furthermore,we also prove that the period of the periodic solutions of this system is function of wave velocity c.Lastly,by applying orbital stability theory,which was established by M.Grillakis,J.Shatah and W.Strauss[2],we study orbital stability of this exact periodic solution of(n+1)dimension coupled nonlinear Klein-Gordon equations.Our paper is divided into five chapters.In the first chapter,we mainly introduced the development of nonlinear science and soliton,some significant methods of solving nonlinear evolution equations,historical background and research development stability of nonlinear evolution equations.At last,we give a brief introduction of the main content of this paper.In the second chapter,we state some definitions and some basic facts.In the third chapter,we consider the following equations,Zakharov equation Klein-Gordon-Zakharov equation and Zakharov-Rubenchik equation By Jacobian ellipse function method,we obtain exact periodic solutions of above these equations.Furthermore,in a neighborhood of the wave velocity c,we prove that the peri-ods of the periodic solutions of Zakharov equation and Klein-Gordon-Zakharov equation are the function of wave velocity c,respectively.Next,by a simple analysis method,we can obtain that the period of the periodic solution of Zakharov-Rubenchik equation is the function of wave velocity c.In the fourth chapter,we discuss the following(n+1)dimension coupled nonlinearKlein-Gordon equations At first,we obtain exact periodic solution Moreover,we also prove that the period of the periodic solutions of this system is function of wave velocity c in its neighborhood.Next,we prove that this solution is orbital stable.In the fifth chapter,we summarized the whole paper,and propose some problems for the future research and explore.更多还原