几类非线性波的轨道稳定性与不稳定性

发布时间：2018-06-13 01:39

本文选题：非线性色散偏微分方程 + 孤立波解　；参考：《广州大学》2016年博士论文

【摘要】：本文主要结合GSS方法、经典方法以及详细的谱分析研究了现代数学物理中出现的一些非线性色散偏微分方程(组)孤立波解、椭圆函数周期行波解的轨道稳定性和不稳定性.第一章主要介绍轨道稳定性的研究方法,以及某些非线性偏微分方程组的研究现状,并给出了本文的主要研究内容和目的.在第二章中,运用Grillakis等提出的轨道稳定性理论和详细的谱分析,证明了耦合组合KdV和MKdV方程具有零渐近值和非零渐近值的六种孤波的轨道稳定性和不稳定性.第三章首先证明耦合非线性波方程具有一固定正周期为L的dn型周期行波解.然后,利用Lam′e方程和Floquet理论给出某线性算子的谱性质,并结合Grillakis等提出的轨道稳定性理论,证明了周期为L的dn型周期行波解的轨道稳定性.在第四章中,运用Grillakis等提出的轨道稳定性理论和Lopes提出的谱分析方法,证明了耦合Klein-Gordon-Schr¨odinger方程组sech2型孤立波解的轨道稳定性.第五章和第六章分别证明了广义长短波方程组和广义Zakharov方程组具有固定周期L的dn型正周期波解光滑曲线的存在性,并运用Benjamin,Bona等提出的经典方法分别证明了在周期为L的扰动下,相应的周期波解是轨道稳定的.本文的工作不仅完善和补充了已有的稳定性结果,而且还将轨道稳定性的研究方法应用到具有非零渐近值的孤立波的轨道稳定性,以及广义非线性色散偏微分方程组的周期行波解的轨道稳定性中.
[Abstract]:This paper mainly studies the orbital stability and instability of solitary wave solutions of nonlinear dispersive partial differential equations (systems) and periodic traveling wave solutions of elliptic functions in modern mathematics and physics by using GSS method classical method and detailed spectral analysis. The first chapter mainly introduces the research methods of orbit stability and the research status of some nonlinear partial differential equations, and gives the main contents and purposes of this paper. In chapter 2, using the orbital stability theory and detailed spectral analysis proposed by Grillakis et al, we prove the orbital stability and instability of six solitary waves with zero asymptotic value and non-zero asymptotic value for coupled combined KDV and MKdV equations. In chapter 3, we first prove that the coupled nonlinear wave equation has a DN type periodic traveling wave solution with fixed positive period L. Then, the spectral properties of some linear operators are given by means of Lamber equation and Floquet theory, and the orbital stability of DN type periodic traveling wave solutions with periodic L is proved by combining with the orbital stability theory proposed by Grillakis et al. In chapter 4, the orbital stability of sech2 solitary wave solutions of coupled Klein-Gordon-Schr? odinger equations is proved by using Grillakis' orbital stability theory and Lopes' spectral analysis method. In chapter 5 and chapter 6, we prove the existence of smooth curves of DN type positive periodic wave solutions for generalized long-short wave equations and generalized Zakharov equations with fixed period L, respectively. By using the classical method proposed by Benjamin Bona and others, it is proved that the corresponding periodic wave solutions are orbital stable under the perturbation of the period L, respectively. The work of this paper not only perfects and complements the existing stability results, but also applies the research method of orbital stability to the orbital stability of solitary waves with nonzero asymptotic value. And the orbital stability of periodic traveling wave solutions of generalized nonlinear dispersive partial differential equations.
【学位授予单位】：广州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.29

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