KAM理论与高维哈密顿偏微分方程

发布时间：2018-01-26 15:41

本文关键词： KAM Birkhoff标准型梁方程 Schrodinger 方程组 Toplitz-Lipschitz性质拟周期解　出处：《南京大学》2017年博士论文　论文类型：学位论文

【摘要】：在本论文中,我们主要研究无穷维KAM理论及其在几类高维哈密顿偏微分方程中的应用。我们主要致力于两类重要的哈密顿偏微分方程:二维环面上的完全共振梁方程,和高维环面上的Schrodinger方程组。通过处理方程的Birkhoff标准型,以及建立抽象无穷维KAM定理,我们证明,在测度意义下,相应的方程存在大量小振幅的拟周期解。在第一章中,我们一方面陈述经典的动力系统与KAM理论,另一方面介绍人们对哈密顿偏微分方程的主要研究兴趣,并对近年来的研究成果做简要回顾。在第二章中,我们研究周期边界条件下二维完全共振梁方程的拟周期解的存在性。我们先对方程的Birkhoff标准型进行处理,通过一个标准的辛变换,我们可以消去标准型中的非共振项。然后利用容许集的特殊结构和零动量条件,我们可以证明在变换后的标准型中,每个整点对应的法变量至多出现在一个不可积项中。这样,我们得到一个分块对角的标准型结构,其中每个分块的阶数至多是2×2的。并且,这个标准型是依赖于角变量的。然后我们应用抽象KAM定理,进行无穷多步迭代并证明这个迭代序列收敛,从而得到拟周期解的存在性。在KAM迭代过程中,参数由解的振幅提供。需要说明的是,由于我们的标准型依赖于角变量,因此我们最终不能得到解的线性稳定性。在第三章中,我们研究非线性高维Schrodinger方程组拟周期解的存在性。我们利用人为施加的傅里叶乘子来提供KAM迭代中需要的参数。首先,我们还是处理标准型,这里由于方程组的耦合性,标准型中仍会出现耦合的不可积项,从而我们的标准型仍然会是2×2分块的分块对角结构。然后我们应用抽象KAM定理,进行无穷多步迭代并证明这个迭代序列收敛,从而得到拟周期解的存在性。这里由于非线性项正则性的缺失,我们需要应用Toplitz-Lipschitz性质来进行测度估计。
[Abstract]:In this thesis. We mainly study the infinite dimensional KAM theory and its applications to several kinds of high dimensional Hamiltonian partial differential equations. We mainly focus on two important classes of Hamiltonian partial differential equations: the complete resonance beam equation on the two-dimensional ring surface. By dealing with the Birkhoff canonical form of the equation and establishing the abstract infinite dimensional KAM theorem, we prove that in the sense of measure. In the first chapter, we present the classical dynamical system and KAM theory, on the other hand, we introduce the main research interests of Hamiltonian partial differential equations. And the research results in recent years are briefly reviewed. In the second chapter. We study the existence of quasi periodic solutions for two-dimensional fully resonant beam equations under periodic boundary conditions. We first deal with the Birkhoff canonical form of the equation and adopt a standard symplectic transformation. We can eliminate the non-resonance term in the canonical form, and then by using the special structure of the admissible set and the zero momentum condition, we can prove that in the transformed canonical form. The normal variable corresponding to each whole point appears at most in a non-integrable term. In this way, we obtain a block diagonal canonical form structure, where the order of each block is at most 2 脳 2, and. This canonical form is dependent on angular variables. Then we apply abstract KAM theorem to infinite iterations and prove the convergence of the iterative sequence. In the KAM iteration process, the parameters are provided by the amplitude of the solution. It is necessary to note that our canonical form depends on angular variables. Therefore, we cannot finally obtain the linear stability of the solution. In Chapter 3. We study the existence of quasi periodic solutions for nonlinear high dimensional Schrodinger equations. We use the artificially imposed Fourier multipliers to provide the necessary parameters in the KAM iteration. We are still dealing with the canonical form, where the coupled non-integrable term will still appear in the standard form due to the coupling of the equations. So our normal form will still be a block diagonal structure of 2 脳 2 blocks. Then we apply abstract KAM theorem to infinitely many iterations and prove the convergence of this iterative sequence. In this paper, we obtain the existence of quasi periodic solutions. Because of the lack of the regularity of nonlinear terms, we need to use the Toplitz-Lipschitz property to estimate the measure.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.2

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