本文關鍵詞： KAM Birkhoff標準型 梁方程 Schrodinger 方程組 Toplitz-Lipschitz性質 擬周期解　出處：《南京大學》2017年博士論文　論文類型：學位論文

【摘要】：在本論文中,我們主要研究無窮維KAM理論及其在幾類高維哈密頓偏微分方程中的應用。我們主要致力于兩類重要的哈密頓偏微分方程:二維環(huán)面上的完全共振梁方程,和高維環(huán)面上的Schrodinger方程組。通過處理方程的Birkhoff標準型,以及建立抽象無窮維KAM定理,我們證明,在測度意義下,相應的方程存在大量小振幅的擬周期解。在第一章中,我們一方面陳述經(jīng)典的動力系統(tǒng)與KAM理論,另一方面介紹人們對哈密頓偏微分方程的主要研究興趣,并對近年來的研究成果做簡要回顧。在第二章中,我們研究周期邊界條件下二維完全共振梁方程的擬周期解的存在性。我們先對方程的Birkhoff標準型進行處理,通過一個標準的辛變換,我們可以消去標準型中的非共振項。然后利用容許集的特殊結構和零動量條件,我們可以證明在變換后的標準型中,每個整點對應的法變量至多出現(xiàn)在一個不可積項中。這樣,我們得到一個分塊對角的標準型結構,其中每個分塊的階數(shù)至多是2×2的。并且,這個標準型是依賴于角變量的。然后我們應用抽象KAM定理,進行無窮多步迭代并證明這個迭代序列收斂,從而得到擬周期解的存在性。在KAM迭代過程中,參數(shù)由解的振幅提供。需要說明的是,由于我們的標準型依賴于角變量,因此我們最終不能得到解的線性穩(wěn)定性。在第三章中,我們研究非線性高維Schrodinger方程組擬周期解的存在性。我們利用人為施加的傅里葉乘子來提供KAM迭代中需要的參數(shù)。首先,我們還是處理標準型,這里由于方程組的耦合性,標準型中仍會出現(xiàn)耦合的不可積項,從而我們的標準型仍然會是2×2分塊的分塊對角結構。然后我們應用抽象KAM定理,進行無窮多步迭代并證明這個迭代序列收斂,從而得到擬周期解的存在性。這里由于非線性項正則性的缺失,我們需要應用Toplitz-Lipschitz性質來進行測度估計。
[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.
【學位授予單位】：南京大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O175.2

【相似文獻】

相關期刊論文 前2條

1 梁建莉;湯龍坤;;復雜單擺的KAM理論[J];華僑大學學報(自然科學版);2011年02期

2 胡志興,管克英;陀螺儀運動的混沌與KAM理論[J];應用數(shù)學學報;2000年02期

相關博士學位論文 前2條

1 常晶;無窮維KAM理論在梁方程中的應用[D];吉林大學;2011年

2 吳健;KAM理論與哈密頓偏微分方程[D];南京大學;2013年

，

本文編號：1465980