本文選題：哈密頓系統(tǒng)　切入點：KAM理論　出處：《東南大學(xué)》2015年博士論文

【摘要】：本文主要利用KAM理論、Brouwer度理論和Banach不動點理論研究了近可積哈密頓系統(tǒng)的預(yù)給頻率方向的不變環(huán)面的保持性、給定勢能的非線性Schrodinger方程在Dirichlet邊界條件下的擬周期解的存在性、具有退化平衡點的周期系統(tǒng)在小擾動下周期解的存在性,以及二維線性擬周期系統(tǒng)在無非退化條件下的可約化性.主要內(nèi)容如下：第一章著重介紹哈密頓系統(tǒng)以及辛幾何的基礎(chǔ)概念以及重要理論,并給出了KAME理論的具體陳述.此外,介紹了有關(guān)周期系統(tǒng)、擬周期系統(tǒng)的可約化性的研究進(jìn)展.第一章的結(jié)尾處,大致陳述了本文的主要工作,并闡釋了創(chuàng)新點.第二章利用改進(jìn)的KAM迭代框架得到了近可積哈密頓系統(tǒng)在滿足Bruno非退化條件時不變環(huán)面的存在性,頻率沿著預(yù)給丟番頻率的方向.此外,這族不變環(huán)面構(gòu)成了單參數(shù)的解析族.第三章研究了實解析的哈密頓系統(tǒng)在無非退化條件下橢圓型不變環(huán)面的擾動.假設(shè)頻率ωo和Ω0滿足下列非共振條件：這里,Ak=1+|k|τ,τ≥n+1.并且,頻率映射ω(ξ)在ω0∈D處具有非零的Brouwer度.本文通過引入外部參數(shù)來調(diào)節(jié)KAM步驟中頻率的漂移以及處理切向頻率與法向頻率的共振.最后拓?fù)涠鹊募僭O(shè)使得參數(shù)化的系統(tǒng)的結(jié)果能應(yīng)用到原系統(tǒng)上.在第四章中,考慮了一類給定解析勢函數(shù)的Dirichlet邊界條件下的非線性Schrodinger方程.本文應(yīng)用了Sturm-Liouville算子特征值的“局部化”性質(zhì)以及一些技巧得到了Birkhoff規(guī)范型,從而給出了參數(shù)化的無窮維哈密頓系統(tǒng).最后KAM理論保證了系統(tǒng)有解析的線性穩(wěn)定的擬周期解.第五章研究一類高維的具有退化平衡點的周期系統(tǒng)的可約化性.考慮n-維實解析的周期系統(tǒng)的擾動：其中f(x,t)是小擾動,N(x)=(x12l1+1,...,xn2ln+1)T,l1,...,ln是非負(fù)整數(shù)h=(h1,...,hn)T表示高階項,滿足hj=O(xj2lj+2),x→0,j=1,...,n本文引入外部參數(shù)并應(yīng)用Banach空間不動點定理,將引入外部參數(shù)后的系統(tǒng)約化成一個具有零平衡點的周期系統(tǒng).最后,利用拓?fù)涠壤碚摰玫搅嗽到y(tǒng)的周期解.在第六章中,研究了二維線性擬周期系統(tǒng)在沒有非退化條件下的約化問題.設(shè)λ1和λ2是A的特征值.假設(shè)(ω,λ1,A2)滿足非共振條件其中,α00,τr-1.則或者對所有的參數(shù)ε∈(0,ε0)系統(tǒng)是可約化的；或者存在一個正Lebesgue測度的Cantor子集E (?)(0,ε0)使得系統(tǒng)對S∈E是可約化的.此外,本文給出了一類特殊的高維線性擬周期系統(tǒng)以及一類特殊的高維非線性擬周期系統(tǒng)在無非退化條件下的約化結(jié)果,作為對二維情形的應(yīng)用.
[Abstract]:In this paper, by using KAM's degree theory and Banach's fixed point theory, we study the preservation of the invariant torus in the pregiven frequency direction of the near integrable Hamiltonian system, and the existence of quasi periodic solutions of the nonlinear Schrodinger equation with given potential energy under the Dirichlet boundary condition. The existence of solutions for periodic systems with degenerate equilibrium points in the next week with small perturbations, The main contents are as follows: in Chapter 1, the basic concepts and important theories of Hamiltonian system and symplectic geometry are introduced, and the specific statements of KAME theory are given. The research progress on the reducibility of periodic systems and quasi periodic systems is introduced. At the end of chapter 1, the main work of this paper is briefly described. In chapter 2, we use the improved KAM iterative framework to obtain the existence of the invariant torus of the near-integrable Hamiltonian system when it satisfies the Bruno nondegenerate condition, and the frequency is in the direction of pregiving the diopsided frequency. In chapter 3, we study the perturbation of elliptic invariant torus of real analytic Hamiltonian systems under nondegenerate conditions. We assume that the frequencies 蠅 o and 惟 0 satisfy the following non-resonance conditions:. K 蟿, 蟿 鈮，

本文編號：1676310