KAM理论及其在微分方程中的应用

【作者】 丛洪滋；

【导师】 袁小平；

【作者基本信息】 复旦大学 ， 基础数学， 2009， 博士

【摘要】 本文主要应用有限维KAM理论证明了一类拟周期系数的Lotka-Volterra模型存在正拟周期解,以及应用无穷维KAM理论证明了高维Ginzburg-Landau方程和一维带有非线性项|u|^2pu的Ginzburg-Landau方程存在2维不变环面,即2维拟周期解。全文共分四章。第一章,主要介绍了KAM理论的背景,意义,国内外研究现状以及本文的主要工作。第二章,证明了一类拟周期系数的Lotka-Volterra模型存在正拟周期解。Lotka-Volterra模型是生态系统中一个很重要的模型,它成功地描述了两个生物种群在各种条件下种群数量随时间变化的情况,如分别是捕食者和食物,两种群竞争,两种群合作,寄生和寄主等。我们假设环境因素是拟周期变化的,这样更能符合现实的生态系统,并用KAM迭代证明该系统具有正拟周期解。第三章,考虑高维Ginzburg-Landau方程。Ginzburg-Landau方程具有十分丰富的物理背景和内涵。它可以描述Banerd对流,Tayor-Couette流,平面Poiseuille流以及化学湍流等问题,在非平衡态的相变,超导和超流理论中均有应用。我们用一个修正的KAM定理证明了高维Ginzburg-Landau方程2维不变环面的存在性。第四章,证明了一维带有非线性项|u|^2pu的Ginzburg-Landau方程存在2维不变环面。更多还原

【Abstract】 We are mainly interested in KAM theory,and apply KAM theory to prove that there exists a positive quasi-periodic solution for Lotka-Volterra system with quasiperiodic coefficients and there exist 2-dimensional invariant tori both for the cubic complex Ginzburg-Landau equation with higher spatial dimension and 1D Ginzburg-Landau equation with nonlinearity |u|^2pu.This paper is divided into four parts.In Chapter 1,we introduce the historical background,some recent results of KAM theory obtained in the literature and our main work in this paper.In Chapter 2,it is proven that the existence of quasi-periodic solutions for Lotka-Volterra system with quasi-periodic coefficients.Lotka-Volterra system plays an important role in the research field of ecosystem.It includes predator-prey system,competitive system,cooperative systems and host-parasite systems.Many work mostly discuss periodic Lotka-Volterra system.In fact,quasi-periodic system describes our world more realistic and more accurate than periodic one.We use KAM iteration to prove the existence of quasi-periodic solutions for the system with quasi-periodic coefficients.In Chapter 3,we consider the cubic complex Ginzburg-Landau equation with higher spatial dimension.Ginzburg-Landau equation is an important equation with very rich physical and mathematical background.It can be described Banerd convection, Tayor-Couette flow,Plane Poiseuille flow,as well as issues such as chemical turbulence, and also applied to non-equilibrium phase transition,theory of superconductivity and superfluidity.It is proven that the existence of 2-dimensional invariant tori for this equation by a modified KAM theorem.In Chapter 4,it is proven that the existence of 2-dimensional invariant tori for 1D Ginzburg-Landau equation with nonlinearity |u|^2pu by a modified KAM theorem.更多还原