无界系统的KAM理论和Birkhoff正规形理论及其应用

项目名称： 无界系统的KAM理论和Birkhoff正规形理论及其应用

项目编号： No.11201147

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 张静

作者单位： 华东师范大学

项目金额： 22万元

中文摘要： 很多科学问题需要用微分方程描述。这些方程解的存在性和有效稳定性是人们关心的热点问题。而KAM理论和Birkhoff正规形理论是刻画微分方程解的存在性和有效稳定性的重要工具，并得到了许多方程解的存在性和有效稳定性。但实际应用中很多方程只能转化为无界近可积反转系统或无界近可积哈密顿系统。到目前为止无界反转系统的KAM理论尚不完善，无界哈密顿系统的Birkhoff正规形理论尚未建立。 本项目将主要研究两个课题，(1)无界反转系统的KAM理论；(2)无界哈密顿系统的Birkhoff正规形理论。在(1)中，我们力争得到法向频率满足弱渐近逼近条件时无界反转系统的KAM理论并应用该理论得到非线性项带有关于时间变量导数的波方程的小振幅拟周期解的存在性。在课题(2)中，我们将重点研究构造有界辛变换来得到无界哈密顿系统的Birkhoff正规形理论，以及通过该理论研究非线性项带导数的薛定谔方程解的有效稳定性。

中文关键词： 无界；哈密顿系统；Birkhoff正规形；；

英文摘要： Many scientific problems can be described by differntial equations. The existence and the long time stabiltiy of solutions to these equations are hot research topics. The KAM theory and Birkhoff normal form theory are imporant tools to study the existence and the long time stability of solutions to these equations. One can show the existence and the long time stability of solutions to some differential equations by the KAM theory and Birkhoff normal form theory. But a wide class of these differential equations can be transformed as unbounded nearly integrable Hamiltonian systems and reversible systems which have good mathematical structures and symmetries, but research on the KAM theory and Birkhoff normal form theory of these systems is still lacking. In this project, we will mainly study two topics,(1)studying the KAM theory of unbounded infinite dimensional nearly integrable reversible system. (2)studying the Birkhoff normal form theory of Hamiltonian systems. In topic (1), we will try to get the KAM theory of reversible systems with frequencies satisfying only weak asymptotic conditions, and we will apply this result to a class of wave equations whose nonlinear term contains derivative with respect to the time variable and prove the existence of quasi-periodic solutions with small amplitude. In topic (2), we

英文关键词： unbounded；Hamiltonian；Birkhoff normal form；；