概周期KAM理论及其应用

Since the late 1980s, the KAM theory for infinite-dimensional Hamiltonian systems makes a rapid advance, and a lot of achievements are obtained in terms of finite-dimensional invariant tori and time quasi-periodic solutions. They are not only in the treatment of bounded perturbations, but also in the treatment of unbounded perturbations, which can be applied to nonlinear partial differential equations containing spatial derivative in their nonlinearity. In contrast, there are much less results of KAM theory dealing with infinite-dimensional invariant tori and time almost-periodic solutions, and all of them are about bounded perturbations. Another important fact is that some basic problems are still open, such as the existence of infinite-dimensional KAM tori with polynomial decay. This project will focus on almost-periodic KAM theory of infinite-dimensional Hamiltonian systems. The first main goal is to establish an almost-periodic KAM theorem with slower or even polynomial decay tori, and apply it to one or more meaningful partial differential equations. The second main goal is the establishment of the almost-periodic KAM theorem for dealing with unbounded perturbations, and its applications to the Hamiltonian partial differential equations with their nonlinearity containing spatial derivative, including the proof of the existence of infinite-gap solutions for KdV equation under Hamiltonian perturbations.

上世纪八十年代末至今，无穷维哈密顿系统的KAM理论蓬勃发展，在有限维不变环面和时间拟周期解方面取得大量成果，不仅广泛应用于有界扰动情况，还发展出处理无界扰动的KAM理论，应用于非线性部分含有空间导数的偏微分方程。相比之下，研究哈密顿偏微分方程无穷维不变环面和时间概周期解的概周期KAM理论发展较慢，至今只有处理有界扰动的结果，并且一些重要的基本问题尚未解决（比如多项式衰减的KAM环面存在问题）。本项目将致力于发展无穷维哈密顿系统的概周期KAM理论，第一个主要目标是建立环面较慢衰减乃至多项式衰减的概周期KAM定理，并应用于一个或多个有意义的偏微分方程，第二个主要目标是建立能处理无界扰动的概周期KAM定理，并应用于非线性部分含有空间导数的哈密顿偏微分方程，其中包括证明KdV方程的infinite-gap解在哈密顿扰动下的保存性。

本项目主要研究了哈密顿偏微分方程的概周期KAM理论以及与其密切相关的有限维KAM环面的存在性和长时间稳定性，包括：用无界KAM理论研究一维周期边条件的导数非线性薛定谔方程，得到小振幅拟周期解构成的不变Cantor流形；对非线性波方程和非线性薛定谔方程，证明了KAM环面的多项式长时间稳定性；对一维周期边条件的非线性薛定谔方程，证明了一般次指数衰减的无穷维KAM环面的存在性和长时间稳定性。这些研究不仅完善和发展了KAM理论本身，还促进人们更深刻地了解这些有重要物理意义的方程的动力学性质。