无穷维KAM理论及其应用于哈密顿偏微分方程

In the late 1980s, KAM theory was generalized to infinite dimensional Hamiltonian systems so as to study quasi periodic solutions of Hamiltonian partial differential equations. Till now, there has been a wealth of achievements for bounded perturbation and unbounded perturbation with simple normal frequency. However, for bounded perturbation, some important problems are still open, such as the linear stability of the quasi periodic solutions of higher dimensional wave equations, while for unbounded perturbation with multiple normal frequencies, there is no essential result at all. This project will devote to infinite dimensional KAM theory and its applications to Hamiltonian partial differential equations, including: (1) in the aspect of bounded perturbation, study higher dimensional Hamiltonian partial differential equations with their frequencies increasing linearly, including higher dimensional wave equations and higher dimensional quantum harmonic oscillator, so as to obtain linearly stable quasi periodic solutions; (2) in the aspect of unbounded perturbation, establish KAM theorems with multiple normal frequencies, and apply to Hamiltonian partial differential equations with their nonlinearity containing spatial derivative.

上世纪八十年代末，KAM理论发展到无穷维，用于研究哈密顿偏微分方程的拟周期解。至今，在有界扰动情况和单重法向频率的无界扰动情况，已经取得丰富成果。然而，对于有界扰动情况，仍有一些重要问题尚未解决，比如高维波方程拟周期解的线性稳定性，而对于多重法向频率的无界扰动情况，则是完全没有本质的结果。本项目将致力于发展无穷维KAM理论及其应用于哈密顿偏微分方程，包括：（1）有界扰动方面，研究频率线性增长的高维哈密顿偏微分方程，包括高维波方程和高维量子调和振子，得到线性稳定的拟周期解；（2）无界扰动方面，建立多重法向频率的无界KAM定理，并应用于带导数非线性项的哈密顿偏微分方程。

本项目主要研究了无穷维KAM理论及其应用于哈密顿偏微分方程，包括有界扰动情形和无界扰动情形，以及与KAM理论密切相关的长时间稳定性理论。研究成果如下：对五次方非线性项的Dirichlet边条件的一维波方程，得到线性稳定的时间拟周期解；在定性非退化条件下建立了无穷维KAM定理，并应用于周期边条件的一维波方程和薛定谔方程，分别在原点附近和平面波解附近得到线性稳定的时间拟周期解；建立了法向频率重数等于2的无界KAM定理，应用于一维周期边条件的导数非线性薛定谔方程，得到任意有限维椭圆不变环面构成的不变Cantor流形；对一维周期边条件的导数非线性薛定谔方程，得到完全的6阶Birkhoff正规形，进而得到长时间稳定性估计。这些研究完善和发展了KAM理论和长时间稳定性理论，促进人们更深刻地了解这些有重要物理意义的偏微分方程的动力学性质。