弱有界扰动下的KAM理论及其应用研究

In this project, we will use KAM methods to study the existence and linear stability of quasi-periodic solutions for the following partial differential equations： 1. the reversible quantum harmonic oscillator; 2. the quasi-periodically forced quantum harmonic oscillators with Liouvillean frequencies; 3. the higher dimensional reversible derivative nonlinear beam equations, which arise widely in Bose-Einstein condensation theory, nonlinear optics and elastic mechanics. The main difficulty in our KAM proof is that their perturbations are all weak bounded ones with bad regularity. Inspired by the decay and Töplitz-Lipschitz properties of Hamiltonian functions in the Hamiltonian KAM theory, we will introduce the more general decay and Töplitz-Lipschitz properties of vector fields, and thus overcome the difficulty. The study of this project will help us to understand the quasi-periodic motions better in infinite-dimensional dynamical systems, and provide the theoretical basis for the related experimental research in mechanics and physics.

本项目拟运用KAM方法研究下面几类偏微分方程的拟周期解存在性及线性稳定性:1.反转量子调和振子；2.具有刘维尔频率的拟周期强迫量子调振子；3.带导数非线性项的高维反转梁方程，它们广泛出现在Bose-Einstein凝聚态理论、非线性光学及弹性力学等领域中。我们在KAM证明中遇到的主要困难是它们的扰动项均为具有较差正则性的弱有界扰动。受已知哈密顿KAM理论中的哈密顿函数衰减性质和Töplitz-Lipschitz性质的启发，我们拟通过引进更一般的向量场衰减性质和Töplitz-Lipschitz性质来克服这一困难。本项目有助于我们更好地认识无穷维动力系统中的拟周期运动，并为力学和物理学中的相关实验研究提供理论基础。

本项目利用无穷维KAM理论研究了下面几类偏微分方程拟周期解的存在性问题: (1) 高维偏微分方程，包括带导数非线性项的反转型完全共振梁方程、带外参数的反转型薛定谔方程组和反转型完全共振薛定谔方程组；(2) 具有弱有界扰动和无界扰动的一维非线性薛定谔方程, 包括非线性Heisenberg铁磁链方程和反转型耦合量子调和振子方程组；(3)具有Liouville外频的拟周期强迫偏微分方程，包括拟周期强迫哈密顿波方程、反转型导数梁方程及反转型导数波方程。研究结果发表(包括在线)在 Math.Z., DCDS-A, JDDE, ZAMP 等国际期刊上。