哈密顿动力系统的KAM理论和长时间稳定性

在数学、力学和物理等学科中，很多模型都是用Hamilton动力系统来描述的。例如，无摩擦的弹簧的振动，地球、太阳和月球构成的三体系统，光在光纤中传播，以及浅水波的传播等等。无论是有限维的还是无穷维的哈密顿动力系统的研究在最近几十年取得了长足的进展，但还留下一些尚未解决的基本问题，例如Herman猜测。我们将研究：具有丢番图频率的线性哈密顿系统在解析的哈密顿扰动下的具有正测度的不变环面族的存在性；反转系统的不稳定性；具有零法向频率的以及具有法向频率以零为聚点的哈密顿偏微分方程的KAM理论；具有无界扰动的哈密顿偏微分方程的不变环面的长时间稳定性。

Many models in mathematics,mechanics, physics and so on are governed by Hamiltonian systems. Those models include such as the vibration of a spring without friction, the spread of light in optical fiber and the diffusion of shallow water wave. The study of both finite and infinite dimensional Hamiltonian systems have seen a great progress in recent decades. However, there are still many basic problems to remain unsolved, for example, the Herman conjecture. In the research plan, we will study the following problems: (a) The existence of invariant tori for a linear Hamiltonian of Diophantine frequencies subject to an analytic perturbation;(b) The instability for reversible system;(c) The KAM theory for Hamiltonian partial differential equations of a zero-frequency and of the frequencies which cluster at zero point;(d) The long time stability for the invariant tori of Hamiltonian partial differential equations with unbounded perturbations.