关于拟周期驱动Hamilton偏微分方程Liouville不变环的存在性研究

本项目拟在已有工作的基础上深入研究几类拟周期驱动偏微分方程Liouville不变环面的存在性，即具有Liouville频率的拟周期解的存在性。主要研究下列问题：1、在Dirichlet边界条件或周期边界条件下研究一类拟周期驱动非线性薛定谔方程Liouville不变环的存在性；2、研究在周期边界条件下具有Liouville驱动频率的一类拟周期驱动梁方程的Whiskered环的存在性；3、在周期边界条件下研究拟周期驱动Kirchhoff方程和Airy方程的拟线性扰动的Liouville不变环的存在性；4、在铰链边界条件下研究一类病态的拟周期驱动Boussinesq方程Liouville不变环的存在性。本项目试图在驱动频率是Liouvillean的条件下，通过改进正规形KAM方法和Nash-Moser迭代方法来证明上述几类拟周期驱动偏微分方程Liouville不变环的存在性。

This project investigates the existence of Liouville invariant tori for some quasi-periodically forced partial differential equations. We mainly consider the following questions : 1. the existence of Liouville invariant tori for quasi-periodically forced nonlinear Schrodinger equation with Dirichlet boundary condition or periodic boundary condition ; 2. the existence of Whiskered invariant tori for quasi-periodically forced beam equation with Liouville frequency in periodic boundary condition; 3. the existence of Liouville invariant tori for quasi-periodically forced Kirchhoff equation and quasi-linear perturbation of Airy equation with periodic boundary condition ; 4. the existence of Liouville invariant tori for ill-posed quasi-periodically forced Boussinesq equation with hinged boundary conditions. The existence of Liouville invariant tori for these classes of quasi-periodic forced partial differential equations is proved by improving the normal form KAM method and the Nash-Moser iteration method.