几类无穷维哈密顿系统拟周期解问题的研究

The Schrödinger equation, wave equation,KDV equation,beam equation, shallow water wave equation and harmonic oscillator equation share the structure of infinite dimensional Hamiltonian system and they come from physics or mechanics. The research on their periodic solutions, quasi-periodic solutions or almost-periodic solutions contributes to the better understanding of their dynamic behavior, and have some sense in theory and reality. . The project devotes to the study of quasi-periodic solutions for generalized Boussinesq equation and high dimensional forced beam equation, the quasi-periodic solutions and almost periodic solutions for a class of infinite dimensional Hamiltonian system with dense spectra is also considered. Details are as following: Firstly, the existence and stability of quasi-periodic solutions for generalized Boussinesq equation with general polynomial nonlinearity is considered. Secondly，the existence and stability of quasi-periodic solutions for generalized Boussinesq equation with quasi-periodic forcing is considered. Thirdly, the existence and stability of quasi-periodic solutions for high dimensional forced beam equation under periodic boundary conditions is considered. At last, the existence and stability of quasi-periodic solutions and the existence of almost periodic solutions for a class of infinite dimensional Hamiltonian system with dense spectra is considered.. The proposing and solving of this project contributes to widening the research of infinite-dimensioanl KAM theory in some related fields.

Schrödinger方程、波方程、KDV方程、梁方程、浅水波方程以及调和振子方程等均具有哈密顿结构而且都有一定的物理背景或力学背景。为理解其动力学行为，人们需要研究它们的周期解、拟周期解以及概周期问题，这具有重要的理论意义和应用价值。. 本课题拟研究广义Boussinesq方程和高维受迫梁方程的拟周期解问题以及一类具有稠谱的无穷维哈密顿系统的拟周期解、概周期解问题。具体内容如下：首先研究具有一般多项式非线性项的广义Boussinesq方程拟周期解的存在性与稳定性；其次研究具有拟周期强迫振动项的广义Boussinesq方程拟周期解的存在性与稳定性；然后研究周期边界条件下，高维受迫梁方程拟周期解的存在性和稳定性。最后研究一类具有稠谱的无穷维哈密顿系统拟周期解的存在性与稳定性以及概周期解的存在性。. 本课题的提出和解决，有助于拓展无穷维KAM理论在相关领域的研究。

该项目通过无穷维KAM理论以及Birkhoff规范形，研究了几类无穷维哈密顿偏微分方程：一类拟周期受迫的广义Kaup系统的小振幅、线性稳定的拟周期解；一类梁方程系统、一类分数阶非线性薛定谔方程系统以及二维修正的Boussinesq方程的小振幅、拟周期解的存在性与稳定性；一类具有刘维尔频率的非线性波方程拟周期解的存在性问题。此外，还研究了催化剂-抑制剂系统的Hopf分歧；本项目已发表8篇SCI期刊收录论文，相关研究成果丰富、拓展了无穷维KAM理论。