两类哈密顿偏微分方程拟周期解问题的研究

The Schrödinger equation, wave equation,KDV 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 a class of shallow water wave equation and a class of water wave equation, details are as following:The existence and stability of quasi-periodic solutions for generalized Boussinesq equation with quasi-periodic forcing is considered; and the existence and stability of quasi-periodic solutions of GKDV-4 equation under periodic boundary conditions is considered.. The proposing and solving of this project contributes to widening the research of infinite-dimensioanl KAM theory in some related fields. Our approach involves infinite-dimensional KAM theory, normal form technique and so on.

Schrödinger方程、波方程、KDV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、拟周期解或概周期问题，这具有重要的理论意义和应用价值。.本课题拟研究一类浅水波方程以及一类水波方程的拟周期解问题，具体内容如下：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；以及周期边界条件下，GKDV-4方程拟周期解的存在性和稳定性。.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。我们的研究将涉及到无穷维KAM理论、正规形方法等方面。

本项目以KAM理论为主要研究工具，研究了GKDV-4方程，广义Boussinesq方程和一类可逆系统的拟周期解问题。研究绞合边界条件下，拟周期受迫的广义Boussinesq方程小振幅拟周期解的存在性及稳定性；研究一类二阶可逆系统解的有界性以及拟周期解的存在性；研究周期边界条件下，GKDV-4方程拟周期解的存在性及稳定性。