带概周期强迫项的Schrödinger方程和梁方程的概周期解

It is a common phenomenon about almost periodic phenomenon in the nature. Schrödinger equations and beam equations are the fundamental equations of dynamic system. It is an important and difficult problem of the existence of almost periodic solutions of those equations in dynamic system. This project will focus on the existence of almost periodic solutions of Schrödinger equations and beam equations with almost periodic forcing, which is great significant in both theory and practice. We state them in detail: 1) Obtain the forms of almost periodic forced item and nonresonance conditions of infinite dimensional frequency. 2)Improve KAM theorem to prove the existence of almost periodic solutions of Schrödinger equations and beam equations with almost periodic forcing, which include that we estimate the measure of the small divisors, coordinate transformation，new perturbation, and the convergence of coordinate transformation. 3) Overcome the difficulty of KAM iteration in high dimension problem, and improve the obtained KAM theorem in 2) to prove the existence of almost periodic solutions of 2-dimension beam equations with almost periodic forcing.

概周期现象是自然界中一种常见的现象，Schrödinger方程和梁方程是动力系统中的基本方程，两类方程的概周期解的存在性是动力系统领域内的难点和重点。本课题主要针对带概周期强迫项的Schrödinger方程和梁方程的概周期解的存在性展开研究，具有重要的实际意义和理论价值。具体内容为：1) 分析确定方程中的概周期强迫项的形式，以及无穷维频率的非共振条件。2) 改进KAM理论来证明这些方程的概周期解的存在性，包括分析构造迭代序列，处理KAM迭代中的无穷维频率，估计更复杂的小除数的测度，更精确的估计典则变换和新扰动项，证明无穷多个典则变换复合的收敛性等。3) 克服高维问题中KAM迭代的难点，推广和改进2)中得到的KAM定理，使得能够证明2维的具有概周期强迫的梁方程的概周期解的存在性。

本项目研究的是动力系统中的三类基本方程薛定谔方程、波动方程和梁方程，主要围绕薛定谔方程、波动方程的概周期解的存在性和高维梁方程的拟周期解的存在性展开研究。具体的研究内容是：1）证明了具有概周期强迫的线性薛定谔方程的概周期解的存在性，其中概周期强迫项的级数形式、无穷维频率的处理、精确估计典则变换等是主要创新点；2）得到了拟周期强迫的高维的线性梁方程的拟周期解的存在性；3）证明了具有概周期强迫的线性波动方程的概周期解的存在性，解决了概周期强迫项含有空间变量产生的困难，扰动项的迭代分步、估计复杂的小除数测度、无穷多个典则变换复合的收敛性等难点；4）构造了KAM定理，并证明了具有概周期强迫的薛定谔方程的概周期解的存在性。.