具有高阶非线性项的二维薛定谔方程的拟周期解

Integrable Hamiltonian systems and its perturbed systems arise widely in all areas of mechanical, acoustic, optical, life sciences and social sciences. The existence of quasi-periodic solutions of nonlinear Hamiltonian partial differential equations is an important branch of nonlinear science. So far, there exist many results about quasi-periodic solutions of Hamiltonian partial differential equations, but little is known about quasi-periodic solutions for high-dimensional equations with higher order nonlinearity. Based on the applicant's research work on KAM theory which prove the existence of quasi-periodic solutions of one dimensional nonlinear wave equations and two dimensional Schrödinger equations with quasi-periodic forcing, this project will focus on quasi-periodic KAM theory of two-dimensional Schrödinger equations. The first main goal is to establish quasi-periodic KAM theorems for two-dimensional Schrödinger equations with higher order nonlinearity under periodic boundary conditions. The second main goal is to establish quasi-periodic KAM theorems for two-dimensional Schrödinger equations with coefficients which depend on spatial variable under periodic boundary conditions. Also, by constructing a local normal form in a neighborhood of the obtained solutions is helpful to understand the dynamics. For example, one sees the linear stability and zero Lyapunov exponents.

可积哈密顿系统及其摄动系统广泛地出现在力学、声学、光学、生命科学及社会科学的各个领域。非线性哈密顿偏微分方程的拟周期解的存在性是非线性科学研究的一个重要分支。到目前为止，已有了大量的哈密顿偏微分方程拟周期解的结果，但有关具有高阶非线性项的高维哈密顿偏微分方程拟周期解的研究不多。基于申请人已完成的利用KAM理论证实带拟周期强迫项的一维非线性波动方程和二维非线性薛定谔方程的拟周期解的存在性，本项目将致力于发展二维薛定谔方程的拟周期KAM理论。第一个主要目标是建立具有高阶非线性项的二维薛定谔方程在周期边界条件下的拟周期KAM定理；第二个主要目标是建立系数依赖空间变量的二维薛定谔方程处于周期边界条件下的拟周期KAM理论。同时，还将在解的邻域内构造正规型，进而探索拟周期解的动力学性质，比如解的稳定性和零Lyapunov指数。

在天体力学、等离子物理、航天科技以及生物工程中很多模型都以哈密顿方程或者其摄动方程的形式出现，因此哈密顿方程一直是数学家和理论物理学家关心的热点，使得对该领域的研究多年来长盛不衰。本项目主要研究了具有五次非线性项的二维空间中完全共振薛定谔方程在周期边界条件下的拟周期解的存在性；具有一般非线性项的二维空间中完全共振薛定谔方程在周期边界条件下的拟周期解的存在性；具有五次非线性项的系数依赖空间变量的二维薛定谔方程在周期边界条件下KAM环面的存在性；二维空间中具有拟周期强迫的梁方程在周期边界条件下的可化性和KAM环面的存在性；具有概周期强迫的梁方程在周期边界条件下全维数不变环面的存在性。通过引入算子的正交特征函数，将方程转化为无穷维哈密顿系统。然后针对选取的特殊切向频率集合，经由KAM理论，证明了上述几类方程均有很多拟周期解或概周期解。除了证明解的存在性，KAM理论还在解的邻域内构造了标准型，得到了解的一些动力学性质，比如解的稳定性和零Lyapunov指数。