带变系数非线性项的哈密顿偏微分方程不变环面存在性研究

In the perturbation theory of Hamiltonian systems, the existence of invariant tori is an important part in studying the geometrical stability and is of great significance for understanding the stability of dynamics. Presently in the research of infinite dimensional Hamiltonian systems, researchers mainly focus on the equations with constant-coefficient nonlinearities. When the coefficient of nonlinear term depends on t and x, the system is forced, and the momentum is not conserved, the coefficients of the perturbation are variable after using the infinite dimensional coordinates expansion, and the measure of all indices need to be estimated, which complicate the transformation of normal form and measure estimation. Three equations with variable-coefficient nonlinearities are selected, including the derivative nonlinear Schrödinger equation, the wave equation with quintic nonlinearity, and the quantum harmonic oscillator. This project will discuss the invariant tori of the equations with variable-coefficient nonlinearities, estimate the measure of infinite small divisor conditions, improve the techniques of normal forms and establish KAM theorems in three different cases, including the unbounded perturbation problem, the higher order perturbation problem, and the bounded limiting case. This work will extend applicable scopes of the KAM method.

在哈密顿系统的摄动理论中，不变环面的存在性是研究系统几何稳定性的重要组成部分，对认识系统的稳定性等动力学行为具有重要意义。以往在对无穷维哈密顿系统的研究中，以带常系数非线性项的方程为主。当非线性项的系数依赖于t和x时，系统受到强迫作用，动量守恒不再成立，扰动项用无穷维坐标展开后系数为变量，需要对所有指标估计小分母的测度，增大了化正规形和测度估计的复杂程度。项目选取了具有变系数非线性项的带导数薛定谔方程、带五次非线性项的波动方程、量子谐振子三个方程为代表展开研究，旨在探讨在具有变系数非线性项的情况下，无界扰动问题、高阶扰动问题以及有界临界情况三种不同类型问题的不变环面，在不同情况中估计无穷多个小分母条件的测度、改进正规型技术、构建KAM定理，进一步丰富KAM方法的应用范围。

哈密顿系统的不变环面对于认识其稳定性等动力学行为具有重要意义。本项目利用KAM方法研究非线性项拟周期依赖于时间变量且周期依赖于空间变量的哈密顿偏微分方程拟周期不变环面的存在性，构造实解析的辛变换，把方程的哈密顿函数转化为Birkhoff标准形，建立无穷维KAM定理应用于标准形，证明对大多数频率向量，方程在其相应边界条件下存在小振幅、线性稳定、实解析的拟周期解。本项目对于带有高阶扰动项的系统，证明对于大多数频率向量和容许指标集合，薛定谔方程存在拟周期的不变环面；对于无势能项的系统，证明对于大多数频率向量和容许指标集合，梁方程存在拟周期的不变环面，且拟周期解的频率和未扰系统的频率非常接近；对于带有重谱的边值问题，证明对于大多数频率向量和大多数指标，波动方程存在周期和拟周期偶解，并给出解的解析表达式；在基本完成工作目标的基础上，进一步拓展了研究内容和方法，证明对于非线性项仅依赖于空间变量的系统，对于任意固定的势能常数和充分大的指标集合，梁方程存在给定频率的不变环面。