具有拟周期强迫扰动的非线性偏微分方程不变环面的存在性研究

本项目在已有研究工作的基础上，进一步深入研究具有拟周期强迫扰动的非线性偏微分方程不变环面的存在性，即拟周期解的存在性。主要研究下列问题：1、具有拟周期有界扰动的非线性哈密顿偏微分方程不变环面的存在性（包括薛定谔（Schrodinger）方程，波方程以及梁方程）；2、具有拟周期无界扰动的非线性哈密顿偏微分方程不变环面的存在性（包括薛定谔（Schrodinger）方程，非线性KdV方程，Benjamin–Ono型方程）；3、具有拟周期有界扰动或无界扰动的金兹堡-朗道（Ginzburg-Landau）方程不变环面的存在性。本项目试图利用改进的Nash-Moser迭代格式结合线性化算子的多尺度分析方法以及KAM迭代结合Birkhoff正规形技巧研究上述几类方程的频率与拟周期项的频率相同的拟周期解和具有附加频率的拟周期解的存在性。

This project investigates the existence of invariant tori for nonlinear partial differential equations with quasi-periodically forced perturbations. We consider the following problems: (1) the existence of invariant tori for nonlinear Hamiltonian partial differential equations with quasi-periodically bounded perturbations, including Schrodinger equation, wave equation and beam equation; (2) the existence of invariant tori for nonlinear Hamiltonian partial differential equations with quasi-periodically unbounded perturbations, including Schrodinger equation, KdV equation and Benjamin–Ono type equation; (3) the existence of invariant tori for Ginzburg-Landau equations with quasi-periodically bounded or unbounded perturbations. We will use Nash-Moser iterative scheme and multiscale analysis for the inverse linearized operators , KAM iteration and Birkhoff normal form to study the existence of quasi-periodic solutions with frequency vector which is same as the forced term and with additional frequencies for the above equations.