两类带导数的非线性Schrodinger方程拟周期解的存在性

The study of Hamiltonian perturbation theory is an important part in nonlinear science. The KAM theory is one of the important tool in the study of nearly integrable Hamiltonian systems: we can investigate the existence of quasi-periodic solutions and almost periodic solutions of Hamiltonian partial differential equations via infinite dimensional KAM theory. We consider the following two classes of derivative nonlinear Schrodinger equations: Chen-Liu-Lee equations and Gerdjikov-Ivanov equations. By constructing KAM iteration, we investigate the real analytic quasi-periodic solutions of Chen-Liu-Lee equation and the smooth quasi-periodic solutions of Gerdjikov-Ivanov equations. This project will not only have important theoretical significance, but also through the study of the dynamic behavior of derivative nonlinear Schrodinger equations, it will have some guidance for practical applications.

Hamilton系统的摄动理论是非线性科学的一个重要研究方向。KAM理论是研究近可积Hamilton系统的重要工具之一，特别地可以利用无穷维KAM理论研究Hamilton型偏微分方程拟周期解及概周期解的存在性问题。本项目研究以下两类带导数的非线性Schrodinger方程：Chen-Liu-Lee方程和Gerdjikov-Ivanov方程。通过构造KAM迭代，我们将研究Chen-Liu-Lee方程实解析拟周期解的存在性问题和Gerdjikov-Ivanov方程光滑拟周期解的存在性问题。本项目不仅具有重要的理论意义，而且通过研究带导数的非线性Schrodinger方程的动力学行为，也对实际应用有一定指导。